[Slfm 118] theory of relations roland fraisse (nh 1986)(t)

STUDIES IN LO-GIC AND

THE FOUNDATIONS OF MATHEMATICS

VOLUME 118

Editors

J. BARWISE, Stanford D. KAPLAN, Los Angeles H. J . KEISLER, Madison

P. SUPPES, Stanford A. S. TROELSTRA. Amsterdam

NORTH-HOLLAND AMSTERDAM NEW YORK OXFORD

THEORY OF

RELAmONS

R. FRAISSE Universitk de Provence

Marseille France

1986

NORTH-HOLLAND AMSTERDAM 0 NEW YORK OXFORD

ELSEVIER SCIENCE PUBLISHERS B.V., 1986

All rights reserved. N o part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or

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ISBN: 0 444 87865 3

Translation of ThLorie des relations Translated by P. Clote

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Library of conppg CataloginginPubliition Data

F r a i s d , Roland. Theory of relations.

(Studies i n logic a d the foundations of matheratics ; v. 118)

Translation of: Thhrie des relations. Bibliography: p. Includes index. 1. Set theory. I. Title. 11. Series.

QA248.FT75 1986 511.3'22 85-20701 ISBN 0-444-87865-3

PRINTED IN THE NETHERLANDS

V

INTRODUCTION

Relat ion theory goes back t o the 1940's ; i t o r ig ina tes i n the theory o f order types, due t o HAUSDORFF (Grundzuge der Mengenlehre 1914), S I E R P I N S K I ( Le~ons sur l es nombres t r a n s f i n i s 1928, taken up again i n Cardinal and ord ina l numbers 1958), SZPILRAJN (Sur l ' ex tens ion de l ' o r d r e p a r t i e l 1930), DUSHNIK, MILLER (Concerning s i m i l a r i t y t ransformat ions of 1 i n e a r l y ordered sets 1940), GLEYZAL (Order types and s t ruc tu re o f orders 1940), and t o HESSENBERG (Grundbegr i f fe der Mengenlehre 1906, in t roduc ing the negat ive and r a t i o n a l o rd ina l s ) . A t t h a t t i m e , r e l a t i o n theory j u s t extended t o a r b i t r a r y r e l a t i o n s the elementary no t ions o f order type and embeddabil ity.

Relat ion theory i n te rsec ts on ly weakly w i t h graph theory, w i t h which i t i s sometimes s t i l l confused. F i r s t l y , echniques i n r e l a t i o n theory on ly r a r e l y d i s t i ngu ish between graphs, i .e. s j m t r i c b inary re la t i ons , and re la t i ons o f a r b i t r a r y a r i t y . Add i t i ona l l y , as opposed t o graph theory, i n r e l a t i o n theory one considers equa l ly t he two t r u t h values (+) and ( - ) taken on by a r e l a t i o n w i t h base E f o r each element o f E2 ( o r o f En f o r the a r i t y n ) .

On the o ther hand, r e l a t i o n theory uses techniques espec ia l l y from combinatorics, the l a t t e r which can be def ined as f i n i t e se t theory. Anything concerning re la t i ons w i t h f i n i t e bases, o r count ing isomorphism types o f f i n i t e r e s t r i c t i o n s o f a given re la t i on , o r again the study o f permutations o f the base which preserves a given r e l a t i o n ( i .e. automorphisms o f the r e l a t i o n ) , makes use o f combinatorics. From a more techn ica l viewpoint, see the combinator ia l lemnas i n ch.3 5 4, and the study o f the incidence ma t r i x i n ch.3 5 5.

' a As f o r mathematical l og i c , i t s i n te rsec t i on w i t h r e l a t i o n theory i s ra the r impor- t an t . One can even say t h a t the two p r i n c i p a l sources f o r r e l a t i o n theory are the study o f order types, already mentioned, and l i n e a r l og i c , i . ? . f i r s t - o r d e r one-quant i f ie r l o g i c ; t h a t i s the study o f un iversa l formOlas (prenex formulas only having un iversa l q u a n t i f i e r s ) , and boolean combinations thereof, w i t h the p a r t i c u l a r case o f q u a n t i f i e r - f r e e formulas. From a semantic, o r model-theoretic viewpoint, t h i s i s the study o f un iversa l classes o f TARSKI,' VAUGHT 1953, and o f boolean combinations thereo f .

I f one presents mathematical l o g i c from a r e l a t i o n a l t heo re t i c viewpoint, the basic no t i on i s t h a t o f l oca l isomorphism, i . e . isomorphism o f a r e s t r i c t i o n o f the f i r s t r e l a t i o n onto a r e s t r i c t i o n o f the second one: see ch.9 5 1.4. For example, the f ree i n t e r p r e t a b i l i t y o f a r e l a t i o n S i n another r e l a t i o n R w i t h the same base, i s a l g e b r i c a l l y def ined by the cond i t ion t h a t every l oca l automorphism o f R ( l o c a l isomorphism from R i n t o R ) i s a lso a l oca l automorphism o f S . Equ iva len t ly , f ree i n t e r p r e t a b i l i t y i s l o g i c a l l y def ined by the existence o f a q u a n t i f i e r - f r e e formula which def ines S i n the s t ruc tu re of R . For example, i f R i s a chain, o r t o t a l order ing, then the betweenness r e l a t i o n S(x,y,z) = + i f f z i s between x and y , i s def ined by the quan t i f i e r - f ree formula (Rxz h Rzy) v (Ryz A Rzx) . This equivalence between algebraic and l o g i c a l no t ions e x i s t s even above the f ree -quan t i f i e r and the one-quant i f ie r cases; s ince l o g i c a l ( o r elementary) equivalence between R and S , saying t h a t R and S s a t i s f y the same f i r s t - o r d e r formulas, i s equivalent t o t h e i r being (k,p)-equivalent f o r a l l in tegers k and p , which i s a pure ly algebraic no t ion : see my Course o f mathematical l o g i c 1974 vol . 2 . Coming back t o the l i n e a r case (one-quant i f ie r l o g i c ) , as common not ions and techniques i n both mathematical l o g i c and r e l a t i o n theory, we have those o f 1-isomorphism, 1-extension, p ro jec t i on f i l t e r ( a va r ian t o f u l t rap roduc t ) : see ch.10 5 1 . And f o r each o rd ina l o( , the 4 -morphism (ch.10 5 4), which i s no t

vi Introduction

a one-quantifier notion, b u t i s ind ispens ib le in r e l a t ion theory f o r the study of embeddability: see ch. 10 5 4 and 5 5.3.

From the 1970's , an important connection appears between r e l a t ion theory and the theory of permutations. See the study of o r b i t s (ch.11 § 2 ) , the theorem on the increasing number of o r b i t s (LIVINGSTONE, WAGNER, ch.11 5 2.8) and the theorem on s e t - t r a n s i t i v e , o r homogeneous groups (CAMERON, ch.11 5 5.10) .

Let us mention, a l s o from the 1970's, some unexpected connections between r e l a t ion theory and topology (ch .1 5 8 and ch.7 5 2 ) ; and even connections w i t h l i n e a r algebra (ch.11 5 2.6) .

We sha l l now b r i e f ly present the pr inc ipa l notions s tud ied , by mentioning f i r s t t h a t chapters 1 through 8 concern the theory of p a r t i a l and t o t a l orderings ( o r cha ins ) , while chapters 9 through 12 concern the general study of r e l a t ions .

In chapter 1, we review bas ic set theo re t i ca l r e s u l t s , i n general without proofs, which allow the reader t o know, f o r ins tance , i n which prec ise sense we use the notion of f i n i t e set (TARSKI's sense r a the r than OEDEKIND's), o r the notion of ca rd ina l i t y of a s e t . T h i s allows us t o p rec i se , throughout the r e s t of t he book, which axioms a r e used f o r each proof: ZF a lone , the axiom of choice, dependent choice, t he u l t r a f i l t e r axiom, the continuum hypothesis, e t c . Moreover i t seems t h a t even among logic ians , t he re a r e few who a re aware t h a t , while O1 > W

i s provable in ZF a lone , y e t the countable axiom of choice, f o r ins tance , i s used t o prove t h a t W 1 i s regular . Or t h a t KONIG's theorem (ch.1 0 1.8), even in the very p a r t i c u l a r case of two ordered pairs of s e t s , i s not provable in ZF alone. Or t h a t t he poss ib le equivalence between the axiom o f choice and the statement t h a t t he range of a function is subpotent w i t h i t s domain, i s s t i l l an open problem, already p u t f o r th by RUBIN 1963. T h u s th i s chapter could be useful as a memory brush-up f o r the axiomatic s e t t heo re t i c i an .

In chapter 2 , in addi t ion t o a review of bas ic r e l a t ion theo re t i ca l no t ions , s imi l a r in s p i r i t t o chapter 1, we introduce some notions which a re no longer c l a s s i c a l , y e t which extend well-known concepts. For example the coherence lemma ( 5 1 .3 ) , a not well-known version of the u l t r a f i l t e r axiom. Another example, the co f ina l i t y of a pa r t i a l o rder ing , as well a s the r e l a t ed notion of cof ina l height ( 5 5.4 and 5 7 ) . C las s i ca l ly , the notion of c o f i n a l i t y i s re lega ted t o the s ing le case of cha ins , o r t o t a l o rder ings , which while in t e re s t ing i s too much r e s t r i c - t i v e .

In chapter 3 , we present RAMSEY's theorem and important refinements of i t , due t o GALVIN and t o NASH-WILLIAMS ( 5 2 ) . Furthermore, the " i n i t i a l i n t e rva l theorem" o r GALVIN's theorem i s presented twice , with very d i f f e r e n t proofs: POUZET's proof i n 5 2 and L O P E Z ' S proof using the c l a s s i ca l Ramsey s e t s of r e a l s , i n 5 6. Then we a re led t o the p a r t i t i o n theorems of DUSHNIK, MILLER and of ERDOS, RADO. We a l so present a combinatorial study of the incidence matrix, w i t h the l i n e a r independence lemma due t o KANTOR.

In chapter 4 , we begin the study of p a r t i a l o rder ings , w i t h the notions of good and bad sequence, of a f i n i t e l y f r e e p a r t i a l o rder ing , and t h a t of a well p a r t i a l ordering. We present HIGMAN's charac te r iza t ion of a well p a r t i a l ordering (the s e t of i n i t i a l i n t e rva l s is well-founded under inc lus ion) ; a l so HIGMAN's theorem on words i n a well pa r t i a l o rder ing , and RADO's well p a r t i a l ordering ( 5 4 ) . Also the notions of i dea l , t r e e , dimension, bound of an i n i t i a l i n t e rva l . We present the theorem of the maximal reinforced chain f o r a well p a r t i a l ordering, due t o D E JONGH, PARIKH ( 5 9 ) . The chapter ends ( 5 10) w i t h POUZET's theorem on regular ( o r f i n i t e ) c o f i n a l i t y of any f i n i t e l y f r e e p a r t i a l ordering.

Introduction vii

In chapter 5, we consider embeddability between orderings, the well par t ia l ordering of f i n i t e trees ( K R U S K A L ) , the existence of immediate extensions and of faithful extensions ( H A G E N D O R F ) , Cantor's theorem for par t ia l orderings (DILWORTH, G L E A S O N ) . Then the existence of s t r i c t l y decreasing inf in i te sequences of chains of reals: the denumerable sequence due t o DUSHNIK, M I L L E R and the continuum length sequence due t o SIERPINSKI. Finally a brief study of SUSLIN's chain and t ree , in connection with SUSLIN hypothesis; also ARONSZAJN t r e e , SPECKER chain.

In chapter 6 , we introduce the scattered chain, which does not admit any embedding of the chain Q of the rat ionals . Also the indecomposable, as well as the right and the l e f t indecomposable chain. We present HAGENDORF's theorem of unique decomposition of an indecomposable chain ( 5 3.3) and some connected resul ts (JULLIEN, L A R S O N ) . We begin t o study the covering of a chain by r l g h t or l e f t indecomposable intervals , or by doublets of indecomposable intervals . We present the hereditarily indecomposable chain with L A V E R ' S r esu l t s , and f ina l ly the indivisible relation o r chain.

I n chapter 7 , we proof supplementary resul ts abou t f in i te ly free par t ia l orderinas and the i r reinforcements by chains. We extend t o the s e t of i n i t i a l intervals the topology already introduced in ch.1 5 8 , and give some applications, namely BONNET'S resul ts . Then we prove the following important theorem of POUZET: every directed well par t ia l ordering has a cofinal res t r ic t ion which i s a direct product of f in i te ly many d is t inc t regular alephs. The chapter ends with a short study of Szpilrajn chains ( B O N N E T , JULLIEN); two interesting resul ts due t o TUKEY and t o KRASNER are presented as exercises.

I n chapter 8 , we introduce the important notion of barr ier due t o NASH-WILLIAMS; the par t i t ion theorem ( 5 1.4) , the theorem on the minimal bad barrier sequence ( 5 2 . 2 ) ; the forerunner and successor barr ier . This i s the main t o o l i n the proof of the very important theorem of LAVER: every se t of scattered chains forms a well quasi-ordering under embeddability ( 5 4.4). In other words, there exis ts neither an inf in i te s t r i c t l y decreasing sequence nor an in f in i te s e t of mutually incomparable scattered chains. L A V E R proved even more, in extending his result to chains formed from a countable union of scattered chains. However his proof has n o t yet been suff ic ient ly simplified t o be presented in a textbook of a reasonable s ize . In t h i s chapter, we also study the bet ter partial ordering, a notion due t o NASH-WILLIAMS, both for i t s in t r ins ic interest and for i t s applications t o chains.

In chapter 9 , we begin the general theory of re la t ions, w i t h the notions of local isomorphism, free interpretabi l i ty and free operator (which i s the relat ionis t version of a logical f ree formula, and links relation theory t o logic) . We study constant, chainable, monomorphic re la t ions. I n the case of a binary relation with cardinality p , we present the deep resul t due t o JEAN: (~-2)-monomorphy implies (p-1)-monomorphy ( 5 6 . 7 ) . We present the prof i le increase theorem (POUZET, 5 7 ) . Finally we extend t o arbi t rary relations the homomorphic image ( 5 8 ) , and in 5 9 we introduce the bivalent table , which apparently yields d i f f icu l t problems, one of them being very par t ia l ly solved by L O P E Z . Most of re la t ion is t researchers seem t o be discouraged by th i s branch of relation theory, which i s s t i l l a marginal study inside relation theory, considered i t s e l f as being marginal d u r i n g t o o long a time.

I n chapter 10, we classify relations according t o the i r age: two representatives of the same age have the same f i n i t e res t r ic t ions , up t o isomorphism. This i s equivalent t o classifying relations by the s e t of universal formulas which they sat isfy. Then we study maximalist or exis tent ia l ly closed relations ( 3 3.8), rich relat ions, inexhaustible relations ( 5 4 and 5) , and relations which are rich for the i r age. This notion, connected t o saturated relations, leads t o the existence cr i ter ion of POUZET, VAUGHT ( 5 7 ) . The f i n i t i s t and almost chainable relations are presented in 5 8 and 9.

viii Introduction

Chapter 11 i s concerned w i t h correspondence between r e l a t i o n theory and permutations, the l i n k between them being the homogeneous r e l a t i o n s and r e l a t i o n a l systems. We already mentioned the theorem o f inc reas ing number o f o r b i t s , due t o LIVINGSTONE, WAGNER. I n 5 3 and fo l low ing , we introduce the compa t ib i l i t y modulo a permutation group, which y i e l d s a marginal study i ns ide permutation group theory, w i th many open problems. The n o t i o m o f i n d i c a t i v e group and i n d i c a t o r lead t o FRASNAY's reduc t ion theorem ( 5 4) . The p a r t i c u l a r case o f Q- ind i ca t i ve groups leading t o the se t - t rans i t i ve group theorem o f CAMERON. F i n a l l y we study the pseudo-homogeneous re la t i ons , the prehomogeneous r e l a t i o n s w i t h POUZET's existence c r i t e r i o n ( 5 7), the set-homogeneous r e l a t i o n s .

I n chapter 12, we introduce the bounds o f a r e l a t i o n R : f i n i t e r e l a t i o n s non- embeddable i n R bu t whose proper r e s t r i c t i o n s are embeddable i n R . We present several important theorems due t o FRASNAY: the reassembling theorem ( 5 3); the existence o f an i n tege r p such tha t , from t h i s p o i n t on, p-monomorphism imp l ies cha inab i l i t y ; and the f i n i t eness o f the number o f bounds f o r a chainable r e l a t i o n . This study uses the method o f permuted chains, o r c o m p a t i b i l i t y modulo a permuta- t i o n group, already presented i n chapter 11. Proofs have been s i m p l i f i e d by using, as a powerful t o o l , the p-wel l r e l a t i o n due t o POUZET. The chapter, and the book, are ending w i t h the study o f reduct ion, reassembling, monomorphism and chainabi- l i t y thresholds: ca lcu la ted f i r s t by FRASNAY, they were improved by HODGES, LACHLAN, SHELAH, then proved again by FRASNAY t o have the smal lest poss ib le value. I n 5 6 we added some easy considerat ions about un iversa l classes.

I n order t o keep t h i s book t o a reasonable s ize , we suppressed two planned chapters. One about the celebrated problem o f recons t ruc t ion , i . e . the problem t o know i n what cases a r e l a t i o n w i t h base E i s completely determined, up t o isomorphism, by the isomorphism types o f i t s r e s t r i c t i o n s t o proper subsets o f E . The reader may consu l t BONDY, HEMMINGER 1977, LOPEZ 1978, 1982, 1983, POUZET 1979', STOCKMEYER 1977, ULAM 1960 (see B ib l iography) . The o thgr y i s s i n g chapter concerned the no t i on o f i n t e r v a l i n r e l a t i o n theory: see FRAISSE 1984 i n Bibl iography.

I would l i k e t o thank those among my colleagues - professors, researchers, students and ex-students - who solved o r cont r ibu ted t o the so lu t i on o f a l l problems presented here; and t o those who, by s i m p l i f y i n g the i no rd ina te l y long o r d i f f i c u l t p roo f o f the o r i g i n a l paper, have made these r e s u l t s accessible, hence su i tab le f o r p resenta t ion i n t h i s textbook. The i r names are mentioned together w i t h t h e i r con t r i bu t i on . As f o r myself, I have the f r e e conscience o f having accomplished my work as "chef d 'eco le" : namely the presentat ion, i n a form accessible t o a wide audience, o f r e s u l t s obtained by those who loved my research area and surpassed me.

1

CHAPTER 1

REVIEW OF AXIOMATIC SET THEORY

The purpose o f t h i s chapter i s t o s i t u a t e p rec i se l y " theory o f re la t i ons " w i t h i n the framework o f axiomatic se t theory, which i n i t i a l l y w i l l be t h a t o f ZERMELO-

FRAENKEL. The axioms f o r ZF are introduced below i n 91 and 92. Our i n i t i a l

no ta t ion w i l l be introduced there. I n r e f e r r i n g t o the f i r s t and sometimes

second chapter, we w i l l i nd i ca te throughout the book which statements requ i re

on ly the axioms o f ZF and those which require, t o our knowledge, the axiom o f choice, o r ra the r the weaker u l t r a f i l t e r axiom (boolean prime idea l axiom), o r the axiom o f dependent choice, e tc . Most o f the proofs, as we l l as c lass i ca l de f i n i t i ons from the f i r s t and second chapter, are l e f t t o the reader.

§ 1 - FIRST GROUP OF AXIOMS FOR ZF, F I N I T E SET, AXIOM OF CHOICE,

KONIG'S THEOREM

We begin w i t h the axioms of ex tens iona l i t y , p a i r , union, power se t ( s e t o f a l l subsets o f a se t ) and the scheme o f separation, a l l supposedly known t o the reader. We denote the empty s e t by 0 , inc lus ion C , s t r i c t i nc lus ion C . We denote the union o f the se t a by u a , and the power se t by p ( a ) . I f b s a , we designate the d i f f e rence by a-b . Singletons, unordered pa i r s

(s imply c a l l e d pa i r s ) are denoted by a } , \ a,b) , e tc . The successor se t a v { a ) o f a i s denoted by a+ l . So t h a t 1 = 0+1 = { 0) i s the successor o f the empty set ; 2 = 1+1 = .( 0,l) i s the successor o f 1 , etc . This nota- t i o n coincides w i t h the no ta t i on f o r o rd ina l add i t ion , introduced i n fj 3 below.

-

1.1. FINITE SET

Fol lowing TARSKI 1924', we def ine a s e t a t o be f i n i t e i f f every non-empty se t b o f subsets o f a contains an element which i s minimal w i t h respect t o

inc lus ion , i . e . an element c E b such t h a t no x E b s a t i s f i e s x c c . Taking complements, i t i s equ iva len t t o say t h a t a i s f i n i t e exac t l y when every non-empty se t o f subsets o f a contains a maximal element. A non - f i n i t e

se t i s sa id t o be i n f i n i t e .

The empty set , a s ing le ton , a p a i r are f i n i t e sets. Every subset o f a f i n i t e se t i s f i n i t e . I f a i s f i n i t e , then so i s the se t composed o f a together w i t h an add i t iona l

element. I n p a r t i c u l a r , the successor a+ l o f a i s f i n i t e .

2 THEORY OF RELATIONS

Scheme of-injuction-for f i n i t e se t s . If a condition f i s t rue for the empty je t , and i f for every s e t a satisfying f and every s e t u , the se t a u \ u j s m - s a t i s f i e s f , t+ i s t rue for every f i n i t e s e t . I f a se t a and a l l i t s elements are f i n i t e , then the union u a i s f i n i t e . This i s often expressed in the following form called pigeonhole principle: i f we par t i t ion an inf in i te se t into f i n i t e l y many subsets, then a t l e a s t one of these subsets i s in f in i te .

1 . 2 . COUPLE OR O R D E R E D PAIR, CARTESIAN PRODUCT Given two se t s a , b the couple or ordered pair (a,b) formed of the singleton { a ) and the (unordered) pair i a , b ) . This definition goes back t o KURATOWSKI 1921 (see also AJDUKIEWICZ). The s e t the f i r s t term and b the second term of the couple. Clearly two couples are equal i f f they have the same f i r s t and same second terms. The Cartesian product a x b i s the se t of couples (x,y) where x belongs t o a and y belongs t o b . FUNCTION, DOMAIN, RANGE A function or mapping from a onto b i s a subset f of a x b such that every element x of a appears as f i r s t term in exactly one couple (x.y) belonging t o f and every element y of b appears as a second term in a t l eas t one couple belonging t o f . The s e t a = Dom f i s called the domain, the s e t b = Rng f i s the range of f . For each element x of a , the second term y of the unique couple (x,y) having f i r s t term x i s denoted y = f ( x ) or y = fx and is called the value of f on x , or the image of x under f . For every superset c 7 Rng f we say tha t f i s a function from a into c . THE TRANSFORMATION f " AN0 ITS INVERSE If uc_ Dom f , we denote by f " ( u ) the se t of elements fx where x u . The function t h u s denoted f" i s a function on the se t of subsets of Dom f and i s called the transformation associated with f . This transformation preserves inclusion, in the sense that u c_ v implies f"(u) c f"(v) . However s t r i c t inclusion i s not preserved. I f v C Rng f , then the inverse image of v by f , denoted ( f - ) " (v) , is the s e t of elements x such that fx belongs to v . So we define the inverse transformation associated w i t h f , denoted inclusion as well as inclusion. INJECTION, INVERSE FUNCTION, PERMUTATION, TRANSPOSITION The function implies fx # fx ' for a l l x , x ' in Dom f . I f a i s the domain, b the range,

i s the s e t {{a} , {a,b]\

a i s said t o be

- -

1

(f-')" . I t preserves s t r i c t

f i s said to be a n injection or injective function i f f x # x '

Chapter 1 3

t hen an i n j e c t i o n i s s a i d t o be a b i j e c t i o n f rom a on to b . The i n v e r s e o f an i n j e c t i o n f i s denoted by f - l , so t h a t i n t h e case o f f

i n j e c t i v e , t h e t r a n s f o r m a t i o n assoc ia ted w i t h f - l c o i n c i d e s w i t h ( f - l ) " ( t h e l a t t e r e x i s t s f o r eve ry f u n c t i o n Given a f u n c t i o n f , i n j e c t i v e o r n o t , i f Oom f i s f i n i t e , t hen Rng f i s f i n i t e . F o r f i n j e c t i v e , t h e converse i s t r u e .

A pe rmuta t i on o f a i s a b i j e c t i o n f rom a on to a . Given two elements x, y

o f a , t h e t r a n s p o s i t i o n (x,y), i s t h e pe rmuta t i on o f a which in terchanges x and y and i s t h e i d e n t i t y on eve ry o t h e r e lement o f a . FIXED POINT LEMMA (KNASTER 1928, g e n e r a l i z e d by TARSKI 1955)

L e t a be a s e t and h a f u n c t i o n which takes each subset x o f a t o a subset hx o f a . Suppose t h a t h i s i n c r e a s i n g under i n c l u s i o n : x c y i m p l i e s t h a t hx C- hy f o r eve ry x, y 5 a . Then: (1) t h e r e e x i s t s e t s x a m a j o r i z e d by h , i n t h e sense t h a t x 5 hx ; f o r

example x can be taken as t h e empty se t ; ( 2 ) if x i s m a j o r i z e d A h , then hx i s m a j o r i z e d & h ; ( 3 ) t h e un ion u o f a l l m a j o r i z e d s u b s e t s - s a t i s f i e s hu = u .

f ) .

1.3. RESTRICTION, EXTENSION, COMPOSITION

Given a f u n c t i o n f w i t h domain a and a subset b o f a , we c a l l t h e r e s t r i c -

t& o f f t o b , denoted f!b , t h e s e t o f o rde red p a i r s be long ing t o f o f which t h e f i r s t t e rm belongs t o b . P u t t i n g g = f / b , we say t h a t f i s an e x t e n s i o n o f g t o t h e domain a . We leave i t t o t h e reader t o d e f i n e t h e compos i t i on g,f o f t h e f u n c t i o n s

f and g , w i t h Dom(g,f) = Dom g n Rng f EQUIPOTENCE, SUBPOTENCE

A s e t b i s s a i d t o be e q u i p o t e n t w i t h a on to b . A s e t b i s s a i d t o be subpotent w i t h a

p o t e n t w i t h b I A s e t b i s s t r i c t l y subp w i t h a b u t a i s n o t subpo ten t w i t h b .

i f f t h e r e e x i s t s a b i j e c t i o n o f a

f f t h e r e e x i s t s a subset o f a equi -

k t w i t h a i f f b i s subpotent By theorem 1.4 below, t h i s i s equ i -

v a l e n t t o say ing t h a t b i s subpo ten t b u t n o t e q u i p o t e n t w i t h a . Every s e t e q u i p o t e n t w i t h a f i n i t e s e t i s i t s e l f f i n i t e . Every f i n i t e s e t i s s t r i c t l y subpo ten t w i t h eve ry i n f i n i t e s e t . Two f i n i t e s e t s a r e always comparable, one b e i n g subpo ten t w i t h t h e o t h e r . I f a and b a r e f i n i t e , t hen t h e Car tes ian

p roduc t a x b i s f i n i t e . I f a i s f i n i t e , t h e n so i s t h e power s e t p ( a ) . A f i n i t e s e t i s n o t e q u i p o t e n t w i t h any o f i t s p roper subsets. E q u i v a l e n t l y , i f a i s f i n i t e , t hen eve ry i n j e c t i o n o f a i n t o a i s a pe rmuta t i on o f a . 0 Suppose t h a t f i s an i n j e c t i o n s a t i s f y i n g f " ( a ) c a . Take a subset m o f a which i s min imal among a l l subsets x o f a s a t i s f y i n g f o ( x ) c x . Then


f " ( f " (m)) c f " (m) by the i n j e c t i v i t y o f f : t h i s cont rad ic ts the min ima l i t y . 0

A se t a i s sa id t o be Dedek ind- f in i te i f f a i s no t equipotent w i t h any proper

subset o f i t s e l f (DEDEKIND 1888); i t i s Dedek ind- in f in i te i n the opposi te case. Every f i n i t e se t i s Dedekind-f in i te. The converse w i l l be proved i n 2.6 by using

the denumerable subset axiom (weaker than the axiom o f choice).

DEDEKIND-FINITE SET

1.4. BERNSTEIN-SCHRODER THEOREM Given sets a and b ,if a i s subpotent w i t h b and b subpotent w i t h a , - then a i s equipotent w i t h b . The fo l l ow ing proo f i s i n FRAENKEL 1953 and a t t r i -

buted t o WHITAKER. It does no t use the no t i on o f in teger , which i s used i n the c lass i ca l "m i r ro r p roo f " ; see a1 so SUPPES 1960. 0 L e t f be an i n j e c t i o n from a i n t o b , and g be an i n j e c t i o n from b

i n t o a . It su f f i ces t o f i n d a subset u o f a such t h a t b - f " (u ) i s sent t o a-o by the func t ion go , o r equ iva len t l y u = a - g" (b- f " (u ) ) . To do t h i s , consider the func t i on which takes each subset x o f a i n t o a - g" (b- f " (x ) ) . This func t i on i s inc reas ing under inc lus ion . By the f i x e d p o i n t lemma, the union u o f a l l x such t h a t x 5 a - g" (b - f " (x ) ) s a t i s f i e s the above.

1.5. CANTOR'S LEMMA Le t a be a set. There i s no func t ion , i n j e c t i v e o r otherwise, w i t h domain a and range 9 (a ) ( s e t o f subsets o f a ) . CANTOR'S THEOREM (1) Every se t a i s s t r i c t l y subpotent w i t h 9 (a ) . (2) I f a i s non-empty, then every se t o f mutua l l y d i s j o i n t subsets o f a s t r i c t l y subpotent w i t h 9 (a ) .

1.6. EXPONENTIAL

Given sets a and b , the exponential o r power ab i s the se t o f func t ions from a However aO = 0 f o r each non-empty s e t a . For each se t a , the se t 7 (a ) o f

subsets o f a i s equipotent w i t h a2 , where 2 = { O , l ) ) . We have the fo l l ow ing equipotences. For b and c d i s j o i n t , (b "c )a i s equipotent w i t h the Cartesian product (ba)x(ca) . The se t '(a x b) i s equipotent

w i t h the product (Ca)x(cb) . F i n a l l y C(ba) i s equipotent w i t h (bxc)a .

i n t o b . Thus 'b = {O) = 1 f o r each b . I n p a r t i c u l a r '0 = 1 .

1.7. CHOICE SET AND CHOICE FUNCTION

L e t a be a s e t o f non-empty mutua l l y d i s j o i n t se ts x . A choice s e t f o r a i s a se t whose in te rsec t i on w i t h each element x o f a i s a s ingleton.

Chapter 1 5

I f a i s f i n i t e , there i s always a choice se t f o r a (p roo f by induc t ion) . L e t a be a se t o f non-empty sets x . A choice func t i on f o r a i s a func t ion

which t o every element x o f a associates an element f x o f x . I f a i s f i n i t e , then there i s a choice func t i on f o r a . A X I O M OF CHOICE (ZERMELO 1908) Every set , even i n f i n i t e , o f non-empty mutual ly d i s j o i n t sets admits a choice set. Equ iva len t ly every s e t o f non-empty sets admits a choice func t ion .

An immediate consequence o f the axiom o f choice i s the fo l low ing . Given a func t ion f , i n j e c t i v e o r otherwise, Rng f i s subpotent w i t h Dom f . I n o ther words,

given a non-empty s e t a , every s e t o f mutual ly d i s j o i n t subsets o f a i s subpotent w i t h a . Problem. Does the preceding statement imply the axiom o f choice (problem mentioned i n RUBIN 1963 p. 5 note 1). A seemingly weaker consequence o f the axiom o f choice i s the asser t ion t h a t i s never s t r i c t l y subpotent w i t h Rng f . This does no t fo l low from ZF alone i .e. from the axioms mentioned i n 5 1, 5 2 and 2.4; see ch.10 exerc. 2, where a FRAENKEL-MOSTOWSKI model i s constructed w i t h Dom f s t r i c t l y subpotent w i t h Rng f , a r e s u l t which i s t rans ferab le t o ZF v i a the

theorem o f JECH-SOCHOR (observat ion due t o HODGES).

.____

Dom f

1.8. GENERALIZED CARTESIAN PRODUCT L e t a a i s the se t o f choice func t ions which, t o each element ai o f a associate

an element o f ai . I f a reduces t o the p a i r \b ,c ) , we have again the Cartesian product b x c o f 1.2. I f a i s i n f i n i t e , i t fo l lows from the axiom o f choice t h a t the Cartesian product o f a i s non-empty. KONIG 'S THEOREM

L e t I be a non-empty s e t o f elements i ( c a l l e d i nd i ces ) , t o each o f which i s associated a p a i r o f se ts ai, bi w i t h ai s t r i c t l y subpotent w i t h bi . Then the union o f the

product o f the bi (axiom o f choice i s used). 0 Suppose there e x i s t s a b i j e c t i o n h from u ai the bi . For each i and each x o f ai , take the func t i on hx E 17 bi and

take i t s value ( h x ) ( i ) . Thus we de f ine a func t i on from ai i n t o bi . By the axiom o f choice, the range o f t h i s func t i on i s subpotent w i t h ai , thus s t r i c t l y subpotent w i t h bi . Hence there i s an element ui o f bi which i s not the

value o f hx on i f o r any x i n ai . The choice func t i on which t o each i associates ui i s no t i n h " ( u ai) : cont rad ic t ion . We leave i t t o the reader

t o see t h a t the union o f t he i s subpotent w i t h the product o f the bi . 0

be a non-empty s e t whose elements a re non-empty. The Cartesian product o f

ai ( i 6 I ) i s s t r i c t l y subpotent w i t h the Cartesian

onto Tr bi , the product o f

ai


Problem. Can the above theorem be proved from only the axioms of ZF in the case where the se t Note that i f , in addition t o I being f i n i t e , we have for each index i that

T ( a i ) suffice for the proof. For asser ts t h a t Dom f i s never s t r i c t l y subpotent with Rng f , or of ZF plus the apparently weaker axiom which asser ts t h a t if a (resp. a ' ) i s s t r i c t l y subpotent with b (resp. b ' ) b , b ' d i s jo in t , then a u a ' i s s t r i c t l y subpotent with b u b ' .

I of indices i s f i n i t e w i t h cardinality greater than or equal t o 2.

i s subpotent with bi , then by C A N T O R ' S lemma 1.5, the axioms o f ZF

Card I = 2 , KONIG's theorem i s a consequence of ZF plus the axiom which

§ 2 - SECOND GROUP OF A X I O M S FOR ZF: FOUNDATION, I N F I N I T Y , SUBSTITUTION; O R D I N A L , I N T E G E R , COUNTABLE SET

AXIOM OF FOUNDATION The axiom of foundation i s the statement t h a t every non-empty s e t element dis joint from a . I t follows that x $ x for any x . Moreover f o r any XY Y i t i s impossible that x E y and Y E x , e tc . The axiom of foundation was introduced by ZERMELO 1930, inspired by a statement of von NEUMANN 1929. As t o i t s consistency, supposing t h a t a l l other axioms of ZF are consistent, see exercise 1. PREDECESSOR Given a s e t a , the successor a+l = a v j a } i s d i s t inc t from a , since a $! a . Moreover i f a+l = b+l then a = b ; otherwise we would have a € b + l with a + b , so a E b and similarly b E a , contradicting the axiom of foundation. Given a se t c , the s e t whose successor i s c (which i s unique i f i t ex is t s ) i s called the predecessor of c , denoted by c-1 . Finally, given a s e t a and i t s successor a t 1 , there i s no s e t x such t h a t a E x E a+l . TRANSITIVE SET, TOTALLY ORDERED SET A se t a i s t ransi t ive i f f , for every x, y, conditions y 6 x E a imply y E a . If a i s t ransi t ive and non-empty, then every element o f a i s a proper subset of a . Also 0 E. a (0 i s the only element o f a which i s dis joint from a ) . Every union and intersection of t ransi t ive sets i s t ransi t ive. I f a i s t rans i t ive , then so i s a+l , A s e t a i s to ta l ly ordered (by membership relation) i f f , for every x, y of a , ei ther X E . y or y E x or x = y . For example, a l l singletons are to ta l ly ordered. However the singleton o f 1, i .e . 11) ={{O)): i s not t ransi t ive. The se t { O , l , { l ) } i s t ransi t ive b u t n o t to ta l ly ordered. The s e t {0,(1)) neither t ransi t ive nor to ta l ly ordered. Every intersection of to ta l ly ordered se t s i s to ta l ly ordered. A union of such sets i s not necessarily to ta l ly ordered;

a admits an

i s

Chapter 1 I

however i f t h e s e t o f t o t a l l y o rde red s e t s i s d i r e c t e d under i n c l u s i o n ( i . e . any two such s e t s a r e i n c l u d e d i n a t h i r d such s e t ) , t hen t h e un ion i s t o t a l l y ordered.

F i n a l l y i f a i s t o t a l l y o rde red by E , then so i s a + 1 .

2.1. ORDINAL An o r d i n a l i s a t r a n s i t i v e s e t which i s t o t a l l y o rde red by E . F o r example 0 ,

Every element o f an o r d i n a l i s an o r d i n a l . The successor s e t o f an o r d i n a l i s an

o r d i n a l . The predecessor ( i f i t e x i s t s ) o f an o r d i n a l i s an o r d i n a l .

The i n t e r s e c t i o n o f any s e t o f o r d i n a l s i s an o r d i n a l . An o r d i n a l a i s s a i d t o be l e s s than o r equal t o an o r d i n a l b , denoted a .4( b , i f f a € b o r a = b ; an o r d i n a l a i s s t r i c t l y l e s s than b , denoted a ( s t r i c t l y g r e a t e r than) a re de f i ned . I f a , c b + l , then a s b o r a = b + l . Given two o r d i n a l s a fi b , t h e c o n d i t i o n a E b ( o r a < b ) i s e q u i v a l e n t t o s t r i c t i n c l u s i o n a c b . Hence a & b i s e q u i v a l e n t t o a c_ b . 0 By t r a n s i t i v i t y a E b i m p l i e s a c b . Conversely, suppose t h a t a c b . L e t d E b-a be an element d i s j o i n t f rom b-a . As d e b , t h i s d i s an o r d i n a l and d c b . A lso d c_ a s i n c e d i s d i s j o i n t

f rom b-a . So e i t h e r d = a ( y i e l d i n g a € b ) , o r d c a . I f t h e l a t t e r occurs, l e t u E a-d so t h a t u c a c b . As b i s an o r d i n a l and u E b and d E b , we have e i t h e r u E d o r d E u o r u = d . I f U E d , t h i s c o n t r a d i c t s u E a-d . I f d E u , then s i n c e u E a , we have d E a which c o n t r a d i c t s d e b-a . I f u = d , then d E a-d so d E a , again c o n t r a d i c t i n g d E b-a . 0

TRICHOTOMY Given any two o r d i n a l s a, b, e i t h e r a € b b E a a = b . 0 As we know, t h e i n t e r s e c t i o n a n b i s an o r d i n a l . E i t h e r a r\ b = a o r a n b = b o r a n b i s s t r i c t l y i n c l u d e d i n b o t h a and b . I n t h e f i r s t

case a s b so a = b o r a c b and thus a € b . A s i m i l a r conc lus ion i s reached i n t h e second case. I n t h e t h i r d case, we have a A b a and

a 0 b E b , so t h a t a n b belongs t o i t s e l f , c o n t r a d i c t i n g t h e axiom o f f ounda t ion . 0

L e t a and b be two o r d i n a l s ; i f b b a then b 3 a + l o r b = a . We leave i t t o t h e r e a d e r t o d e f i n e t h e maximum o r minimum o r d i n a l o f a s e t o f o r d i n a l s , denoted Max, Min . Every non-empty s e t a o f o r d i n a l s admi ts a m i n i -

- mum: t a k e b be long ing t o a and d i s j o i n t f rom a . More g e n e r a l l y we have t h e f o l l o w i n g scheme o f s ta tements: g i v e n a c o n d i t i o n ‘8 which i s s a t i s f i e d by a t l e a s t one o r d i n a l , t h e r e i s a minimum o r d i n a l s a t i s f y i n g f

Every t r a n s i t i v e s e t o f o r d i n a l s , e v e r y un ion o f a s e t o f o r d i n a l s i s an o r d i n a l .

1 = t o } , 2 = {0,1) .

We leave i t t o the reader t o de f ine upper bound and lower bound o f a s e t o f o r d i - nals. Given a se t u o f o rd ina ls , we denote the union o f u by Sup u . It i s

the supremum o f u , i . e . t he l e a s t upper bound o f u . I f o( i s an ord ina l and u a se t o f o rd ina ls such t h a t /3 e u imp l ies f i s , then Sup u 6 o( . I n o ther words i f a q .

2.2. ORDINAL-INDEXED SEQUENCE, o( -SEQUENCE; EXTRACTED SEQUENCE

Given an o rd ina l o( , an d - z e q u e x e , o r ordinal- indexed sequence, i s a func t ion

w i t h domain o( . I n t h i s case oc i s the length o f the sequence. Given a sequence u , the elements o r = o f u are a l l ordered p a i r s (i,ui) f o r which the f i r s t term i i s an o rd ina l s t r i c t l y less than OC . The i ' s are c a l l e d ind ices o f u , o r u i s indexed by i < o( . The second terms o f the ordered pa i r s (which are a r b i t r a r y se ts ) are c a l l e d the values o f u and denoted

ui o r u ( i ) . I n the p a r t i c u l a r case o f an h -sequence w i t h o rd ina l values, we leave i t t o the

reader t o de f ine increasing, decreasing, s t r i c t l y inc reas ing and s t r i c t l y decrea-

sing sequences. Given an o rd ina l o( and an 4 -sequence u , we def ine an ex t rac ted sequence

from u t o be a sequence w i t h length 1364 , obtained by composition v o f u w i t h h , where h i s a s t r i c t l y inc reas ing &-sequence w i t h values i n O( ;

so v = u,h and v. = u f o r each i < f j . The no t ion o f ex t rac ted sequence i s r e f l e x i v e and t r a n s i t i v e , bu t no t antisynnnetric. For instance, by the axiom o f

i n f i n i t y introduced i n 2.4 below, given two d i s t i n c t sets a, b, we w i l l de f ine the a-sequences a,b,a,b,.. and b,a,b,a,.. , each ex t rac ted from the other.

i h ( i )

2.3. INTEGER, n-ELEMENT SET, WORD, n-TUPLE

By non-negative integer, o r in teger , o r na tura l number, we mean a f i n i t e o rd ina l . Every element o f an i n tege r i s an in teger . Every non-zero ( i . e . non-empty) i n tege r has an i n tege r predecessor. I f a i s an i n tege r and b E a ( o r b < a ) , then b i s s t r i c t l y included i n a . As a i s f i n i t e , b i s s t r i c t l y subpotent w i t h a . Thus equipotent in tegers are i d e n t i c a l .

We thus have the scheme o f induct ion: i f a cond i t ion f holds f o r 0 , and i f f o r each in tege r a the cond i t i on f ( a ) imp l ies f? (a+ l ) , z/f holds f o r every in teger .

Every f i n i t e se t i s equipotent w i t h an in teger . Given a se t a , t h i s i n tege r i s

c a l l e d the card ina l , o r c a r d i n a l i t y o f a and denoted Card a . A se t equipotent w i t h an i n tege r n i s c a l l e d an n-element se t .

A f i n i t e sequence o r word i s an n-sequence, where n of leng th n i s c a l l e d an n-t=.

i s an in teger ; such a word -

Chapter 1 9

When r e s t r i c t e d t o words, the no t i on o f ex t rac ted sequence becomes antisymmetric;

i . e . two words each o f which i s ex t rac ted from the o ther are i d e n t i c a l .

2.4. A X I O M OF INFINITY, SUCCESSOR AND LIMIT ORDINAL

The axiom o f i n f i n i t y asserts the existence o f an i n f i n i t e set . A more useful and stronger version asserts the existence o f a Dedek ind- in f in i te se t . More

p rec i se l y the existence o f a se t a conta in ing the element 0 and such t h a t - i f x belongs t o a , then the successor x+ l = x u ( x ) belongs t o a . With

an appropr iate app l i ca t i on o f the separat ion scheme, the axiom o f i n f i n i t y y i e l d s the existence o f the se t o f integers, denoted by 0 . The se t w i s an

i n f i n i t e o rd ina l , the smal lest o rd ina l > 0 w i thout a predecessor.

A limit ord ina l i s an o rd ina l w i thout a predecessor. A successor o rd ina l i s an ord ina l w i t h a predecessor.

SUBSTITUTION SCHEME

A t t h i s p o i n t we replace the separat ion scheme by the more general subs t i t u t i on scheme (due t o FRAENKEL 1925), o f which the reader i s assumed t o be f a m i l i a r . With t h i s scheme we can def ine, f o r example, L3+ W = w . 2 : beginning w i t h the

se t w o f integers, associate t o each in tege r i the o rd ina l w+i def ined

below i n sec t ion 3.1. Then using the s u b s t i t u t i o n scheme def ine the se t o f ~ + i as i runs through . Another example: denote by No the se t G) o f integers, and f o r each in tege r i s u b s t i t u t i o n scheme al lows one t o de f ine the s e t o f Ni f o r i belonging t o w . The axioms prev ious ly introduced, from 5 1 t o the present 5 2.4 (no t i nc lud ing the axiom of choice nor i t s weakened versions such as choice among f i n i t e sets) , are c a l l e d the axioms of ZF. I f no special assumption i s e x p l i c i t e l y mentioned i n a theorem, then t h i s ind ica tes t h a t the theorem i s proved i n ZF alone.

If, however, the axiom o f choice o r o ther supplementary axioms ( f o r the most

p a r t weakened versions o f the axiom o f choice, s ta ted below) are used, then we

c l e a r l y i nd i ca te such. We have already done t h i s f o r KONIG's theorem i n 1.8.

Recall t h a t the axiom of choice has been proved cons is ten t w i t h ZF ( i f ZF i t s e l f i s cons is ten t ) by GODEL 1938. The negat ion o f the axiom o f choice has been proved equiconsistent w i t h ZF by COHEN 1963 (see the Bibl iography COHEN 1966).

l e t Ni+l = y ( N i ) ( t he s e t o f subsets o f Ni) ; the

2.5. DENUMERABLE SET, COUNTABLE SET, COUNTABLE AXIOM OF CHOICE A se t i s sa id t o be denumerable, resp. countable, i f i t i s equipotent, resp.

subpotent w i t h w , the s e t o f in tegers . ZF alone su f f i ces t o show t h a t the union o f two denumerable sets, t he Cartesian product o f two denumerable sets, and the se t o f a l l f i n i t e subsets o f w are a l l denumerable.

Fol lowing 2.2, we c a l l an &-sequence a sequence o f leng th w , hence indexed by the se t o f in tegers .


The coun tab le axiom o f cho ice i s a p a r t i c u l a r case o f t h e axiom o f cho ice . It s t a t e s t h a t f o r e v e r y coun tab le s e t o f non-empty d i s j o i n t se ts , t h e r e i s a cho ice s e t . T h i s axiom i s s t r i c t l y weaker than t h e axiom o f choice; i . e . i f ZF i s c o n s i s t e n t , t hen t h e r e i s a model o f ZF and coun tab le cho ice which s a t i s f i e s t h e n e g a t i o n of t h e genera l axiom o f cho ice (JECH 1973). Note t h a t t h e coun tab le axiom o f cho ice i m p l i e s t h a t eve ry denumerable un ion o f denumerable s e t s i s denumerable. Indeed, t h i s axiom a l l o w s one t o choose, f o r each o f t h e denumerable s e t s i n t h e union, a b i j e c t i o n f rom t h a t s e t o n t o t h e i n t e g e r s .

On t h e o t h e r hand, t h e above s tatement i s n o t p rovab le f rom ZF a lone: t h e r e i s a model o f ZF itn which t h e cont inuum i s a denumerable un ion o f denumerable s e t s ( A z r i e l LEVY, unpub l i shed) .

2.6. DENUMERABLE SUBSET A X I O M

T h i s axiom s t a t e s t h a t eve ry i n f i n i t e s e t has a denumerable subset. It f o l l o w s

f rom t h e coun tab le axiom o f cho ice . 0 L e t a be an i n f i n i t e s e t . Fo r each i n t e g e r i , assoc ia te t h e s e t o f i - t u p l e s

o f elements f rom

one o f these i - t u p l e s . It remains t o take t h e W-sequence formed f rom t h e terms

o f t h e chosen 1 - t u p l e , 2 - tup le , ... . 0

The denumerable subset axiom i s s t r i c t l y weaker than t h e coun tab le cho ice (JECH 1973). L e t a

(1) t h e r e e x i s t s a denumerable subset o f a ; ( 2 ) t h e r e e x i s t s a b i j e c t i o n o f a a i s D e d e k i n d - i n f i n i t e (see 1.3);

(3 ) t h e r e e x i s t s a cho ice f u n c t i o n f which t o each f i n i t e subset x of a assoc ia tes an e lement f x i n t h e complement a-x . Consequently, t hedenumerab le subset axiom i s e q u i v a l e n t t o say ing t h a t f i n i t e n e s s c o i n c i d e s w i t h Dedek ind - f i n i t eness . However w i t h ZF a lone, t h e r e can e x i s t an

i n f i n i t e s e t hav ing f o r each i n t e g e r i a subset equ ipo ten t w i t h i , y e t hav ing no denumerable subset .

a . By coun tab le choice, we can a s s o c i a t e t o each i n t e g e r i

be an i n f i n i t e se t ; t h e f o l l o w i n g t h r e e c o n d i t i o n s a re e q u i v a l e n t :

o n t o a p r o p e r subset o f a ; i n o t h e r words,

2.7. Having d e f i n e d t h e i n t e g e r s , we can now complete t h e i n i t i a l remarks f rom 5 2 by adding t h a t , w i t h t h e axiom o f f ounda t ion , t h e r e a r e no O-sequences u w i t h ui+l be long ing t o ui f o r each i n t e g e r i . I n p a r t i c u l a r , f o r eve ry i n t e -

ge r r , t h e r e i s no c y c l e u1C u2 E . .. E ur E u1 .

2.8. The axiom o f f o u n d a t i o n i s e q u i v a l e n t t o t h e f o l l o w i n g axiom scheme.

L e t f be a c o n d i t i o n which ho lds f o r 0 and such t h a t , i f ho lds f o r each e l e - ment o f a g i v e n s e t a , then f ho lds f o r a . Under these hypotheses, f holds f o r eve ry s e t . Note t h a t we can e l i m i n a t e t h e hypo thes i s " f h o l d s f o r 0 "; t h i s

-

Chapter 1 11

be ing a p a r t i c u l a r case o f t h e second hypo thes i s , made p r e c i s e as f o l l o w s : e i t h e r

t h e r e e x i s t s an element o f a s a t i s f y i n g " n o t f I' , o r a s a t i s f i e s f . 0 L e t a be a s e t which f a l s i f i e s t h e axiom o f f ounda t ion , and l e t '6 be t h e c o n d i t i o n h o l d i n g f o r 0 and f o r eve ry s e t which does n o t be long t o a . Then

(e s a t i s f i e s o u r hypotheses, b u t 'if does n o t h o l d f o r eve ry e lement o f a . Conversely, l e t be a c o n d i t i o n s a t i s f y i n g o u r hypotheses, b u t such t h a t t h e s e t a does n o t s a t i s f y . L e t al be t h e s e t o f elements o f a s a t i s f y i n g " n o t f". L e t a2 be t h e s e t o f e lements o f t h e un ion u al s a t i s f y i n g " n o t If: " . L e t a3 be t h e s e t o f elements o f u a2 s a t i s f y i n g " n o t I$" , e t c . Then t h e un ion o f t h e

ai (i i n t e g e r ) f a l s i f i e s t h e axiom o f f ounda t ion . 0

2.9. A necessary and s u f f i c i e n t c o n d i t i o n f o r a s e t f o r eve ry f i n i t e sequence ao, al, ... , an w i t h a. = a and ai+l be long ing t o

ai f o r i < n , every ai i s t r a n s i t i v e . I n o t h e r words, an o r d i n a l i s a s e t which i s h e r e d i t a r i l y t r a n s i t i v e ; use 2.1: e v e r y t r a n s i t i v e s e t o f o r d i n a l s i s

an o r d i n a l . E q u i v a l e n t l y , a i s an o r d i n a l i f f a and a l l e lemen tso f a a re t r a n s i t i v e (see f o r i n s t a n c e POWELL 1975 p. 223).

Analogously , we leave i t t o t h e reader t o p rove t h a t a s e t a i s an i n t e g e r

i f f a i s empty o r a i s a successor s e t , and eve ry element o f a i s e i t h e r empty o r a successor s e t (communicated by HATCHER 1977).

a t o be an o r d i n a l i s t h a t

2.10. A X I O M OF C H O I C E FOR FINITE SETS Now t h a t i n f i n i t e s e t s have been in t roduced , we i n d i c a t e he re an i m p o r t a n t

weakening o f t h e axiom o f cho ice , which a s s e r t s t h e e x i s t e n c e o f a cho ice s e t f o r e v e r y s e t o f non-empty, f i n i t e , m u t u a l l y d i s j o i n t se ts . T h i s weakened fo rm

i s n o t i m p l i e d by and does n o t i m p l y t h e coun tab le axiom o f cho ice f rom 0 2.5, n o r t h e denumerable subset axiom o f 2.6. I n f a c t t h e axiom o f dependent choice, which i s s t r o n g e r than coun tab le cho ice , does n o t i m p l y cho ice among f i n i t e

se ts : see ch.2 5 1.6.

2.11. We s h a l l c a l l i n d u c t i o n , o r t r a n s f i n i t e i n d u c t i o n , t h e f o l l o w i n g reasoning. Suppose t h a t i f a c o n d i t i o n 'f h o l d s f o r e v e r y o r d i n a l s t r i c t l y l e s s than o( t hen If h o l d s f o r 4 ; under t h i s hypo thes i s , f ho lds f o r e v e r y o r d i n a l . Th i s i s a fo rm o f t h e scheme s t a t e d i n 5 2.1: i f " n o t " i s s a t i s f i e d by a t l e a s t one o r d i n a l , t h e r e i s a l e a s t o r d i n a l s a t i s f y i n g " n o t e'' . Often, i n d u c t i o n i s broken

up i n t o a p r o o f f o r 0, a p r o o f f o r t h e t r a n s i t i o n between an a r b i t r a r y o r d i n a l o(

and i t s successor o( + 1 , and a p r o o f f o r o(

A d e f i n i t i o n by r e c u r s i o n i s made by i n t r o d u c i n g a s tatement f ( o ( ,a) which u n i q u e l y assoc ia tes a s e t a t o each o r d i n a l o( . T h i s s ta temen t w i l l u s u a l l y be

a l i m i t o r d i n a l .

o f the fo l l ow ing form. "There e x i s t s one and on ly one func t ion f w i t h domain d + 1 ( the successor o f o(), such t h a t the i n i t i a l ordered p a i r f (where u i s a r b i t r a r i l y given), the f i n a l ordered p a i r (o( ,a) belongs t o f , and such t h a t f o r each (3 o( the p a i r ( (3 ,b) belongs t o f , provided t h a t b has been obtained i n a c e r t a i n ( s u i t a b l y defined) manner from the se t o f ordered pa i r s belonging t o f w i t h f i r s t term < 13 ' I . Because o f the uniqueness o f f , when q'>d, , the func t i on f ' corresponding t o oc' w i l l be an exten- s ion o f f t o the domain a( '+1 . Some examples o f d e f i n i t i o n by recursion:

sum, product, exponent iat ion f o r o rd ina l s i n 0 3; aleph rank i n 0 6.4. The d e f i n i - t i o n o f fundamental rank i n 0 5.2 i s a lso by recursion, i f one begins by associa- t i n g t o each o rd ina l o( the se t o f a l l sets w i t h fundamental rank o( . Note t h a t d e f i n i t i o n by recurs ion using the axioms o f ZF i s eas ie r t o j u s t i f y than d e f i n i t i o n by simple recurs ion i n f i r s t - o r d e r Peano ar i thmet ic , such as i s general ly presented today (however, the o r i g i n a l t e x t o f PEANO 1894 i s w r i t t e n

i n second-order l o g i c ) . I n order t o j u s t i f y d e f i n i t i o n by recurs ion i n f i r s t - order a r i thmet ic , one i s led, i n the manner o f GOOEL 1931, t o use the "Chinese remainder theorem". For instance, one def ines b = a! as an abbrev ia t ion f o r

the fo l low ing : " there e x i s t two in tegers u, v such t h a t the remainder a f t e r

d i v i s i o n o f u by v+ l i s 1 , the remainder a f t e r d i v i s i o n o f u by (a+ l )v + 1 i s b , and fo r each i ( 1 s i d a) one obtains the remainder a f t e r

d i v i s i o n o f u by ( i + l ) v + 1 from the remainder a f t e r d i v i s i o n o f u by i v + 1 by m u l t i p l y i n g the l a t t e r by i+l " .

(0,u) belongs t o

5 3 - REVIEW OF ORDINAL ALGEBRA, CANTOR NORMAL FORM, INDECOMPOSABLE ORDINAL

3.1. SUM

We say t h a t o( +/3 = Y (where d,b, 8 are o rd ina l s ) i f f t h e r e e x i s t s a func t ion

f w i t h domain f i +1 (hence, f o r each u & f i there i s one and only one ordered p a i r belonging t o f w i t h f i r s t term u ), such t h a t the i n i t i a l p a i r (0, m ) and f i n a l p a i r ( / 5 , ) belong t o f ; i f (u,v) belongs t o f where u < fs , then (u+l,v+l) belongs t o f ; and f i n a l l y such t h a t i f f contains as elements ordered p a i r s (x,y) f o r which the f i r s t terms x admit a supremum Sup x 4 (5, then the p a i r (Sup x, Sup y ) belongs t o f . Given o( and /3 , the reader can prove by induc t ion on /3 the existence and

uniqueness o f the preceding func t ion , hence the existence and uniqueness o f the ord ina l 5 = A + f i . I n the same manner, one proves f o r every o( , /s the equa-

l i t i e s d + O = O+o( = % and oC+( /5+1) = (* +f3)+1 , and f o r every o( and

every s e t o f o rd ina l s u the supremum e q u a l i t y o( +(Sup u) = Sup(& + u) . For a l l o( , 13 P 0 we have o(+ /3, > o( . For a1 1 o( , p we have 4+ >, (3 where

Chapter 1 13

equa l i t y i s poss ib le w i t h non-zero o( : f o r instance 1 + 4) = G, . The supremum equa l i t y does no t ho ld on the l e f t : i f i i s an a r b i t r a r y i n tege r then S u p ( i + w ) = ~d # (Sup i ) + LJ = ~ d + t.3.

Ordinal add i t i on i s assoc ia t i ve . Commutativity holds f o r integers, o r f i n i t e ordinals; however 1 + CJ = c d # id+ 1 . For a l l O( and f i 2 o( , there e x i s t s one and on ly one s a t i s f y i n g CA + 8 = 0 ; t h i s 2( i s c a l l e d the d i f f e rence /3 - o( .

The inequa l i t y 6 imp l ies o(+ /3$ o< + &' and conversely. Also the

same r e s u l t f o r s t r i c t i n e q u a l i t y < . Hence add i t i on i s l e f t cancel lable, i .e.

The i n e q u a l i t y % $ impl ies o( + 6 + I f . This does no t hold i n general f o r < , as 0 + W = 1 + W . Hence add i t i on i s no t r i g h t cancel lable. F i n a l l y the ord ina l 1 and consequently every f i n i t e o rd ina l i s absorbed by

every i n f i n i t e o rd ina l , i n t he sense t h a t 1 + o( = o( f o r q i n f i n i t e .

w + /3 = O( + I I imp l ies b = 8 .

3.2. PRODUCT

We say t h a t o( . /s = ?f i f f there e x i s t s a func t i on f w i t h domain f i + 1 , such t h a t the i n i t i a l p a i r (0,O) and the f i n a l p a i r ( 0 , g ) belong t o f , and such t h a t i f (u,v) f where u < fs then (u+l,v+o( ) 6 f , and such t h a t i f (x,y) E f f o r a l l x belonging t o a se t which admits a supremum

Sup x ,< f i then (Sup x , Sup y ) E f . For a l l o( , p we have o( .O = 0 . N = 0 For every o( and every se t o f o rd ina ls u , we have the supremum equa l i t y d .(sup u) = sup(@ .u) . Moreover d.0 = O i s equivalent t o o ( = O o r /3= O . Ordinals o f the form o(.u , w i t h o( f i x e d and u an a r b i t r a r y o rd ina l , are

ca l l ed the mu l t i p les o f o( . For example 0 i s a m u l t i p l e o f every o rd ina l .

Every m u l t i p l e o f MI augmented by o( , y i e l d s a m u l t i p l e o f o( . The supremum

o f a se t o f mu l t i p les o f d i s a m u l t i p l e o f o( . F i n a l l y every m u l t i p l e o f o(

i s obtained from 0 by these two ind i ca ted orocesses. More r i go rous l y i f a condi-

t i o n i s t r u e f o r 0 and i s preserved i n the passage from an ord ina l u t o u+w as we l l as i n the passage t o supremum, then t h i s cond i t ion i s t r u e f o r every m u l t i p l e o f o( . The supremum e q u a l i t y on the r i g h t , given above, does no t ho ld on the l e f t : i f

i designates an a r b i t r a r y in teger , then Sup(i.2) = W # (Sup i ) . 2 = CJ .2 . M u l t i p l i c a t i o n i s assoc ia t i ve and d i s t r i b u t i v e on the r i g h t : r . ( M + / 3 ) =r.q+r,(,p I

D i s t r i b u t i v i t y on the l e f t and commutativity ho ld f o r integers; however ( w + l ) . w = c J . o # w . b + 1 . ~ 3 and 2 . ~ = ~ # ~ 3 . 2 . I t c a n h a p p e n t h a t c ( . p

i s not a m u l t i p l e of (3 , e.g. (0+1).2 = w . 2 + 1 : i t i s no t a m u l t i p l e o f 2 . For 4 # 0 the i n e q u a l i t y 1) 4 $ impl ies & . (3 6 &.$ and conversely. The same r e s u l t holds f o r s t r i c t i nequa l i t y . Thus m u l t i p l i c a t i o n i s cancel lable on

and d .( &+1) = o(. 0 + o( .

the l e f t except f o r 0 ; i . e . f o r O( non-zero g. fs = q . v imp l ies 0 = d . The inequa l i t y o( & f s imp l ies o( .y,r 0 . 8 since 1.a = 2 . w = W . M u l t i p l i c a t i o n i s thus no t cancel lable on the r i g h t . Given two o rd ina l s d and f i # 0 , there i s a unique o rd ina l c a l l e d the quot ient , and a unique o rd ina l E c a l l e d the remainder i n the d i v i s i o n o f o( by f i , w i t h g = f i $ + and E <' 0 : consequence o f t he existence o f a maximum o rd ina l u such t h a t /s u 4 4 .

. This does no t subs is t f o r <

3.3. POWER OR EXPONENTIATION We say t h a t &Is =

t h a t the i n i t i a l p a i r (0 , l ) and the f i n a l p a i r ((3,&) belong t o f , and such t h a t i f (u ,v )E f where u < (3 then ( u t l . v . 4 ) f , and such t h a t i f (x,y) E f f o r x belonging t o a se t which admits a supremum Sup x 6 (3, then

For a l l cx,p we have oCo = 1 and cc ('+'I = 4'. 4 . For a l l o( and every se t o f o rd ina ls u , we have the supremum e q u a l i t y o( (sup Moreover W p = 0 i s equ iva len t w i t h o( = 0 and f i # 0 . The e q u a l i t y M p = 1 i s equivalent t o o( = 1 Ordinals o f the form o( , w i t h cx f i x e d and u an a r b i t r a r y o rd ina l , are c a l l e d powers o f O( . For example 1 i s a power o f every r% . I f v i s a power o f o( then so i s v.* . The supremum o f a s e t o f powers o f i s a power o f 4 . F i n a l l y , every power o f o( i s obtained from 1 by these two processes. The supremum e q u a l i t y on the r i g h t g iven above does no t subs i s t on the l e f t : i f i designates an a r b i t r a r y in teger , then Sup(i ) = W # (Sup i)' = W

We have o( (*+'

We have ( ~ & f i ) ~ = &(b' r ) f o r a l l o<,p, 8 . It i s impossible i n general t o interchange terms o f the product i n the exponent: The e q u a l i t y (ac).(bc) = (a.b)c which holds f o r integers, does n o t subs is t i n general, even f o r f i n i t e exponents: = o .2 . This e q u a l i t y does no t subs is t f o r a, b f i n i t e and an i n f i n i t e exponent: ( 2 L J ) . ( Z W ) = L 3 2 # 4 W = w . For o( >/ 2 , the inequa l i t y f34 21 imp l ies NP< c ( ~ and conversely; same r e s u l t with < . Thus we have cance l la t ion : The i n e q u a l i t y o( 4 -fir ; t h i s does no t subs i s t f o r < since z W = 3 w = w . F i n a l l y f o r o(( f s the o rd ina l w4 i s absorbed by &* , i . e . aa+ W f i = k)?

i f f there e x i s t s a func t ion f w i t h domain f i +1 , such

(SUP X I SUP Y) E f .

= Sup (oi ') .

o r /3 = 0

2 2 . = &fi.4r f o r a l l O( ,P, r . It i s impossible i n general

t o interchange terms i n the product; f o r example 2( w+l) =w.2 f 2 1 . 2 w = w .

(2 ")' = W2 # 2('*@) =

2 2 ( Q ~ ) . Z ~ = u 2 . 4 # ( w . 2 )

= o( ' impl ies fs = b' .

3.4. Given two o rd ina l s o( and /3 >/ 2 , there i s a unique o rd ina l which i s the maximum exponent among the o rd ina l s u s a t i s f y i n g f i \< 4 .

Chapter 1 15

3- Given o( and (3 2 2 and t h e maximuA exponent 8 such t h a t /3 6 o( , there e x i s t s a maximum o r d i n a l s such t h a t ( f i r ) . $,< o( . Moreover

Given b , 2 < (-5

and a s t r i c t l y decreas ing, t hus f i n i t e sequence o f o r d i n a l s

8 2 $ ( I ) '> g(2) , ... and a co r respond ing se uence o f o r d i n a l s t ( l ) , $ ( 2 ) , ... each s t r i c t l y l e s s than /3 , we have bx> f i9 ( ' ) . $ (1 ) t f i g ( * ) . $(2 ) t ... ( p r o o f by i n d u c t i o n on 8 ) .

3.5. CANTOR NORMAL FORM Given 4 and f3 >, 2 , t h e r e e x i s t s a decomposi t ion o f o( i n t o a f i n i t e sum o f - terms f i r , 6 , w i t h c o e f f i c i e n t s $< /3 and exponents s t r i c t l y decreas ing.

Fur thermore t h i s decomposi t ion i s unique. It i s c a l l e d t h e Cantor decomposi t ion

o f o( i n t o powers o f 0 o r Cantor normal f o rm o f 4 i n base f , . I n t h e case t h a t f i = u , t h e c o e f f i c i e n t s 6 a r e i n t e g e r s .

3.6. DECOMPOSABLE AND INDECOMPOSABLE ORDINAL An o r d i n a l o( i s c a l l e d decomposable i f f t h e r e e x i s t & < q and r'c.C with

then eve ry sum o f two non-zero o r d i n a l s which i s equal t o o( has second te rm

equal t o d , and converse ly . A non-zero o r d i n a l o( i s indecomposable i f f O( i s a power o f a. T h i s f o l l o w s

f rom t h e e x i s t e n c e and uniqueness o f t h e Cantor decomposi t ion i n t o powers o f LC: , t o g e t h e r w i t h t h e a b s o r p t i o n s tatement (end o f 3.3).

o( = f l+ 8 ; othe rw ise o( i s c a l l e d indecomposable. I f o( i s indecomposable,

5 4 - EQUIPOTENT WITH THE CONTINUUM, CONTINUUM HYPOTHESIS, REAL 4.1. EQUIPOTENT WITH THE CONTINUUM

A s e t i s s a i d t o be e q u i p o t e n t w i t h t h e continuum i f f i t i s e q u i p o t e n t w i t h ?(a), t h e power s e t o f t h e i n t e g e r s , o r e q u i v a l e n t l y w i t h " 2 , t h e s e t o f f u n c t i o n s

on w t a k i n g va lues 0 o r 1 . By CANTOR'S theorem 1.5, eve ry coun tab le s e t i s

s t r i c t l y subpo ten t w i t h 9 ( IC) ) . L e t a , b be two d i s j o i n t denumerable se ts . By 1.6 we have t h a t a2 x b2 i s equ ipo ten t w i t h ( a " b ) 2 . Hence t h e C a r t e s i a n p r o d u c t o f two s e t s each equi - p o t e n t w i t h t h e cont inuum i s i t s e l f e q u i p o t e n t w i t h t h e continuum. The same

r e s u l t ho lds f o r t h e C a r t e s i a n p r o d u c t o f a coun tab le s e t w i t h a s e t which i s e q u i p o t e n t w i t h t h e continuum. S i m i l a r l y 2) i s e q u i p o t e n t w i t h ( w w ) 2 . Hence a i s a s e t

e q u i p o t e n t w i t h t h e continuum, then t h e s e t o f &-sequences w i t h va lues i n i s a l s o e q u i p o t e n t w i th t h e continuum.

4.2. I f we s u b t r a c t an a r b i t r a r y denumerable subset a f rom a s e t c equ ipo ten t

w i th t h e continuum, t h e n t h e d i f f e r e n c e c-a i s e q u i p o t e n t w i t h t h e continuum.

a


This i s a special case o f the fo l l ow ing propos i t ion .

- Let a be an i n f i n i t e se t which i s equipotent w i t h the Cartesian product 2xa , and l e t c = y ( a ) . Then the d i f f e rence set, obtained by removinq from c an a r b i t r a r y subset which i s e q u i p o t e n t w i g a , i s equipotent w i t h c . 0 Since a i s equipotent w i t h 2xa , the se t c , which i s equipotent w i t h a2 , i s a lso equipotent w i t h cxc by 1.6. Hence the d i f fe rence o f c and a subset which i s equipotent w i t h a i s equipotent w i t h the d i f fe rence o f cxc and the range o f a b i j e c t i o n f on a . Each element x o f a i s associated t o an ordered p a i r f x = (y,z) o f elements y, z o f c . Le t us associate t o each x

the f i r s t term y o f t h i s p a i r . The func t ion thus obtained has domain a and cannot have range c = ? (a) , by CANTOR'S lemma 1.5. Thus there e x i s t s an element u o f c f o r which (u,z) i s no t the value by f o f an element o f a , f o r any z belonging t o c . Hence the d i f f e rence o f cxc and f " ( a ) includes a subset which i s equipotent w i t h

w i t h c . 0

c , and so by BERNSTEIN-SCHRODER 1.4 i s equipotent

4.3. Le t a be a se t equipotent w i t h the continuum. For every p a r t i t i o n o f a i n t o denumerably many subsets, one o f the subsets i s equipotent w i t h the

continuum (uses the axiom of choice).

0 Suppose on the cont ra ry t h a t there i s a p a r t i t i o n o f ai ( i in teger ) , and t h a t every theorem 1.8 (axiom o f choice), the union a o f the ai i s s t r i c t l y subpotent w i t h the Cartesian product o f an a-sequence o f sets, each equipotent w i t h the continuum. But t h i s Cartesian product i s equipotent w i t h the continuum: contra- d i c t i on . 0

a i n t o d i s j o i n t subsets

ai i s s t r i c t l y subpotent w i t h a . Then by KONIG's

4.4. CONTINUUM HYPOTHESIS, GENERALIZED CONTINUUM HYPOTHESIS

The axiom c a l l e d continuum hypothesis asserts the non-existence o f a se t which i s s t r i c t l y intermediate, w i t h respect t o subpotence, between o and y( a ) . This axiom i s l o g i c a l l y independent o f ZF, and even o f ZF p lus the axiom o f choice (COHEN 1963, see B ib l iography 1966). The axiom c a l l e d general ized continuum hypothesis asserts the non-existence o f a se t s t r i c t l y intermediate, w i t h respect t o subpotence, between a and p ( a ) , f o r every i n f i n i t e s e t a . When added t o the axioms o f ZF, t h i s imp l ies the

axiom o f choice (see ch.2 exerc. 1).

4.5. REAL

We leave i t t o the reader t o redef ine p o s i t i v e and negat ive in teger , and then

rea l , as an ordered p a i r formed from an in tege r which i s c a l l e d the i n tege r pa r t , and an i n f i n i t e se t o f non-negative in tegers . The l a t t e r s e t w i l l be

i d e n t i f i e d w i t h an W-sequence o f terms ui ( i non-negative in teger ) w i t h

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Chapter 1 17

ui = 0 o r 1 according t o whether i belongs t o the i n f i n i t e se t o f in tegers

o r not. This sequence i s c a l l e d the b inary expansion o f the rea l , which always contains i n f i n i t e l y many occurrences o f zero. The not ions o f r a t i o n a l rea l and dyadic rea l , i . e . r a t i o n a l whose denominator i s a power o f 2 , are assumed t o be f a m i l i a r , as we l l as the denumerabil ity o f the se t o f ra t i ona ls .

The se t o f rea l s i s equipotent w i t h the continuum: remove from the se t o f a l l sets

o f in tegers , the denumerable se t o f f i n i t e sets o f in tegers , and use 4.2. We leave i t t o the reader t o de f ine the order ing on the rea ls : less than o r equal

t o ( 3 ) , and the re la ted s t r i c t i nequa l i t i es . Also the reader can de f ine the not ions o f dense, c o f i n a l , c o i n i t i a l se t o f rea l s

(an example being the r a t i o n a l s o r the dyadic r e a l s ) . The reader can def ine a closed, open, hal f -open i n t e r v a l o f rea l s , an i n i t i a l , f i n a l i n t e r v a l , an _upper - bound and lower bound o f a se t o f rea l s , the maximum, the minimum, a rea l valued sequence which i s s t r i c t l y ( o r otherwise) increasing, decreasing.

Every se t o f mutual ly d i s j o i n t i n t e r v a l s o f rea l s which are no t reduced t o

s ing le tons i s countable: enumerate the r a t i o n a l s and associate t o each i n t e r v a l the f i r s t r a t i o n a l which belongs t o i t .

Consequently, every s t r i c t l y inc reas ing ( o r s t r i c t l y decreasing) ordinal-indexed

sequence o f rea l s i s countable.

( 3 ) , greater than o r equal t o

4.6. DEDEKIND'S THEOREM

I f we p a r t i t i o n the rea l s i n t o an i n i t i a l i n t e r v a l a and i t s complement the f i n a l i n t e r v a l b , both non-empty, then e i t h e r a has a maximum element o r b has a minimum element.

Consequently, f o r any s e t a o f rea l s , i f there e x i s t s an upper bound, then there e x i s t s a l e a s t upper bound c a l l e d the supremum o f a and denoted Sup a . Analogous d e f i n i t i o n o f the infimum which i s denoted I n f a . I n o ther words,

f o r every s e t a o f reals, there e x i s t s a smal lest i n t e r v a l (w i th respect t o

i nc lus ion ) i nc lud ing a : the i n t e r v a l ( I n f a , Sup a) which i s closed, open o r half-open, i n i t i a l , f i n a l o r con ta in ing a l l t he rea l s , depending on the case.

When usefu l , we w i l l use the presumed t o know.

and p roduc t o f rea ls , which the reader i s

4.7. To see some i n i t i a l d i f f i c u l t i e s provided by the axiom o f choice, which

i nd i ca te t h a t t h i s axiom i s no t "obvious", note t h a t i t i s impossible i n ZF p lus

the axiom o f choice, t o def ine and prove uniqueness o f a choice func t ion which

associates t o each non-empty se t o f rea l s one o f i t s elements. S i m i l a r l y i t i s

impossible t o uniquely de f ine a choice se t p i ck ing one func t ion from each p a i r

of r e a l func t ions h,-h , where f o r each rea l x , the value o f -h

on x i s t he a d d i t i v e inverse o f h (x ) . To obta in a p roo f o f uniqueness, com- p l e t i n g the existence (which i s guaranteed by the axiom o f choice), i t i s necessary f o r example t o add t o ZF the axiom o f c o n s t r u c t i b i l i t y o f GODEL 1940.

§ 5 - TRANSITIVE CLOSURE, HEREDITARILY F I N I T E SET, FUNDAMENTAL RANK, CARD I NAL

5.1. TRANSITIVE CLOSURE For every se t a , there e x i s t t r a n s i t i v e supersets o f a , and among these there ex i s t s one which i s included i n a l l t he others. Th is se t i s formed from the values o f a l l f i n i t e sequences xl, ... ,xh (h i n tege r ) such t h a t x1 € a and xi+l E xi f o r each i (1 6 i < h) . We s h a l l c a l l t h i s s e t the t r a n s i t i v e c losure of a . For each non-empty s e t a , t h e t r a n s i t i v e c losure o f a i s t he union of a together w i t h the t r a n s i t i v e c losures o f the elements o f a . I f a s b then (c losure o f a ) 5 (c losure o f b ) ,

HEREDITARILY FINITE SET A h e r e d i t a r i l y f i n i t e se t i s a se t whose t r a n s i t i v e c losure i s f i n i t e . For ins tance, every f i n i t e t r a n s i t i v e s e t i s h e r e d i t a r i l y f i n i t e . I n p a r t i c u l a r every i n t e - ger ( i . e . every f i n i t e o rd ina l ) i s h e r e d i t a r i l y f i n i t e . The s ing le ton o f 1 i s non - t rans i t i ve y e t h e r e d i t a r i l y f i n i t e . Every h e r e d i t a r i l y f i n i t e se t i s f i n i t e , as it i s included i n i t s t r a n s i t i v e c losure which i s f i n i t e . Every element and every subset o f a h e r e d i t a r i l y f i n i t e s e t i s h e r e d i t a r i l y f i n i t e . Every f i n i t e s e t o f h e r e d i t a r i l y f i n i t e se ts i s here- d i t a r i l y f i n i t e . S i m i l a r l y f o r f i n i t e unions, f i n i t e Cartesian products, and the power s e t o f h e r e d i t a r i l y f i n i t e sets. A necessary and s u f f i c i e n t cond i t i on f o r a s e t a t o be h e r e d i t a r i l y f i n i t e i s tha t , f o r every f i n i t e sequence xO, ..., xh ( h i n tege r ) w i t h xo = a and xi+l E xi f o r each i < h , the terms xi are f i n i t e .

-

5.2. FUNDAMENTAL RANK Le t a be a se t and c be the t r a n s i t i v e c losure o f the s ing le ton i a 1 . We say t h a t the o rd ina l o( i s the fundamental rank o f a , i f there e x i s t s a func t i on f w i t h domain c , tak ing o rd ina l values 4 o( , such t h a t the i n i t i a l ordered p a i r (0,O) and the f i n a l ordered p a i r (a, cx ) belong t o f : so t h a t f ( 0 ) = 0 and f ( a ) = d ; and such t h a t i f u E c then the value f ( u ) i s the smal les t o rd ina l s t r i c t l y g rea ter than f ( x ) f o r a l l x belonging t o u . It fo l lows from the axiom o f foundat ion t h a t every s e t has a unique fundamental rank. Indeed, the empty se t 0 has rank 0 . Suppose t h a t a i s non-empty and t h a t every element o f a has a rank. Then by the preceding d e f i n i t i o n , a has rank equal t o the smal les t o rd ina l which i s s t r i c t l y g rea ter than the ranks of

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Chapter 1 19

a l l i t s elements. The exis tence of rank r e s u l t s from the axiom of foundation i n the form of scheme 2.8. For every ordinal oc the fundamental rank i s g .

5.3. For every ordinal o( , there i s a s e t t t . 4 . Moreover V, has fundamental rank o( . 0 Obvious f o r 0 s ince Vo i s empty. I f t h i s i s t r u e f o r cl( , then i t i s t rue f o r @ + l with Vq +1 = s e t of elements and subse ts of V,. Fina l ly f o r q a l i m i t o rd ina l , V, i s the union of t he Vi f o r i s t r i c t l y l e s s than o( . 0

Note t h a t f o r each ordinal & , the s e t Vatl - V, of s e t s of rank o( i s non- empty, s ince V, and 4 belong t o t h i s s e t . For i an in t ege r , o r f i n i t e o rd ina l , the s e t of sets of rank i i s f i n i t e . I t follows t h a t every i n f i n i t e s e t has rank a t l e a s t equal t o w . Note t h a t a s e t i s he red i t a r i l y f i n i t e i f f i t s fundamental rank i s f i n i t e . The set of he red i t a r i l y f i n i t e sets i s the in t e r sec t ion of a l l sets which contain 0 and which, i f they contain x and y , a l so contain x u { y ) as an element.

Vd o f a l l s e t s of ranks s t r i c t l y l e s s

-

5.4. CARDINAL, OR CARDINALITY Given a set a , consider sets equipotent w i t h a and among these , those of minimum fundamental rank. By the preceding, these form a non-empty s e t which we c a l l the card ina l o r ca rd ina l i t y of a , denoted by Card a : def in i t i on from SCOTT 1955. T h u s every set has a ca rd ina l , and two s e t s a r e equipotent i f f they have the same card ina l . Note t h a t every s e t a i s equipotent, not t o Card a , but t o an a rb i - t r a r y element o f

Given two card ina ls a and b , the ordering of l e s s than o r equal t o , o r grea te r than o r equal t o , means t h a t every s e t of cardinal a i s subpotent w i t h every s e t of cardinal b . Obvious de f in i t i on of s t r i c t ordering; notations 6 , < .

Card a . T h i s i s only a minor inconvenience i n the de f in i t i on .

5.5. CARDINAL SUM, CARDINAL PRODUCT AND EXPONENTIATION Let a and b be card ina ls ; the cardinal sum a + b i s defined as the cardinal of the union of two d i s j o i n t sets of cardinal a , respec t ive ly b . We denote the cardinal sum by + (boldface) t o avoid confusion w i t h the ordinal sum + i n 3.1. Thus we can iden t i fy , i n 5 6 below, Card w w i t h W i t s e l f , and wr i te a+ 1 = W and y e t W + ~ > L S . To be rigorous, we should a l s o d is t inguish bet-

ween the ordering r e l a t ion f o r card ina ls and f o r ord ina ls . In p rac t i ce the context will always permit t he d i s t i n c t i o n . Since cardinal mul t ip l ica t ion and exponentia- t i on a re denoted by a x b and a b (no ta t ions from 1 . 2 and 1.6), there wi l l be no confusion w i t h the operations of ordinal mul t ip l ica t ion and exponentiation a.b and ba . In p a r t i c u l a r the cardinal notation “ i s not necessary: ~3~

wil l be s u f f i c i e n t .

The sum a + b does no t depend upon the choice o f d i s j o i n t se ts o f card ina l and cardinal b . Cardinal add i t i on i s commutative and assoc ia t i ve . We have

a

a + O = a . F i n a l l y a b a ' and b & b ' imply a + b , ( a ' + b ' .

The cardinal product a x b i s def ined as the card ina l o f t he Cartesian product o f a set o f cardinal a with a se t o f card ina l b . There w i l l be no inconvenience i n using the same symbol f o r card ina l m u l t i p l i c a t i o n and f o r the Cartesian product o f two sets (see 1.2). The card ina l product does no t depend upon the choice o f t he sets o f cardinal a , resp. b . Cardinal m u l t i p l i c a t i o n i s commutative, associa- t i ve , and d i s t r i b u t i v e over card ina l add i t ion : We have a % 0 = 0 and a x 1 = a . F i n a l l y a G a ' and b ,<b ' imply a x b s a ' x b '

The cardinal power ab i s def ined as the card ina l o f t he power between sets o f cardinal a , resp. b (no ta t i on from 1.6). Cardinal exponent iat ion does no t depend upon the choice o f t he se ts o f card ina l a , resp. b . We have 'a = 1 , 'a = a , aO = 0 f o r a # 0 , and al = 1 . Moreover f o r b # 0 , condit ions a,< a ' and b,< b ' imply ab C a 'b ' . F ina l l y the equipotences i nd i ca ted i n 1.6 become card ina l equa l i t i es :

(b+c)a = (ba)$(Ca) ; then '(ash) = ('a)x('b) ; and C(ba) = ( b * c ) a .

(a+b)%c = ( a x c ) + ( b x c ) .

§ 6 - ALEPH, HARTOGS, ALEPH RANK

6.1. ALEPH I n the case o f a se t a which has an o rd ina l equipotent t o it, we take as the d e f i n i t i o n o f the card ina l o f a , denoted s t i l l by Card a , the smal lest o rd ina l equipotent t o a . In s p i t e o f the very d i f f e r e n t no t i on o f card ina l as def ined i n 5.4, t h i s new Card a b i j e c t i v e l y corresponds t o the o l d not ion, a t l e a s t f o r sets a which are equipotent t o an o rd ina l . Such sets a re c a l l e d wel l -orderable i n ch.2 5 2.5 below. The card ina l o f such a se t i s c a l l e d an aleph. D e f i n i t i v e l y , we have the fo l l ow ing a r t i f i c i a l bu t general and r igorous d e f i n i t i o n : i f a i s equipotent t o an ord ina l , then the smal lest such i s Card a ; otherwise Card a i s the se t o f a l l sets o f minimum fundamental rank which are equipotent w i t h a . Another d e f i n i t i o n o f aleph, which i s equ iva len t t o the preceding one: an aleph i s an ord ina l o( which i s equipotent t o no o rd ina l < o( ( l e s s than w i t h respect t o the order ing o f the o rd ina l s ) . I n p a r t i c u l a r , the f i n i t e alephs are the integers, the f i r s t i n f i n i t e aleph i s 0 . We w i l l see i n ch.2 5 2.5 tha t , with the axiom o f choice, every card ina l i s an aleph (equ iva len t ly every se t i s wel l -orderable). Notice t h a t i f o( and are two equipotent o rd ina ls , then every intermediate o rd ina l i s equipotent t o them. Moreover, f o r every i n f i n i t e o rd ina l a( , the

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Chapter 1 21

successor o( +1 i s e q u i p o t e n t w i t h iA . It f o l l o w s t h a t eve ry i n f i n i t e a leph i s a l i m i t o r d i n a l .

6.2. HARTOGS SET, OR HARTOGS ALEPH Le t a be an i n f i n i t e s e t . We say t h a t an o r d i n a l u i s i n j e c t a b l e i n a , i f f the re e x i s t s an i n j e c t i o n o f u i n t o a , o r e q u i v a l e n t l y i f a subset o f a i s

equ ipo ten t w i t h u . I f u i s i n j e c t a b l e i n a , then eve ry o r d i n a l s u and

every o r d i n a l e q u i p o t e n t w i t h u i s i n j e c t a b l e i n a . Since a i s i n f i n i t e , every i n t e g e r i s i n j e c t a b l e i n a . However i n o r d e r t h a t c3 be i n j e c t a b l e i n a , i t i s necessary t h a t a be D e d e k i n d - i n f i r i t e : see 2.6.

Given a s e t a , t h e o r d i n a l s i n j e c t a b l e i n a fo rm a s e t .

0 For each o r d i n a l u and each i n j e c t i o n f o f u i n t o a , t h i s f , and conse- q u e n t l y u , i s d e f i n e d by t h e s e t o f o rde red p a i r s ( f x , f y ) f o r which x < y < u . Such a s e t i s a r e l a t i o n , i n t h e sense o f ch.2 5 1 below; and a l l these r e l a t i o n s

form a s e t by t h e axioms o f ZF. 0

Hence, t h e s e t o f a l l o r d i n a l s i n j e c t a b l e i n a g i v e n s e t i s an a leph, which we

s h a l l c a l l t h e Har togs se t , o r t h e Har togs a leph o f a . Th is i s a l s o t h e s m a l l e s t

o r d i n a l which i s n o t i n j e c t a b l e i n a (HARTOGS 1915).

a

6.3. SUCCESSOR ALEPH, LIMIT ALEPH

I f a i s i t s e l f an a leph, t hen t h e Har togs a leph o f a i s t h e unique a leph imme- d i a t e l y g r e a t e r t han a , i n t h e sense t h a t t h e r e i s no s t r i c t l y i n t e r m e d i a t e o r d i -

n a l ( w i t h r e s p e c t t o subpotence) between a and i t s Har togs a leph . We s h a l l denote the l a t t e r by a+ and c a l l i t t h e successor a leph o f a . For example

countable o r d i n a l s ; o r aga in t h e l e a s t uncountable o r d i n a l . L e t t i n g , f o r each i n t e g e r i we l e t i+l = ( (3 i)+ . The un ion o r supremum o r d i n a l o f an a r b i t r a r y s e t o f a lephs i s aga in an a leph. For

example, f rom t h e p reced ing W i ( i i n t e g e r ) , we l e t W U = Sup( w i ) , which i s

an a leph . We c a l l a non-successor a leph, such as &J o r Urn , a l i m i t a leph.

w 1 = LJ+ denotes t h e successor a leph o f w , and i s t h e s e t o f a l l

W o =

6.4. ALEPH RANK We g e n e r a l i z e t h e p reced ing n o t a t i o n . Given an a r b i t r a r y o r d i n a l u , f o r an i n f i - n i t e a leph a we w r i t e a = W u i f t h e r e e x i s t s a f u n c t i o n f w i t h domain u + l ( t h e successor o r d i n a l o f u ) , such t h a t t h e i n i t i a l o rde red p a i r (0, W ) and

t h e f i n a l o rde red p a i r (u,a) be long t o f ; and such t h a t , i f t h e o rde red p a i r

(x,y) be longs t o f w i t h x an o r d i n a l < u and y an a leph, t hen (x+l,y+)

belongs t o form (x,y) w i th Sup x ,C u t hen t h e o rde red p a i r (Sup x, Sup y) belongs t o f .

f ; and f i n a l l y such t h a t i f f c o n t a i n s a s e t o f o rde red p a i r s o f t h e

Thus f o r each o r d i n a l eve ry i n f i n i t e a leph a t h e r e e x i s t s a un ique o r d i n a l u such t h a t a = Ou . We c a l l u t h e a leph rank o f a . We have u ,< W ; e q u a l i t y i s p o s s i b l e : see f o r i n s t a n c e ch.2 5 6.6.

Fo r eve ry o r d i n a l u we have Wu+l = (G) u)+ , t h e successor a leph o f (3 . Moreover, f o r eve ry s e t o f o r d i n a l s x , we have t h e supremum e q u a l i t y :

u , t h e r e e x i s t s a un ique a leph G,, . Conversely f o r

(SUP x ) . sup wx = w

Hence an i n f i n i t e a leph i s a successor o r a l i m i t a leph, acco rd ing t o whether i t s

a leph rank i s a successor o r d i n a l o r l i m i t o r d i n a l ( i n c l u d i n g 0, s i n c e a,, = id).

It f o l l o w s f rom t h e correspondence between a lephs and a leph ranks t h a t , g i v e n a

s e t o f a lephs ai and an a leph b , i f a i 4 b f o r a l l ai t hen Sup ai 6 b .

6.5. I n t h e presence o f t h e axiom o f choice, t h e cont inuum hypo thes i s i s equiva-

l e n t t o t h e e q u a l i t y &2 = O1 . However, w i t h t h e axioms o f ZF a lone i n t h e absence o f t h e ax iom o f choice, t h e e q u a l i t y L32 = c d 1 i s , a p r i o r i , a s t r o n g e r

a s s e r t i o n than t h e cont inuum hypo thes i s . Indeed, t h e r e may e x i s t a model o f ZF

w i t h o u t choice, where t h e r e i s no s t r i c t l y i n t e r m e d i a t e s e t ( w i t h r e s p e c t t o sub-

potence) between G) and " 2 , y e t where W 1

The s i t u a t i o n i s d i f f e r e n t w i t h t h e g e n e r a l i z e d cont inuum hypo thes i s , which i m p l i e s t h e axiom of cho ice (see ch.2 exe rc .1 ) . Thus t h e g e n e r a l i z e d cont inuum hypo thes i s

i m p l i e s t h e e q u a l i t y a 2 = a+ f o r each i n f i n i t e a leph a . However, i t seems p o s s i b l e t o c o n s t r u c t a model o f ZF s a t i s f y i n g

eve ry i n f i n i t e a leph

thus nega t ing t h e ax iom o f cho ice . Elementary p r o p e r t i e s concern ing a d d i t i o n , m u l t i p l i c a t i o n and e x p o n e n t i a t i o n of i n f i n i t e a lephs ch.2 5 3.8 t o 3.10.

i s incomparable w i t h & 2 .

a2 = a+ f o r

a , and y e t hav ing non-aleph c a r d i n a l s which a r e incomparable,

w i l l be o b t a i n e d wi th t h e h e l p o f r e l a t i o n s and isomorphisms, i n

§ 7 - FILTER, ULTRAFILTER A X I O M

7.1. FILTER ,--

Given a s e t a , r e c a l l t h a t a f i l t e r on a i s a s e t f o f non-empty subsets o f a , such t h a t ( 1 ) i f ~ € 5 and x s y s a , t h e n Y E T ; (2) i f x,y E , t h e n t h e i n t e r s e c t i o n x n y E F; thus e v e r y f i n i t e i n t e r - s e c t i o n o f elements o f F i s an element o f F . Forhxamp le , t h e s e t o f complements o f f i n i t e subsets o f k) i s a f i l t e r on c . 3

Every i n t e r s e c t i o n o f f i l t e r s on a i s a f i l t e r on a . L e t be a f i l t e r on a , and b 5 a such t h a t b n x i s non-empty f o r eve ry

Chapter 1 23

element x of LT . Then the s e t of intersections b n x constitutes a f i l t e r on b , cal'led the f i l t e r induced by on b . Let & be a s e t of subsets of a for which every f i n i t e intersection of i t s e le . ments i s non-empty. Then the s e t of supersets of these intersections constitutes a f i l t e r on a , called the f i l t e r generated by h! . Let 3 , 3 be two f i l t e r s on the same se t . 9 t o extend 3 , i f i t includes ; s t r i c t l y f iner i f i t s t r i c t l y includes F . Given a f i l t e r on a and b C_ a , e i ther b E or a-b E , or every intersection of any x 6 with b and with a-b i s non-empty. Hence there exists a f i l t e r f iner t h a n 7 which contains which contains a-b . Let a be a s e t . If a s e t of f i l t e r s on a i s to ta l ly ordered by the comparison relation "f iner than", or more generally i f th i s comparison relation i s a directed partial ordering ( i . e . given two f i l t e r s , there i s a third f i l t e r which i s finer than b o t h ) , then the union of the f i l t e r s i s a f i l t e r on a .

i s said t o be finer than 3 , or

b , or a f i l t e r f iner t h a n 3

7 . 2 . ULTRAFILTER, ULTRAFILTER AXIOM Given a s e t a , an u l t r a f i l t e r on a i s a f i l t e r for which there i s no s t r i c t l y finer f i l t e r on a . For example i f u E a , then the s e t of subsets of a containing u i s an u l t r a f i l t e r , said t o be t r i v i a l . Already for the s e t w , the axioms of ZF alone are n o t suff ic ient t o prove the existence of a non-trivial u l t r a f i l t e r . I t i s necessary t o add, for instance, the u l t r a f i l t e r axiom (also called boolean prime ideal axiom), which asserts that for every s e t a and every f i l t e r on a , there exis ts an u l t r a f i l t e r on a which i s f iner t h a n 9 . For example th i s implies the existence of an u l t r a f i l t e r on 0 which contains as elements a l l complements of f i n i t e subsets of w . We will see in ch.2 5 2.8 tha t the axiom of choice implies the u l t r a f i l t e r axiom. For a model of ZF having no u l t r a f i l t e r s other t h a n the t r iv ia l u l t r a f i l t e r s , see BLASS 1977. A necessary and suff ic ient condition that a f i l t e r 7 on a be an u l t r a f i l t e r , i s that for every subset x o f a , e i ther x E 7 or a-x e 3 . Let be an u l t r a f i l t e r and-x E F . Then for every par t i t ion of x 1%

f in i te number o f disjoint subsets, one and only one of these subsets belongs-

Every f i l t e r 7 on a i s the intersection of a l l u l t r a f i l t e r s on a which are f iner than 7 (uses the u l t r a f i l t e r axiom). To calculate the number of f i l t e r s on a s e t , see ch.2 exerc. 2 . To see the impossibility of countably generating an u l t r a f i l t e r on CS, see ch.2 5 8.1.

-

-

t.3.


§ 8 - TOPOLOGY ON SETS OF INTEGERS

S t a r t w i t h t h e s e t N o f t h e n a t u r a l i n t e g e r s . F o r each o rde red p a i r o f f i n i t e s e t s F, G o f N , l e t UF denote t h e s e t o f those subsets o f N which i n c l u d e

F and a re d i s j o i n t f rom G . For F and G empty, we o b t a i n t h e e n t i r e s e t Y ( N ) Note t h a t UF i s non-empty i f f F and G a r e d i s j o i n t .

The i n t e r s e c t i o n U F n U:: i s U~~~~

any un ion o f p reced ing U s e t s , t hen t h e i n t e r s e c t i o n o f any two open s e t s i s

s t i l l an open se t ; so t h a t we o b t a i n a topo logy on T (N) . The complement o f a U s e t i s a un ion o f U s e t s , t hus an open se t ; so t h a t each U s e t i s b o t h open and c losed, i . e . t h e complement o f an open se t ; more b r i e f l y each U i s a c lopen s e t . Th i s topo logy i s Hausdor f f : g i v e n two subsets A and B o f N , supposed t o be d i s t i n c t , t a k e a f i n i t e s e t F which i s i n c l u d e d i n A y e t n o t i n B , and a

f i n i t e s e t G i n c l u d e d i n B y e t n o t i n A , w i t h F and G n o t b o t h empty: t hen UF and UG separa te A f rom B .

G

G

. Consequently, d e f i n i n g an o p e n s e t t o be

G F

8.1. CONVERGENT SEQUENCE, CLOSURE

Consider an a - s e q u e n c e o f subsets Hi o f N ( i n a t u r a l i n t e g e r ) . We say t h a t

t h i s sequence converges, and t h a t a subset H o f N i s t h e limit o f t h e Hi , i f f o r eve ry i n t e g e r x , e i t h e r x belongs t o H and then x be longs t o Hi f r om some index on ; o r x be longs t o t h e complement o f H and then x be longs t o t h e

complement o f Hi f r om some index on. The c l o s u r e o f a s e t l i m i t elements f o r a l l convergent ra-sequences o f elements o f S . T h i s c l o s u r e i s a l s o t h e s m a l l e s t c losed superse t o f S , w i t h r e s p e c t t o i n c l u s i o n .

S o f s e t s o f i n t e g e r s , i s d e f i n e d as be ing t h e s e t of a l l

8.2. THE TOPOLOGY I S COMPACT

I n o t h e r words, if T ( N )

by a f i n i t e number o f t hese s e t s . 0,Suppose we have an a - s e q u e n c e o f o rde red p a i r s ( F ( i ) , G ( i ) ) (i i n t e g e r ) of f i n i t e subsets o f N , and l e t Ui = UG(!) . Assume t h a t f o r each i , t h e un ion

U o u U1 u ... u U . i s s t r i c t l y i n c l u d e d i n Q(N) . We w i l l show t h a t t h e un ion

o f a l l t hese Ui i s d i s t i n c t f rom T(N) . We see t h a t t h e r e e x i s t s a f u n c t i o n h which, t o each i , assoc ia tes an element

hi o f F ( i ) u G ( i ) , which can be assumed t o be non-empty. G i v i n g hi t h e s i g n (+) o r ( - ) acco rd ing t o whether i t belongs t o F ( i ) o r G ( i ) , we can choose h

t o v e r i f y t h e f o l l o w i n g c o n d i t i o n . F o r each i , t h e r e e x i s t s an element i n t h e

d i f f e r e n c e T ( N ) - (Uo u ... u Ui) , which c o n t a i n s as an e lement a l l t hose

hl,...,hi

i s covered by a un ion o f open s e t s , t h e n i t i s covered

F (1 )

1

ho,

hj o f s i g n ( - ) , and none o f s i g n (+). It f o l l o w s t h a t t h e i d e n t i t y hi =

Chapter 1 25

( w i t h i # j) i m p l i e s t h a t hi and h . have t h e same s ign . F i n a l l y t h e s e t o f a l l

hi o f s i g n ( - ) , does n o t be long t o t h e un ion o f t h e Ui f o r a l l i n t e g e r s i . 0

It f o l l o w s immediate ly f rom t h i s compactness, t h a t t h e c lopen s e t s a r e e x a c t l y t h e f i n i t e un ions o f t h e p reced ing U s e t s .

J

8.3. Consider an a - s e q u e n c e o f o rde red p a i r s ( F ( i ) , G ( i ) ) o f f i n i t e subsets o f N

( i i n t e g e r ) , and assume t h a t F ( i ) and G ( i ) a r e b o t h non-empty and d i s j o i n t , f o r each i . L e t h be a f u n c t i o n which, t o each i , assoc ia tes an element hi i n

F ( i ) u G ( i ) ; w i t h t h e c o n d i t i o n t h a t i f hi = h . (i # j) , then e i t h e r hi both t o F ( i ) and F ( j ) , o r b o t h t o G ( i ) and G ( j ) . Then we assoc ia te t h e s i g n (t) o r ( - ) t o hi , accord ing t o whether i t belongs t o F ( i ) o r t o G ( i ) . Le t Vh be t h e open s e t formed by t h e subsets X o f N f o r which t h e r e e x i s t s an i such t h a t hi has s i g n (+) and hi E X , o r hi has s i g n ( - ) and hi X . Then t h e open s e t , un ion o f t h e UG(i) , i s t h e i n t e r s e c t i o n o f t h e Vh f o r a l l t h e

f u n c t i o n s h p r e v i o u s l y d e f i n e d .

belongs J

F ( i )

8.4. DENSE SET, BAIRE'S CONDITION

A s e t i s c a l l e d dense, i f i t s i n t e r s e c t i o n w i t h eve ry non-empty open s e t i s non-

empty.

For eve ry f u n c t i o n h a s s o c i a t i n g t o each i n t e g e r i an e lement hi o f N , t a k i n g i n f i n i t e l y many va lues hi w i t h any s i g n , t h e open s e t Vh p r e v i o u s l y de f i ned i s dense. Thus t h e r e e x i s t i n f i n i t e l y many dense open se ts , a l t hough t h e on ly dense c losed s e t i s 9 (N) . No t i ce t h a t t h e i n t e r s e c t i o n o f two dense open s e t s i s a dense open s e t .

Every compact topo logy s a t i s f i e s BAIRE's c o n d i t i o n : eve ry coun tab le i n t e r s e c t i o n o f dense open s e t s i s non-empty, and even dense. We l e t t h e p r o o f t o t h e reader ,

f o r i n s t a n c e by t a k i n g a f i n i t e s e t o f dense ooen se ts ; and then i n f i n i t e l y many.

0 L e t us ske tch a d i r e c t p r o o f f o r t h e p resen t topo logy . Every open s e t i s a un ion o f s e t s UF , thus by 8.3, a coun tab le i n t e r s e c t i o n o f open s e t s Le t us prove BAIRE's c o n d i t i o n f o r t h e dense Vh , hence those co r respond ing t o the h t a k i n g i n f i n i t e l y many va lues. To do t h i s , r e p l a c e each Vh , by reduc ing h t o o n l y those va lues w i t h s i g n (t), o r o n l y those o f s i g n ( - ) , assumed t o be

i n f i n i t e i n number. Denote by W t h e i n f i n i t e s e t o f these va lues. The open s e t V h i s reduced t o t h e s e t o f subsets X o f N such t h a t W n X i s non-empty, o r t o

the s e t o f subsets X such t h a t W n (N-X) i s non-empty. We s h a l l see i n ch.2

5 8.1 ( coun tab le case), t h a t t h e r e e x i s t s a s u i t a b l e s e t X f o r a l l t h e W , w i t h W n X and W n (N-X) i n f i n i t e . Thus t h e r e e x i s t s a s e t X i n c l u d i n g any g i ven

f i n i t e s e t and e x c l u d i n g any g i v e n f i n i t e s e t ( d i s j o i n t each f rom t h e o the r ) ; hence t h e r e e x i s t s an

G Vh .

X b e l o n g i n g t o any g i v e n open s e t . 0


I n conclusion, an open se t def ined by an a-sequence (F( i ) ,G(

f o r every ordered p a i r o f f i n i t e d i s j o i n t subsets F, G o f N

i w i t h F ( i ) d i s j o i n t from F and G ( i ) d i s j o i n t from G .

) ) i s dense i f f , there e x i s t s an

§ 9 - NATURAL SUM AND PRODUCT FOR ORDINALS

From the unique decomposition o f an o rd ina l i n t o a sum o f decreasing powers o f 0 , one def ines the commutative operat ions o f na tura l sum and na tura l product f o r

o rd ina ls : t h i s goes back t o HESSENBERG 1906 ; see a lso BACHMANN 1967 p. 107.

For the sum, we begin w i t h O( = w0( (l).ml + ... t W w(h).mh , where the

c o e f f i c i e n t s ml, ..., mh are in tegers , and where cx (1 ) 7 ... 7 o( (h ) are

o rd ina l exponents, and p = Wo((l).nl + . .. + G a ( h ) . n h . We can always assume t h a t the o rd ina l exponents are the same f o r both decompositions, i f necessary by i n s e r t i n g terms w i t h c o e f f i c i e n t zero.

Then the na tura l sum o( @ fs i s def ined as:

c3 M(l).(ml+nl) + ... + ~ c ) ~ ( ~ ) . ( m ~ + n ~ ) . For the na tura l product, one f i r s t def ines the product o f u' and U 'as

being *@ . Then f o r two a r b i t r a r y o rd ina ls , each w r i t t e n i n the form o f

a sum o f powers o f G) , one m u l t i p l i e s them as w i t h polynomials. With respect t o these not ions, one def ines the " r e l a t i v e " ( i . e . p o s i t i v e o r nega- t i v e ) o rd ina ls , by s u b s t i t u t i n g i n the c o e f f i c i e n t s in tegers . Then the r a t i o n a l o rd ina l s by tak ing the quo t ien t f i e l d : see S I K O R S K I 1948, and a lso BACHMANN 1967. More d i f f i c u l t and less "na tura l " , one can de f ine t r a n s f i n i t e rea ls ; see f o r instance KLAUA 1959 and 1960.

m , p o s i t i v e and negat ive

EXERCISE 1 - CONSISTENCY OF THE A X I O M OF FOUNDATION

To see t h a t the add i t i on o f the axiom o f foundat ion does no t imply a cont rad ic t ion ,

i f the o ther axioms o f ZF are cons is ten t , we de f ine an ord ina l t o be a se t no t only t r a n s i t i v e and t o t a l l y ordered by the membership r e l a t i o n , bu t a lso sa t i s - f y i n g the fo l l ow ing foundat ion cond i t ion . For every non-empty subset

o rd ina l , there e x i s t s an element y o f , x which i s d i s j o i n t from x : see BERNAYS 1968 p. 80.

Next, one must redef ine an i n tege r as a f i n i t e o rd ina l . Then say t h a t a se t a i s well-founded i f every sequence xi indexed by in tegers , w i t h xo = a and xi+l

belonging t o xi f o r each i , i s f i n i t e . F i n a l l y , one v e r i f i e s t h a t the we l l - founded sets w i t h the membership r e l a t i o n , s a t i s f i e s a l l the axioms o f ZF, i nc lu - d ing the axiom o f foundation. N.B. It i s no t s u f f i c i e n t t o de f ine a well-founded se t (o the r than an o r d i n a l ) by

x o f an

Chapter 1 21

t he f i r s t f o u n d a t i o n c o n d i t i o n

EXERCISE 2 - ALEXANDROFF-FODOR THEOREM A subset o f

if i t i s equ ipo ten t w i t h element i n t h e subset . We say t h a t a f u n c t i o n f w i t h domain every x i n Dom f . 1 - I f f i s r e g r e s s i v e , t hen t h e r e e x i s t s a coun tab le o r d i n a l o( f o r which t h e

s e t o f x such t h a t f x = d i s c o f i n a l (ALEXANDROFF 1935, coun tab le axiom o f

choice used). S t a r t i n g w i t h a coun tab le o r d i n a l o( (0) , i f f o r eve ry /s < o< (0 ) there are coun tab ly many x where f x = f i , then t h e r e e x i s t s an m ( l ) 7 a( (0) such t h a t f rom t h a t p o i n t on, sequence O( ( i ) ( i i n t e g e r ) , and so 8 = Sup O( ( i ) i s coun tab le by 2 .5 ( t o u n t a - b l e axiom o f cho ice ) , and s a t i s f i e s veness o f f . 2 - We say t h a t a s e t o f o r d i n a l s i s c l o s e d i f , f o r e v e r y s t r i c t l y i n c r e a s i n g

w -sequence o f elements, t h e supremum o r d i n a l be longs t o t h e s e t . C l e a r l y , t h e

i n t e r s e c t i o n o f two c l o s e d c o f i n a l s e t s i s c l o s e d c o f i n a l . A l s o t h e i n t e r s e c t i o n o f a countable s e t o f c l o s e d c o f i n a l s e t s i s c l o s e d c o f i n a l : t h i s reduces t o an 0 -sequence which i s dec reas ing w i t h r e s p e c t t o i n c l u s i o n .

We say t h a t a s e t i s s t a t i o n a r y , i f i t i n t e r s e c t s e v e r y c losed c o f i n a l s e t . Note t h a t every s t a t i o n a r y s e t i s c o f i n a l .

I f f i s reg ress i ve , t hen t h e r e e x i s t s a coun tab le o r d i n a l o( f o r which t h e s e t o f x such t h a t f x = o( , i s s t a t i o n a r y (FODOR 1966, uses axiom o f cho ice ) .

Suppose t h e c o n t r a r y . S ince f o r e v e r y coun tab le o r d i n a l /?J , t h e s e t o f x

( f x = /?I ) i s n o t s t a t i o n a r y , t h e r e e x i s t s a c l o s e d c o f i n a l s e t C ,-, (axiom

of choice) on which f x # /5. S t a r t i n g w i t h an a r b i t r a r y coun tab le o r d i n a l w. (0) , take d(1) > 4 ( 0 ) where 4 (1) belongs t o n C ,-, ( /5 < o ( ( 0 ) ) . I t e r a - t i n g t h i s , we o b t a i n an w-sequence o( (i) (i i n t e g e r ) , and thus 21 = Sup o( ( i ) . Hence belongs t o A C ( /?J < 6 ) , and so f 8 i s d i s t i n c t f rom eve ry

/s < ‘E( , and so f y >, , which c o n t r a d i c t s t h e regress iveness o f f .

m1 , t h e s e t o f coun tab le o r d i n a l s , i s s a i d t o be c o f i n a l i n W , w1 , so t h a t eve ry element o f U1 admits a g r e a t e r

W 1 - { O } i s r e g r e s s i v e , i f f x < x f o r

f x >/ O( ( 0 ) . I t e r a t i n g t h i s , we o b t a i n an GO-

f 8 >/ , thus c o n t r a d i c t i n g t h e r e g r e s s i -

3 - Extend t h i s p r o p o s i t i o n t o t h e case where Dom f i s s t a t i o n a r y .

EXERCISE 3 - A CLASSICAL INTERPRETATION OF THE ORDINAL EXPONENTIATION

Let o( , f i be two o r d i n a l s . Consider f u n c t i o n s f w i th domain /3 , t a k i n g

values i n o( . We say t h a t such an f i s a lmost z e r o i f f ( i ) # 0 f o r a t most f i n i t e l y many elements i o f /3 . Given two a lmost z e r o f u n c t i o n s f and g , l e t f < g i f t h e r e e x i s t s an i i n (3 w i t h f ( i ) < g ( i ) i n t h e usual o r d e r i n g

f o r o r d i n a l s , and f ( j ) = g ( j ) f o r a l l j such t h a t i < j < b . L e t f< g if

f < g o r f = g . I n o t h e r words, t h e s e t o f such a lmost ze ro f u n c t i o n s by l a s t d i f f e r e n c e .

1 - Show t h a t 6 i s a w e l l - o r d e r i n g on t h e s e t o f a lmost ze ro f u n c t i o n s reader i s assumed t o know t h i s n o t i o n ; o t h e r w i s e see ch.2 5 2.4) .

s o rde red

t h e

2 - Show t h a t t h i s w e l l - o r d e r i n g i s i somorph ic w i t h t h e exponen t ia l o(" ( i nduc - t i o n on (3 ) .

EXERCISE 4 - A FAMILY OF SUBSETS I N A DENUMERABLE SET, OR I N THE CONTINUUM 1 - Given a denumerable s e t E , show t h a t t h e r e e x i s t continuum many denumerable

subsets A o f E , hav ing p a i r w i s e a f i n i t e i n t e r s e c t i o n . Indeed t a k e f o r E t h e s e t o f a l l f i n i t e sequences w i t h p o s s i b l e va lues 0 and 1 . Then d e f i n e each A

by an w-sequence o f 0 and 1 ; say t h a t a g i v e n f i n i t e sequence belongs t o A i f f i t i s an i n i t i a l i n t e r v a l o f t h e W-sequence assoc ia ted w i t h A ( t h e n o t i o n o f i n i t i a l i n t e r v a l i s obv ious; i f necessary see ch.4 § 2.1) .

2 - Modulo t h e axiom o f choice, complete t h e g i v e n f a m i l y o f s e t s

t o s a t i s f y t h e f o l l o w i n g c o n d i t i o n : f o r eve ry denumerable subset X o f E , t h e r e e x i s t s an A such t h a t t h e i n t e r s e c t i o n A n X be i n f i n i t e .

3 - Now cons ide r a s e t E w i t h continuum c a r d i n a l i t y . Show t h a t , modulo t h e

axiom o f choice, t h e r e e x i s t ( 2 t o t h e power al) many subsets A o f E , each hav ing c a r d i n a l i t y c.dl , and hav ing p a i r w i s e a coun tab le i n t e r s e c t i o n . Indeed t a k e f o r E t h e s e t o f a l l coun tab le o r d i n a l - i n d e x e d sequences o f 0 and 1 . Then d e f i n e each A by an W1-sequence o f 0 and 1 ; say t h a t a g i v e n coun tab le

sequence belongs t o A i f f i t i s an i n i t i a l i n t e r v a l o f t h e ul-sequence asso- c i a t e d w i t h A ( n o t e t h a t t h e axiom o f cho ice i s needed t o see t h a t t h e s e t E

o f a l l coun tab le sequences i s e q u i p o t e n t w i t h t h e continuum). 4 - Complete t h e g i v e n f a m i l y o f s e t s A i n o r d e r t h a t , f o r eve ry subset X o f E hav ing c a r d i n a l i t y O1 , t h e r e e x i s t s an A such t h a t t h e i n t e r s e c t i o n A X have c a r d i n a l i t y cS1 .

A i n o r d e r

29

CHAPTER 2

RELATI ON, PARTIAL ORDER1 NG, C H A I N , ISOMORPH I Sp1, COFI WALITY

§ 1 - RELATION, MULTIRELATION, RESTRICTION, EXTENSION, COHERENCE LEMMA, AXIOM OF DEPENDENT CHOICE

Le t E be a s e t and n an i n t e g e r . I n ch .1 5 2.3 we d e f i n e d t h e n o t i o n o f n - t u p l e w i t h va lues i n E . We s e t a s i d e two e lements c a l l e d va lues, which a r e denoted + and - ( f o r i n s t a n c e , t hese can be d e f i n e d by 0 and 1). An n-ary

r e l a t i o n . w i t h e E , o r based on E , i s a f u n c t i o n R which assoc ia tes t h e va lue R(xl ,..., xn) = + o r - t o each n - t u p l e x1 ,..., xn i n E ( f o r convenience, we o f t e n denote t h e n - t u p l e b y i t s i n d i c e s 1 t o n i n s t e a d o f f rom 0 t o n -1 ) . The s e t E , t h e base o f R , w i l l be denoted I R I . The i n t e g e r n

w i l l be c a l l e d t h e * o f R . For n = 1,2,3, we w i l l say a unary, b i n a r y , t e r n a r y r e l a t i o n .

For n = 0 , we adopt t h e conven t ion t h a t t h e r e e x i s t two 0 - 9 r e l a t i o n s based on E , which we denote by ( E , t ) and (E,- ) : t h e 0 -a ry r e l a t i o n s w i t h va lue t and va lue - . We adopt t h e conven t ion t h a t , f o r each p o s i t i v e n , t h e r e e x i s t s a unique n-ary r e l a t i o n w i t h empty base. However, t h e r e e x i s t two 0 -a ry r e l a t i o n s w i t h empty base: ( O , + ) and (0,-) . These convent ions agree w i t h t h e c a l c u l a t i o n o f t h e

n number o f n - t u p l e s w i t h va lues taken f rom a base o f f i n i t ' e c a r d i n a l p ; i . e . p . The number o f n-ary r e l a t i o n s based on p elements i s " 2 t o t h e power pn 'I. Here o r d i n a l e x p o n e n t i a t i o n c o i n c i d e s w i t h c a r d i n a l e x p o n e n t i a t i o n np , f o r n and p f i n i t e .

Examples o f r e l a t i o n s . The usua l o r d e r i n g o f t h e i n t e g e r s i s t h e r e l a t i o n R which s a t i s f i e s R(x1,x2) = + when x1 x2 and - when x1 > x2 . A group i s

a t e r n a r y r e l a t i o n t a k i n g t h e va lue t when x1.x2 = x3 and t h e va lue - when

x1.x2 # x3 , where . i s t h e compos i t i on l a w o f t h e group. I n s t e a d o f x1,x2,x3 we s h a l l o f t e n use x,y,z . A m u l t i r e l a t i o n w i t h base E i s a f i n i t e sequence R o f r e l a t i o n s R1, ..., Rh ( h i n t e g e r ) , each w i t h base i s c a l l e d a component o f t h e m u l t i r e l a t i o n R . We c a l l t h e arity o f R t h e sequence (nl, ..., nh) o f a r i t i e s o f t h e components R1, ..., Rh . We say then t h a t t h e m u l t i r e l a t i o n R i s

(nl, ..., n h ) - x . The l e n g t h h o f t h e sequence o f i n d i c e s can be zero: i n t h i s case, t h e m u l t i r e l a t i o n i s reduced t o i t s base E . I n s t e a d o f t h e n o t a t i o n R1, R2,R3 , o f t e n we s h a l l use R,S,T . I n t h e case where h = 2 , we w i l l say

- E . Each Ri (i = 1, ..., h )


a b i r e l a t i o n ; f o r h = 3 a t r i r e l a t i o n , e t c . F i n a l l y , the base o f a m u l t i r e l a t i o n R sha l l be denoted I R I . Example. An ordered group i s a (3,2)-ary b i r e l a t i o n which i s formed o f the te rnary group r e l a t i o n and the b inary o rder ing r e l a t i o n .

A r e l a t i o n o r m u l t i r e l a t i o n w i l l be c a l l e d f i n i t e , i n f i n i t e , countable o r continuum-equipotent, according t o whether i t s base i s f i n i t e ,

countable, denumerable or continuum-equipotent. The card ina l o f the mu R i s the card ina l o f i t s base J R I .

denumera- i n f i n i t e , t i re1 a t i o n

Given two m u l t i r e l a t i o n s R, S wi th common base E , we c a l l the concatenation o f R and S , denoted (R,S) , the sequence o f components o f R fo l lowed by the components o f S , i n which case f o r the l a t t e r the ind ices a re increased by the number of terms i n R .

1.1. n-ARY RESTRICTION, n-ARY EXTENSION Le t R be an n-ary r e l a t i o n w i t h base E , and l e t F be a subset o f E . We c a l l the n - a 3 r e s t r i c t i o n o f R t o F , denoted by R/F , the n-ary r e l a t i o n tak ing the same value f o r each n- tup le w i t h values i n F . The no t ion o f r e s t r i c - t i o n o f a func t i on i n ch.1 5 1.3, i s more general than t h a t o f n-ary r e s t r i c t i o n : t he former would a l l ow one t o r e s t r i c t R t o an a r b i t r a r y subset o f the s e t 'E

o f n- tuples w i t h values i n E , and no t necessar i l y t o a subset o f the form 'F

w i t h F S E . However i n p rac t i ce , the context w i l l make the meaning o f the ad- j e c t i v e "n-ary" obvious: we t a c i t l y assume t h i s . For the a r i t y 0, the r e s t r i c t i o n t o F o f the 0-ary r e l a t i o n (E,+) w i l l be

(F,+) ; s i m i l a r l y w i th - ; t h i s remains v a l i d f o r empty F . Given a r e l a t i o n R w i t h base E and a superset E+ o f E , we c a l l an 9- - sion o f R t o E+ any r e l a t i o n w i t h base E+ whose r e s t r i c t i o n t o E i s R . Le t R, R ' be two n-ary r e l a t i o n s w i t h common base E . I f f o r every subset X 3 E w i t h card ina l 6 n , we have R/X = R ' / X , then R = R ' . Given a m u l t i r e l a t i o n R = (R1,-..,Rh) w i t h base E and a subset F o f E , we def ine the r e s t r i c t i o n o f R t o F , denoted by R/F , t o be the m u l t i r e l a - t i o n (R1/F, ..., Rh/F) . Given a m u l t i r e l a t i o n R w i t h base E and a superset E+ o f E , we c a l l an extension o f R t o Ef any m u l t i r e l a t i o n w i t h base E+ whose r e s t r i c t i o n t o E i s R . Equ iva len t ly , any sequence (R;,.. . ,Rh) where each Ri

L e t R, R ' be two m u l t i r e l a t i o n s o f common a r i t y (nl, ..., nh) and w i t h common base E . I f f o r each subset X o f E w i t h card ina l 6 Max(nl, ..., nh) , have R/X = R ' / X , then R = R ' .

+ +

i s an extension o f Ri t o E+ ( i = 1, ... ,h) .

- - -

Chapter 2 31

1.2. COMPATIBLE RELATIONS

Two r e l a t i o n s ( o r m u l t i r e l a t i o n s ) w i t h t h e same a r i t y a r e s a i d t o be compat ib le i f f they have t h e same r e s t r i c t i o n t o t h e i n t e r s e c t i o n o f t h e i r bases. L e t 6% be a s e t o f m u t u a l l y compa t ib le r e l a t i o n s ( o r m u l t i r e l a t i o n s ) : ( 1 ) t h e r e e x i s t s a common e x t e n s i o n o f t h e r e l a t i o n s i n &, , based on t h e un ion o f t h e i r bases;

( 2 ) l e t us denote by E t h e un ion o f t h e bases and by n t h e common a r i t y , o r

t he maximum o f t h e common a r i t y ( f o r m u l t i r e l a t i o n s ) ; i f each n-element subset o f E i s covered by one o f t h e bases, t hen t h e common ex tens ion i s unique.

1.3. COHERENCE LEMMA Consider a s e t 9 o f s e t s UF o f m u l t i r e l a t i o n s based on F ( a l l o f t h e same a r i t y ) w i t h t h e f o l l o w i n g hypotheses:

(1) 3 i s a d i r e c t e d system: i f F, F ' belong t o )3 , then t h e r e e x i s t s an F" i n 3 w i t h F " ? F u F ' ; ( 2 ) i f F, F ' belong t o 3 and F ' C F , then eve ry m u l t i r e l a t i o n be long ing t o

U F , when r e s t r i c t e d t o F ' , y i e l d s an element o f U F , ;

i n t h i s case, t h e r e e x i s t s a m u l t i r e l a t i o n R based on t h e un ion o f t h e s e t s F - i n 3 , such t h a t f o r each F t h e r e s t r i c t i o n R/F belongs t o U F (uses t h e u l t r a f i l t e r axiom; ZF s u f f i c e s if t h e F a r e f i n i t e and t h e i r un ion coun tab le ) . 0 Denote by E t h e un ion o f t h e F i n 3 . To each F a s s o c i a t e t h e s e t V F o f ex tens ions t o E o f m u l t i r e l a t i o n s be long ing t o U F . The supersets o f t h e VF c o n s t i t u t e a f i l t e r on t h e s e t o f m u l t i r e l a t i o n s based on E w i t h t h e g i v e n a r i t y . Indeed i f F, F ' belong t o 3 , t h e n t h e r e e x i s t s i n 3 an F " ? F u F ' ; hence VFn V F 8 i s a superse t o f VF,, . Take an u l t r a f i l t e r ex tend ing t h i s f i l t e r .

For each F o f '3 , p a r t i t i o n t h e m u l t i r e l a t i o n s i n V F i n t o a f i n i t e number o f c lasses, each c l a s s de f i ned by t h e r e s t r i c t i o n t o F o f these m u l t i r e l a t i o n s .

One and o n l y one o f t hese c lasses i s an element o f o u r u l t r a f i l t e r : denote by R F t h e co r respond ing r e s t r i c t i o n , so t h a t RF belongs t o U F . Hence t h e RF a re m u t u a l l y compa t ib le i n t h e sense o f 1 .2 above: t h e e x i s t e n c e o f t h e m u l t i - r e l a t i o n R s t a t e d i n o u r p r o p o s i t i o n f o l l o w s . I f E i s coun tab le and t h e F are f i n i t e subsets o f E , then t h e u l t r a f i l t e r becomes super f l uous , so t h a t t h e axioms o f ZF a r e s u f f i c i e n t . 0

F f o r each o f which we have a f i n i t e non-empty s e t

1.4. The coherence lemma i m p l i e s , and hence i s e q u i v a l e n t t o t h e u l t r a f i l t e r axiom.

0 L e t e be a se t , p ( e ) be t h e s e t o f subsets o f e , and 'bQ a f i l t e r on e . L e t F be a f i n i t e s e t o f subsets o f e which i s c l o s e d w i t h r e s p e c t t o un ion,

i n t e r s e c t i o n and t a k i n g complements ( i n e ) . To each F a s s o c i a t e t h e s e t U F o f unary r e l a t i o n s X w i t h base F which s a t i s f y t h e f o l l o w i n g c o n d i t i o n s :


f o r each a & F , i f a E 8 t h e n t h e va lue X(a) = + ; i f e-a E 2 t h e n X(a) = -;

f o r each a E F , we have o p p o s i t e va lues X(e-a) # X(a) ;

i f a, b E F so a n b e F and X(a) = X(b) = + , t h e n X ( a n b ) = + ; i f a, b E F and a c b and X(a) = + , then X(b) = + . The s e t UF i s non-empty f o r each F . The s e t o f t h e F forms a d i r e c t e d system, so we can app ly t h e coherence lemma. Consequently t h e r e e x i s t s a unary r e l a t i o n based on T ( e ) whose r e s t r i c t i o n t o each F belongs t o UF . The subsets o f e which g i v e t h e va lue (+) t o t h i s unary r e l a t i o n c o n s t i t u t e an u l t r a f i l t e r on e

which i s f i n e r t han w . 0

1.5. A v a r i a n t o f t h e coherence lemma i s g i v e n by RADO 1949. Consider a s e t o f

f i n i t e m u t u a l l y d i s j o i n t s e t s a , and f o r each f i n i t e s e t I o f s e t s a , cons ide r a cho ice f u n c t i o n fI which a s s o c i a t e s t o each a o f I an element fI(a) o f a . Then t h e r e e x i s t s a cho ice f u n c t i o n f whose domain i s t h e s e t o f t h e a , and f o r each f i n i t e s e t I o f t h e a , t h e r e e x i s t s a f i n i t e superse t J of I with f / I equal t o t h e r e s t r i c t i o n f,/I . The preceding RADO's lemma p l u s t h e axiom o f cho ice f o r f i n i t e s e t s i s e q u i v a l e n t t o t h e coherence lemma (BENEJAM 1970).

1.6. A X I O M OF DEPENDENT CHOICE L e t R be a b i n a r y r e l a t i o n w i t h base E , such t h a t f o r each x o f E t h e r e e x i s t s a t l e a s t one y o f E s a t i s f y i n g R(x,y) = + . The axiom o f dependent

cho ice a s s e r t s t h a t , g i v e n such an R , t h e r e e x i s t s an w-sequence o f elements ai of E s a t i s f y i n g R(ai,ai+l) = + f o r each i n t e g e r i (MOSTOWSKI 1948). The axiom o f dependent cho ice o b v i o u s l y f o l l o w s f rom t h e axiom o f cho ice . It i s proved t h a t t h e dependent c h o i c e i s s t r i c t l y weaker than t h e axiom o f cho ice : see f o r i n s t a n c e JECH 1973 p. 122 and f o l l o w i n g .

The coun tab le axiom o f choice, s t a t e d i n ch.1 5 2.5, f o l l o w s f rom t h e axiom o f dependent choice.

0 S t a r t f rom an w -sequence o f non-empty m u t u a l l y d i s j o i n t s e t s ai t a k e R t o be t h e b i n a r y r e l a t i o n based on t h e un ion o f t h e ai , d e f i n e d by R(x,y) = + i f f t h e r e e x i s t s an i w i t h x E ai and y e ai+l . 0 The coun tab le axiom o f cho ice i s s t r i c t l y weaker than t h e axiom o f dependent cho ice : see JECH 1973 p. 119 and f o l l o w i n g . F i n a l l y , from t h e axiom o f dependent choice, assumed t o be c o n s i s t e n t , one cannot

deduce t h e axiom o f cho ice f o r f i n i t e se ts , s t a t e d i n ch.1 5 2.10. The p r o o f i s due t o MOSTOWSKI 1948 w i t h o u t t h e axiom o f f ounda t ion , and t o FEFERMAN 1965 w i t h founda t ion .

(i i n t e g e r ) ,

Chapter 2 33

1.7. NEGATION, CONJUNCTION, DISJUNCTION

Given a r e l a t i o n R , i t s n e g a t i o n 1 R i s t h e r e l a t i o n w i t h same base and a r i t y , always t a k i n g t h e oppos i te va lue . Given R, S w i t h t h e same base and a r i t y , t h e c o n j u n c t i o n R A S takes t h e va lue (+) i f f R and S take t h e va lue (+ ) . The d i s j u n c t i o n R v S takes t h e va lue (+) i f f e i t h e r R o r S takes t h e va lue (+).

CONVERSE OF A B I N A R Y RELATION, RETRO-ORDINAL Given a b i n a r y r e l a t i o n R , t h e converse o f R , denoted by R- , i s t h e r e l a t i o n

w i t h t h e same base, such t h a t R-(x,y) = R(y,x) f o r e v e r y x, y . I n p a r t i c u l a r we cons ide r an o r d i n a l o( as t h e b i n a r y r e l a t i o n based on t h e s e t a l ready deno- t e d by o( ( t h e s e t o f o r d i n a l s < o( ) , and t a k i n g t h e va lue (+) i f f x E y o r

x = y a r e t r o - o r d i n a l .

(denoted a l ready b y x ay ) . The converse r e l a t i o n o( w i l l be c a l l e d

2 - QUASI-ORDERING, PARTIAL ORDERING, CHAIN, WELL-FOUNDED PARTIAL

ORDERING, WELL-ORDERING, WELL-ORDERABLE SET, HAUSDORFF-ZORN AXIOM

A q u a s i - o r d e r i n g ( o r p r e - o r d e r i n g ) i s a b i n a r y r e f l e x i v e and t r a n s i t i v e r e l a t i o n

( these n o t i o n s assumed t o be known). I f A i s a q u a s i - o r d e r i n g and x, y two

elements i n t h e base, t h e n x 4 y (mod A) o r x precedes or i s l e s s t h a n o r equal t o y , means t h a t A(x,y) = + . We a l s o w r i t e y >, x (mod A) o r y f o l l o w s o r i s g r e a t e r t han o r equal t o x . We w r i t e x < y (mod A) o r x s t r i c t l y l e s s t h a n y , i f x 6 y and y$ x ; i n o t h e r words A(x,y) = + and A(y,x) = - ;

o r aga in y s t r i c t l y g r e a t e r t han x ; a l s o " s m a l l e r " i s synonymous w i t h " l e s s than" . We say t h a t x i s comparable w i t h y i f x 6 y o r y 6 x ; o the rw ise x i s incomparable w i t h y , denoted b y x l y (mod A) . An equ iva lence r e l a t i o n i s a symmetric quas i -o rde r ing . Given a e lement x o f t h e base, t h e equ iva lence c l a s s of x (mod A) i s t h e s e t o f t hose y such

t h a t A(x,y) = A(y,x) = + . A p a r t i a l o r d e r i n g i s an an t i symmet r i c q u a s i - o r d e r i n g ( n o t i o n assumed t o be known). We a l r e a d y have t h e example o f i n c l u s i o n .

Given a q u a s i - o r d e r i n g A , t h e equ iva lence r e l a t i o n generated by A , i s t h e r e l a t i o n w i th t h e same base, t a k i n g t h e va lue (+) i f f x < y and y 4 x (mod A ) . F o r each e lement x , t h e equ iva lence c l a i s o f x (mod A) i s t h e c l a s s o f x

modulo t h e equ iva lence r e l a t i o n generated by A . Take as a new base t h e s e t o f equ iva lence c lasses , and w r i t e ( e q u i v . c l a s s o f x)

4 (equ iv . c l a s s o f y) , i f x \ < y (mod A) . We thus o b t a i n a p a r t i a l o r d e r i n g c a l l e d t h e p a r t i a l o r d e r i n g generated by t h e q u a s i - o r d e r i n g A . L e t A be a p a r t i a l o r d e r i n g , D a subset o f t h e base I A I . We assume t h a t t h e

n o t i o n of maximum o f D (mod A) , denoted Max D , i s known. S i m i l a r l y f o r t h e

minimum, denoted Min D . R e c a l l t h a t an e lement i s maximal i n D (mod A ) , i f i t


belongs t o D and there i s no element of D which s t r i c t l y follows i t . Analogous notion of a minimal element. The maximum, i f i t ex is t s , i s maximal, b u t the converse i s fa lse . Similarly for the minimum. These notions extend in an obvious manner t o a quasi-ordering. Here there can ex is t several maximums and several minimums, which are equivalent t o each other in the sense of the qenerated equivalence relation. The reader i s assumed t o know the notion of uoper bound of a s e t well as t h a t of lower bound. The supremum of D , denoted by i s the minimum in the se t of upper bounds. Hence x greater than or equal to every element in D . If Sup D belongs t o D , then i t i s the maximum. Analogous definition of the infimum, denoted by Inf D . These notions appeared already in ch.1 5 2.1 for ordinals, in ch.1 5 4.5 and 4.6 for reals.

D (mod A ) Sup D , i f i t ex is t s ,

as

x > r Sup D i s equivalent t o

-

INTERVAL, INITIAL AND FINAL INTERVAL The reader i s assumed t o be familiar with the notion of an element z between x and y (mod A ) , or z intermediate between x and y , as well as t h a t of an element s t r i c t l y intermediate. An interval of A i s a subset of the base which i s closed with respect t o the notion of intermediate (mod A ) . An i n i t i a l interval o r i n i t i a l segment of A i s a subset closed with respect t o " less than" . A final interval i s a subset closed with respect t o "greater t h a n " .

2 .1 . Let A be a par t ia l ordering. Then every subset of the base I A 1 without a minimal element i s in f in i te . Similarly for a subset without a maximal element. 0 To each element x of the subset 0 under consideration, associate the s e t D x of elements of D which are less t h a n or equal t o x (mod A ) . None of the

Dx i s minimal under inclusion (see ch.1 5 1.1, definition of a f i n i t e s e t ) . 0

2 .2 . AMALGAMATION LEMMA Let A , B be two par t ia l orderings having the same restr ic t ion t o the intersection of the bases. Then there ex is t s a partial ordering which i s an extension of b o t h A and B , based on the union of the bases. OWrite x G y when x , y G l A l and x s y ( m o d A ) , o r w h e n w e h a v e t h e s a m e condition for B , or when x belongs t o I A l , y belongs t o I B I and there exis ts an element t in the intersection with x st (mod A) and t $ y (mod B ) , or when we have the same condition when interchanging A and B . Finally write x\y in the other cases. 0

- .-

2.3. C H A I N , ORDERABLE SET

A chain, or total orderinq, i s a par t ia l ordering whose elements are mutually

Chapter 2 35

comparable. For example, we shall denote by Z the chain of the positive and negative integers, and by Q the chain of the rationals. The previous amalgamation lemma 2 . 2 extends t o the case of two chains, the common extension i t s e l f being a chain. However, th i s lemma does not extend t o t rees , defined in ch.4 5 6. 0 Take a t ree on a,b,c,d with a,b,c mutually incomparable, d < a , d c b and d I c ; and another t ree on a,b,c,e with e < b , e < c and e l a . Then ei ther d < e < c or e Q d < a : contradiction. 0

We say that a s e t E i s orderable i f f there exis ts a chain based on E . Using only the axioms o f ZF, every f i n i t e se t i s orderable (induction: see ch.1 5 1.1). ORDERING AXIOM The ordering axiom asser ts t h a t every s e t i s orderable. I t follows from the ultra- f i l t e r axiom, or equivalently from the coherence lemma 1.3. 0 Let E be a s e t ; t o each f i n i t e subset F of E , associate the s e t U F of chains based on F . By 1.3 there exis ts a relation R based on E every of whose f i n i t e res t r ic t ion i s a chain; thus R i s a chain. 0

The ordering axiom i s s t r i c t l y weaker t h a n the u f t r a f i l t e r axiom (JECH 1973 p.100). The ordering axiom implies the axiom of choice for f i n i t e sets (see ch.1 5 2.10). 0 Given a s e t of mutually dis joint f i n i t e se t s , i t suffices to take a chain A based on the union: t o each f i n i t e s e t we associate i t s minimum (mod A ) . 0

The axiom of choice for f i n i t e sets i s s t r i c t l y weaker than the ordering axiom: see LAUCHLI 1964 for ZF without foundation, completed for ZF by PINCUS 1972. The axiom of choice for f i n i t e se t s does not follow from the axiom of deoendent choice: see 5 1 . 6 . Hence the ordering axiom does n o t follow from dependent choice.

2 .4 . WELL-FOUNDED PARTIAL O R D E R I N G OR QUASI-ORDERING; WELL-ORDERING We say that a partial ordering o r quasi-ordering i s well-founded i f f every non-empty subset of i t s base has a t l e a s t one minimal element. A well-founded chain, o r total ordering, i s called a well-ordering. Every f i n i t e par t ia l ordering i s well-founded. Every restr ic t ion of a well-founded par t ia l ordering i s well-founded.

Given a par t ia l ordering A , the reader i s assumed to know the notion of a sequence with values in A which i s incr?asing, decreasing, s t r ic t12 or otherwise.

Every well-founded par t ia l ordering A s a t i s f i e s the following conditions: (1) there i s no s t r i c t l y decreasing (mod A) W-sequence; ( 2 ) every to ta l ly ordered restr ic t ion of A i s well-founded, hence a well-ordering; equivalently every non-empty to ta l ly ordered restr ic t ion of A has a minimum.

Conversely, each of the conditions ( l ) , ( 2 ) implies, hence i s equivalent t o saying t h a t A i s well-founded. This uses the axiom of dependent choice, yet ZF suffices i f A i s countable, or i f the base I A l i s well-orderable, in the sense below. In the general case, apply dependent choice t o the relation y < x (mod A ) .

2 .5 . WELL-ORDERABLE SET We say t h a t E i s well-orderable i f f there exis ts a well-ordering based on E . For example any f i n i t e or denumerable s e t i s well-orderable. - A s e t E i s well-orderable i f f there exis ts a choice function on the s e t < non-empty subsets of E . 0 Let f be a choice function on non-empty subsets. Let a. = f(E) . Let u be a non-zero ordinal, D u the s e t of a l l a i ( i i u ) . Let au = f(E-DU) , as long as possible, thus reaching a s e t

I t follows that the axiom of choice i s equivalent t o saying that every s e t i s well-orderable. Or again that every cardinal i s an aleph, or that every inf in i te cardinal has the form c . ) ~ ( o( ordinal index: see ch.1 5 6.1 t o 6 .4) . The axiom of choice i s equivalent to the trichotomy axiom which says t h a t , given any two cardinals a , b , e i ther a b . 0 If every cardinal i s an aleph, then trichotomy holds. Conversely, given a s e t a and the Hartogs u of a (see ch.1 5 6.2), i f trichotomy holds then necessa- r i l y a i s subpotent t o o( , hence a i s well-orderable. 0

DU = E . 0 WELL-ORDERING AXIOM, TRICHOTOMY AXIOM

2.6. MAXIMAL CHAIN Let A be a partial ordering and C be a to ta l ly ordered restr ic t ion of A . The chain C i s said t o be maximal (under inclusion, mod A) i f f every to ta l ly ordered restr ic t ion of A extending C i s identical t o C . Let E be a s e t ; denote by X any well-ordering based on a subset of E . Write X6 X ' i f f X i s an i n i t i a l interval of X ' . The well-founded par t ia l ordering thus defined on the s e t of X will be called the interval-orderinqon E . (1) A s e t E i s well-orderable i f f there exis ts a maximal chain which i s a res t r ic t ion of the interval-ordering on E . (2) Let A be a partial ordering and C be a to ta l ly ordered res t r ic t ion of A . Let U be any chain which i s both a res t r ic t ion of A and an extension of C . Every function f which t o each U associates f(U) , a to ta l ly ordered res- t r ic t ion of A and extension of U , has a fixed point V such that f(V) = V . 0 Index by ordinals a sequence of chains U i star t ing with Uo = C ; se t U i + l = f (Ui) and, for i a l imit ordinal, l e t U i be the common extension of U ( j < i ) t o the union of the i r bases. 0

j

Chapter 2 37

2.7. MAXIMAL CHAIN AXIOM, OR HAUSDORFF-ZORN A X I O M Th is axiom, go ing back t o HAUSDORFF 1914, then taken up by KURATOWSKI, MOORE and

then ZORN, i s s t a t e d as f o l l o w s . Given a p a r t i a l o r d e r i n g A and a c h a i n C which i s a r e s t r i c t i o n o f A , t h e r e e x i s t s a c h a i n which i s an e x t e n s i o n o f C

and maximal (mod A) . By 2.6. (2) above, t h e axiom o f cho ice i m p l i e s t h e maximal c h a i n axiom. By 2.6.(1),

t h e maximal c h a i n axiom i m p l i e s t h a t every s e t i s we l l -o rde rab le . By 2.5, the maximal c h a i n axiom i s t h e n e q u i v a l e n t t o t h e axiom o f cho ice .

2.8. The u l t r a f i l t e r axiom f o l l o w s f rom t h e axiom o f choice.

0 Consider t h e s e t o f f i l t e r s on a g i v e n se t , w i t h t h e comparison o r d e r i n g " f i n e r f i l t e r t han" . Take a maximal c h a i n ex tend ing t h e c h a i n reduced t o a g i v e n f i l t e r . The u l t r a f i l t e r g i v e n by t h e un ion o f t h e f i l t e r s be long ing t o t h e maximal c h a i n i s f i n e r t han F. 0

The u l t r a f i l t e r axiom i s s t r i c t l y weaker than t h e axiom o f choice: HALPERN, LEVY

1971 p. 83-134.

2.9. FREE SUBSET, ANTICHAIN, MAXIMAL FREE SUBSET, MAXIMAL ANTICHAIN

Given a p a r t i a l o r d e r i n g i t s elements a r e m u t u a l l y incomparable (mod A) . The r e s t r i c t i o n A/D t o such a f r e e subset D i s c a l l e d an a n t i c h a i n (mod A ) . It reduces t o t h e i d e n t i t y r e l a t i o n based on D . A f r e e subset, and t h e co r respond ing a n t i c h a i n , a r e c a l l e d maximal (under i n c l u - s ion ) i f f t h e r e i s no p r o p e r superse t which i s f r e e . Given a p a r t i a l o r d e r i n g A and a f r e e subset D , t h e r e e x i s t s a maximal f r e e subset i n c l u d i n g 0 (uses axiom o f choice; ZF s u f f i c e s i f A i s coun tab le ) : a p p l y t h e maximal c h a i n axiom

t o t h e i n c l u s i o n among f r e e subsets .

A , a subset o f i t s base i s c a l l e d free (mod A) i f f

§ 3 - ISOMORPHISM^ AUTOMORPHISM, HEIGHT OF A WELL-FOUNDED PARTIAL ORDERING, SUM AND PRODUCT OF CHAINS, HOMOMORPHIC IMAGE

Le t n be a non-negat ive i n t e g e r , R an n-ary r e l a t i o n w i t h base E and R ' an n -a ry r e l a t i o n w i t h base E ' ; l e t f be a b i j e c t i o n f rom E on to E ' . We say t h a t f t rans fo rms o r takes R i n t o R ' , denoted R ' 5 f " ( R ) , o r t h a t

f i s an isomorphism o f R o n t o R ' iff R ' ( f x l ,..., fx,) = R(xl ,..., xn)

(= + o r - ) f o r a l l elements x1 ,..., xn i n E . F o r n = 0 , e i t h e r R = (E,+)

and we s e t f " ( R ) = (E l ,+ ) o r R = (E,-) and then f " ( R ) = ( E l , - ) . A r e l a t i o n R ' i s s a i d t o be isomorphic w i t h R i f f t h e r e e x i s t s an isomorphism

from R on to R ' . T h i s c o n d i t i o n i s r e f l e x i v e , symmetric and t r a n s i t i v e , y i e l d i n g an equ iva lence r e l a t i o n on e v e r y s e t o f r e l a t i o n s o f a g i v e n a r i t y .

ISOMORPHISM TYPE, ORDER TYPE Modeled after the definition of "cardinal" in ch.1 0 5.4, we consider the relations isomorphic with R , and among such, those whose base has minimum fundamental rank. These form a set, called the isomorphism type of R (the order type if R is a chain, or total ordering). Thus two relations are isomorphic iff they have the same type. AUTOMORPHISM, EMPTY FUNCTION Given a relation R with base E , a permutation f of E i s called an automorphism of R iff f i s an isomorphism from R onto R . The automorphisms of R form a group of permutations of E . We adopt the convention that the empty function, which is a bijection of the empty set onto itself, is also an automorphism of each relation with empty base. In particular it is an automorphism of the 0-ary relation with empty base and

. However (O,+) and (0,-) are value (+ ) , denoted (O,+) , and also of (0,-) not isomorphic. These definitions and conventions extend to mu R = (R1 ,..., Rh) with base E and R' = (Ri,. f from E onto E' transforms R into R' onto R' , denoted R' = f"(R) , iff for each

tirelations. Given a multirelation .,Rb) with base E ' , a bijection

i = 1, ..., h , the function f is or is an isomorphism from R

an isomorphism of the component Ri onto the component R; . In other words fo(R1 ,..., Rh) = (fo(R1) ... .,fo(Rh)) .

3.1. (1) Let A be a well-ordering and f be an isomorphism from A onto a restriction of A . Then fx & x (mod A ) for each element x of A . (2 ) Given a well-ordering, its unique automorphism is the identity. Given two well-orderings A, B , there exists at most one isomorphism from A onto B . ( 3 ) Given a well-ordering A , no proper initial interval of A is isomorphic - with A . In particular two isomorphic ordinals are identical.

3.2. HEIGHT IN A WELL-FOUNDED PARTIAL ORDERING Let A be a well-founded partial ordering. To each element x of IAI , associate as follows an ordinal called the height of x (mod A) and denoted Ht x . If x is a minimal element, let Ht x = 0 . Let o( be a non-zero ordinal; assume that each ordinal (o( has been associated to at least one element, but that there still remain elements in the base to which no height (q has been associated. Then associate the height o( to minimal elements among these. Given a well-founded partial ordering A , there i s a unique height associated to each element of the base I A l . Moreover, for each element x of height cr( and every ordinal with height /3.

/3 < d , there exists at least one element < x (mod A)

Chapter 2 39

However, g i ven x of h e i g h t o( and an o r d i n a l f5 2 4 , even i f t h e r e e x i s t elements > x (mod A) o f h e i g h t s > /s , i t i s p o s s i b l e t h a t no element x

(mod A) e x i s t s w i t h h e i g h t f s . 0 L e t a < b i c ( d and a < e <.d w i t h e l b and e l c . Then e has h e i g h t 1 and i t s o n l y s t r i c t upper bound i s d , w i t h h e i g h t 3 . 0

- If x ( y (mod A) , then H t x < H t y . Two d i s t i n c t elements of t h e same h e i g h t a r e incomparable. But two incomparable

elements may have d i f f e r e n t h e i g h t s .

3.3. HEIGHT OF A WELL-FOUNDED PARTIAL ORDERING The h e i g h t s of t h e elements o f a we l l - f ounded p a r t i a l o r d e r i n g A c o n s t i t u t e

an o r d i n a l , c a l l e d t h e h e i q h t o f A and denoted H t A . E q u i v a l e n t l y we have

H t A = Sup((Ht x ) + l ) f o r a l l X E I A I . The h e i g h t o f an element x (mod A ) t o elements < x . I n p a r t i c u l a r , t h e fundamental rank o f a s e t a ( c h . 1 5 5.2) i s t h e h e i g h t o f

t he we l l - f ounded p a r t i a l o r d e r i n g based on t h e t r a n s i t i v e c l o s u r e o f a , and de f ined by x < y i f f t h e r e e x i s t s a f i n i t e sequence to = x, ... th = y w i t h

h i n t e g e r and ti E ti+l f o r each i < h .

i s a l s o t h e h e i q h t o f t h e r e s t r i c t i o n o f A

3.4. Every w e l l - o r d e r i n g A i s isomorphic t o t h e o r d i n a l H t A phism t rans fo rms each element x i n t o H t x . Consequently, g i v e n two w e l l - o r d e r i n g s , one i s i somorph ic t o an

t h e isomorphism

n i t i a l i n t e r v a l o f t h e o the r , by ch .1 5 2 .1 ( t r i c h o t o m y ) . Given a w e l l - o r d e r i n g A , each r e s t r i c t i o n o f A i s isomorphic w i th an i n i t i a l i n t e r v a l o f A . I f two w e l l -

o rde r ings a r e each isomorphic t o a r e s t r i c t i o n o f t h e o t h e r , t hen t h e y a r e isomorphic .

3.5. (1) L e t A be a wel l - founded p a r t i a l o r d e r i n g . Then each r e s t r i c t i o n B o f A i s a we l l - f ounded p a r t i a l o r d e r i n g w i t h H t B 6 H t A . Indeed f o r each

element x o f 1 6 1 we have H t x (mod B ) 4 H t x (mod A) . ( 2 ) I f B i s an i n i t i a l i n t e r v a l o f A c o n t a i n i n g x , then H t x (mod B) = H t x (mod A) . Consequence o f 3.2 ( i n d u c t i o n on

Given a we l l - f ounded p a r t i a l o r d e r i n g A , i t can happen t h a t a maximal t o t a l l y ordered r e s t r i c t i o n o f A does n o t reach H t A . Even t h e supremum o f t h e

he igh ts o f t h e maximal cha ins can be s t r i c t l y l e s s than H t A . 0 For each i n t e g e r i , t a k e a c h a i n isomorphic t o i ; these cha ins a r e assumed t o be m u t u a l l y incomparable. We o b t a i n a wel l - founded p a r t i a l o r d e r i n g w i t h

he igh t a, b u t i n which eve ry maximal c h a i n i s f i n i t e .

-

H t x ) .

Another example. Take denumerably many cop ies isomorphic t o t h e p rev ious o r d e r i n g

Above t h e jth copy ( j i n t e g e r ) , p u t a c h a i n isomorphic t o j . L e t t hese d i f f e r e n t o r d e r i n g s w i t h d i s j o i n t bases be m u t u a l l y incomparable. We o b t a i n a wel l - founded p a r t i a l o r d e r i n g w i t h h e i g h t w.2 , i n which eve ry maximal c h a i n i s f i n i t e . 0

3.6. ORDINAL SUM, HOMOMORPHIC IMAGE OF A CHAIN L e t A, B be two cha ins w i t h d i s j o i n t bases. We c a l l t h e o r d i n a l sum, o r s imp ly t h e sum A+B

ex tens ion o f A and B f o r which each element i n I A I precedes each element

i n I B I . T h i s g e n e r a l i z e s t h e n o t i o n o f sum among o r d i n a l s (ch.1 5 3.1). T h i s sum agrees w i t h isomorphism, i n t h e sense t h a t i f A ' i s isomorphic t o A and 6 ' t o B , then A ' + B ' i s isomorphic t o A+B . It i s a s s o c i a t i v e b u t n o t commutative, even up t o isomorphism: w + 1 and l + w a r e n o t i somorph ic . More g e n e r a l l y , l e t A be a cha in . Consider a p a r t i t i o n o f i t s base i n t o mutual -

l y d i s j o i n t i n t e r v a l s Ai , aga in denoted i . Take t h e new base t o be t h e s e t o f t hese i n t e r v a l s , and d e f i n e t h e c h a i n I by p u t t i n g i < j i f f each element

o f \A i l i s s t r i c t l y l e s s than each element o f I A . 1 . T h i s c h a i n I i s c a l l e d J a homomorphic image o f A . The s e t o f t h e i n t e r v a l s i s c a l l e d a decomposi- t&of A . The c h a i n A i s c a l l e d t h e I-= o r t h e sum a l o n g I o f t h e Ai : n o t a t i o n A =r Ai (i E I ) . Each homomorphic image o f a c h a i n A i s isomorphic t o a r e s t r i c t i o n o f A

(uses axiom o f c h o i c e ) : t ake one e lement i n each i n t e r v a l o f t h e decomposit ion. The converse i s f a l s e . F o r example, t h e c h a i n w o f t h e i n t e g e r s i s a r e s t r i c - t i o n o f W + 1 , b u t each decomposi t ion o f ~ + 1 i n t o i n t e r v a l s has a l a s t i n t e r v a l . Hence w i s n o t a homomorphic image o f w+l . Another counterexample. The c h a i n Q o f t h e r a t i o n a l s i s a r e s t r i c t i o n o f t h e c h a i n o f t h e r e a l s , b u t n o t a homomorphic image. Indeed DEDEKIND's theorem (ch .1 5 4.6) would be v i o l a t e d by a decomposi t ion a long Q o f t h e cha in R of t h e r e a l s , and a p a r t i t i o n o f Q , thus a p a r t i t i o n o f R i n t o an i n i t i a l i n t e r v a l w i t h o u t a maximum and t h e complementary i n t e r v a l w i t h o u t a minimum.

Note t h a t a homomorphic image o f a homomorphic image o f A

t h e c h a i n based on t h e un ion o f t h e bases, which i s t h e common

Ai

i s an image o f A .

3.7. ORDINAL PRODUCT OF CHAINS

Given two cha ins A, B t h e o r d i n a l p roduc t A.B i s t h e c h a i n based on t h e

Car tes ian p roduc t o f t h e bases, d e f i n e d by p u t t i n g y < y ' (mod B ) o r y = y ' and x 6 x ' (mod A) . I n o t h e r words, by a s s o c i a t i n g t o each y o f 1 B 1 t h e c h a i n A o b t a i n e d by r e p l a c i n g each x o f A by t h e o rde red p a i r (y,x) , t h e n t a k i n g t h e 8-sum o f t h e A T h i s p roduc t genera l i zes t h e n o t i o n o f p roduc t among o r d i n a l s (ch.1 5 3.2) .

(y,x),( ( y ' , x ' ) i f f

Y

Y '

Chapter 2 41

Th is p roduc t agrees w i t h isomorphism: i f A ' i s isomorphic t o A and B ' t o B, then A ' .B ' i s isomorphic t o A.6 . It i s a s s o c i a t i v e b u t n o t commutative, even up t o isomorphism: ~3 .2 and 2. W a r e n o t isomorphic .

3.8. CARDINAL SUM, PRODUCT AND EXPONENTIATION REVISITED

L e t us r e t u r n t o t h e o p e r a t i o n s between a lephs and between c a r d i n a l s (ch.1 § 5.5) w i t h t h e means now a f f o r d e d by t h e n o t i o n s o f isomorphism and w e l l - o r d e r i n g . (1) a be an i n f i n i t e aleph; t h e n a + a = a x a = a . ( 2 ) k t a, b be two a lephs, a t l e a s t one o f which i s i n f i n i t e ; t hen a + b = a x b = Max(a,b) ( f o r t h e p r o d u c t we assume t h a t a, b # 0 ) . 0 It s u f f i c e s t o e s t a b l i s h a x a = a : indeed a + a = a f o l l o w s by BERNSTEIN-

SCHRODER; by t h e same argument (2 ) f o l l ows f rom (1). L e t E be t h e f o l l o w i n g t o t a l o r d e r i n g r e l a t i o n between two couples o f o rd ina l s ( d , f s ) and ( ~ ' , f i ' ) : e i t h e r M a x ( & , p ) < M a x ( o ( ' , / j ' ) , o r t h e maximums are equal and fi <& or A = 0' anb 4 so(' . F o r coup1 es

o f o r d i n a l s , t h i s comparison i s a w e l l - o r d e r i n g . Thus t h e r e e x i s t s a c o n d i t i o n ( & , f i , r ) which t o each coup le (d , (3 )

associates one and o n l y one o r d i n a l

couples < (6.0) (mod take /3 = 0 : then f ("i ,~0, '6) assoc ia tes t o each o( an o r d i n a l 3 o( This 8 increases as o( increases, and i s e q u i p o t e n t w i t h t h e Car tes ian product o( x o< . Moreover, g i v e n a s e t o f o r d i n a l s 4 and t h e xi which are associated t o them by , then Sup gi i s assoc ia ted t o . Everyth ing comes down t o p r o v i n g t h a t , f o r o( i n f i n i t e , t h e assoc ia ted 8 i s

equipotent w i t h o( . F o r o( = w we have t h e s m a l l e s t

o rd ina l f o r which o( i s s t r i c t l y subpotent t o , thus t o o( x o ( . T h i s o( does n o t have any o r d i n a l which i s e q u i p o t e n t w i t h o( . Indeed we

would have o( ' e q u i p o t e n t w i t h o( ' x 6' , hence o( e q u i p o t e n t w i t h a x 4. Thus q i s n e c e s s a r i l y an a leph. Denote by o( t h e o r d i n a l s < o( and by

equipotent w i t h o( Ti < 4 ; thus Sup Fi 6 4 . By t h e p reced ing d i s c u s s i o n = Sup x i * % : c o n t r a d i c t i o n . 0

Modulo t h e axiom o f choice, o u r p r o p o s i t i o n extends t o a r b i t r a r y c a r d i n a l s . Note t h a t t h e e q u a l i t y a x a = a f o r eve ry i n f i n i t e c a r d i n a l a i s equiva-

l e n t t o t h e axiom o f c h o i c e (TARSKI 1924; see a l s o RUBIN 1963).

which i s isomorphic w i t h t h e s e t o f

) . We have ?f >/ M a x ( d , p ) . I n p a r t i c u l a r ,

= w . Denote by

o( ' <' P(

8 t h e o r d i n a l assoc ia ted t o ct by . By hypo thes i s each ri i s , and so s t r i c t l y subpotent t o o( , so

3.9. k t a be an i n f i n i t e a leph and l e t 2 6 b 6 a ; then ab = a2 . Indeed a = a x b ~ so a2 = (axb)2 = a(b2) >, ab . Modulo t h e axiom o f choice, t h i s i d e n t i t y extends t o a r b i t r a r y c a r d i n a l s .

3.10. Given an i n f i n i t e card ina l a , l e t o( be the Hartogs aleph f o r sets o f card ina l a (ch.1 5 6.2). E i the r N = w ; then a i s i n f i n i t e bu t no t Dedek ind- in f in i te (ch.1 5 2 . 6 ) ; then the card ina l sum a + 1 i s immediately g rea ter than a , i n the sense t h a t i t i s s t r i c t l y g rea ter than a (as a ca rd ina l ) y e t there i s no s e t s t r i c t l y intermedia- t e w i t h respect t o subpotence. O r o( i s a t l e a s t equal t o w 1 ; then the card ina l sum a + d i s immediately g rea ter than a i n the preceding sense (see TARSKI 1954). 0 Consequence o f the f a c t t h a t a r e s t r i c t i o n o f the o rd ina l o( e i t h e r i s isomorph i c w i t h o( o r w i t h an o rd ina l /3 s t r i c t l y l ess than o( ; then a + CardfS= a.D Modulo the axiom o f choice, every card ina l i s an aleph. Hence f o r each card ina l a there e x i s t s a card ina l b which i s immediately g rea ter than a i n the stronger fo l l ow ing sense: every card ina l > a i s b . Put b = a+ , the successor card ina l o f a , I n the mentioned paper, i t i s proved t h a t the preceding statement i s equ iva len t t o the axiom o f choice. I n the absence o f the axiom o f choice, there can e x i s t several card ina ls immedia- t e l y g rea ter than a given card ina l ( i n the weak sense). For example, w i t h the continuum hypothesis w i thout the axiom o f choice, w 1 and a lso the continuum are immediately g rea ter than w and poss ib ly incomparable: see ch.1 5 6.5.

§ 4 - REINFORCED RELATION, REINFORCED PARTIAL ORDERING 4.1. REINFORCEMENT, WEAKENING

Le t A be an n-ary r e l a t i o n w i t h base E . An n-ary r e l a t i o n B w i t h the same base i s c a l l e d a reinforcement o f A , and A i s a weakening o f B i f f every n - tup le i n E having the value (+) i n A s t i l l has the value (+) i n B . Reinforcement i s a p a r t i a l o rder ing on the se t o f n-ary r e l a t i o n s w i t h base E . A s t r i c t o r proper reinforcement o f A i s a reinforcement d i s t i n c t from A . S i m i l a r l y we speak o f a s t r i c t o r p roper weakenin&. L e t A be a p a r t i a l o rder ing which i s n o t a chain. We ob ta in a s t r i c t re in fo rce - ment o f A by tak ing two elements u, v which are incomparable (mod A) , then de f in ing , f o r a l l elements x, y o f t h e base, x d y i f f e i t h e r x d y (mod A) o r x,< u and v g y (mod A ) . Generalize t h i s procedure as fo l lows. Le t A be a p a r t i a l o rder ing w i t h base E . L e t D g : E and B be a p a r t i a l orde- r i n g w i t h base D which i s a reinforcement o f t he r e s t r i c t i o n A/D . Then there e x i s t s a p a r t i a l o rder ing C which i s a reinforcement o f A and an extension o f 6 2 E . Moreover, there e x i s t s a p a r t i a l o rder ing Co which i s minimal among these reinforcements C , i.e. every C which v e r i f i e s the preceding i s a reinforcement o f Co .

Chapter 2 43

0 L e t x, y be two elements o f E . Pu t x + y i f f x 6 y (mod A ) o r i f t h e r e

e x i s t two elements XI, y ' o f D w i t h x 4 x ' (mod A) and x ' d y ' (mod B) and y ' 6 y (mod A ) . The p a r t i a l o r d e r i n g thus o b t a i n e d i s t h e min imal re in fo rcemen t

Co ; t h e C ' s a r e re in fo rcemen ts o f Co . 0

4.2. REINFORCEMENT A X I O M

This axiom a s s e r t s t h a t each p a r t i a l o r d e r i n g has a c h a i n among i t s re in fo rcemen ts . - This f o l l o w s f rom t h e u l t r a f i l t e r axiom (SZPILRAJN 1930); ZF s u f f i c e s i n t h e case o f a countable p a r t i a l o r d e r i n g .

0 L e t A be a p a r t i a l o r d e r i n g w i t h base E . F o r each f i n i t e subset F o f E , we l e t UF denote t h e s e t o f a l l cha ins w i t h base F each o f which i s a r e i n f o r -

cement o f A/F . The UF a r e non-empty and v e r i f y t h e hypotheses o f t h e coherence lemma 1.3. Hence t h e r e e x i s t s a r e l a t i o n C based on E which s a t i s f i e s C/F E UF f o r each F (uses u l t r a f i l t e r axiom). T h i s C i s a chain, s i n c e i t s f i n i t e r e s t r i c t i o n s a r e chains; i t i s a re in fo rcemen t o f A . 0

I n the l i g h t o f t h e p reced ing 4.1, t h e re in fo rcemen t axiom can be s t a t e d equiva- l e n t l y : g i v e n a p a r t i a l o r d e r i n g A and a c h a i n C which i s a re in fo rcemen t o f

a r e s t r i c t i o n o f A , t h e r e e x i s t s a c h a i n which i s s imu l taneous ly a r e i n f o r c e - ment o f A and an e x t e n s i o n o f C ,

The re in fo rcemen t axiom i m p l i e s t h e o r d e r i n g axiom o f 5 2.3. It s u f f i c e s , g i v e n a s e t E , t o r e i n f o r c e i n t o a c h a i n t h e p a r t i a l o r d e r i n g which reduces t o t h e i d e n t i t y on E . Note t h a t t h e re in fo rcemen t axiom i s s t r i c t l y weaker than t h e u l t r a f i l t e r axiom: see FELGNER 1974 p. 375, making re fe rence t o t h e unpubl ished t h e s i s o f t h e au tho r .

Also the o r d e r i n g axiom i s s t r i c t l y weaker than t h e re in fo rcemen t axiom: see MATHIAS 1974.

4.3. Each weakened p a r t i a l o r d e r i n g o f a wel l - founded p a r t i a l o r d e r i n g i s w e l l - founded. On t h e o t h e r hand, a re in fo rcemen t o f a we l l - f ounded p a r t i a l o r d e r i n g i s no t n e c e s s a r i l y wel l - founded: r e i n f o r c e t h e i d e n t i t y w i t h a denumerable base i n t o an w- c h a i n (converse o f w ) .

4.4. For each we l l - f ounded p a r t i a l o r d e r i n g re in forcement o f A (uses axiom o f choice; ZF s u f f i c e s i f A i s coun tab le o r has we l l -o rde rab le base).

For each o r d i n a l i s t r i c t l y l e s s than H t A , t a k e a w e l l - o r d e r i n g Ci based on the f r e e s e t o f elements w i t h h e i g h t i of the Ci acco rd ing t o i n c r e a s i n g i . 0

A , t h e r e e x i s t s a we l l -o rde red

(axiom o f c h o i c e ) . Then take t h e sum

4.5. Le t A, B be two well-founded p a r t i a l order ings w i t h the same base. - If B i s a reinforcement o f A , then f o r each x i n the base, H t x (mod B) >/ H t x (mod A) . Consequently H t B 3 H t A . Proof by induc t ion ,

5 5 - COFINAL SUBSET, CO-INITIAL SUBSET, COFINALITY OF A CHAIN

5.1. COFINAL. CO-INITIAL SUBSET AND RESTRICTION

Le t A be a D a r t i a l order ino. A subset D o f IAI i s sa id t o be c o f i n a l (mod A ) . and A / D i s sa id t o be a co f i na l r e s t r i c t i o n o f A i f f f o r each x i n I A l there e x i s t s a y i n D w i t h y + x (mod A) . Analogous d e f i n i t i o n f o r a

c o - i n i t i a l subset and a c o - i n i t i a l r e s t r i c t i o n . I f the base i s non-empty, then

every co f i na l o r c o - i n i t i a l subset i s non-empty. Each superset o f a c o f i n a l se t i s c o f i n a l ; s i m i l a r l y w i t h c o - i n i t i a l .

Each co f i na l r e s t r i c t i o n K a co f i na l re_z_tt--c_t_ion_ i s a qozinal r e s t r i c t i o n ; s i m i l a r l y w i t h c o - i n i t i a l . Le t A be a p a r t i a l order ing, B a well-founded p a r t i a l o rder ing w i t h the same

base E . For each p a i r \x,y) i n E , i f y < x (mod A) and > x (mod B) , then remove t h i s y . Then the se t D obtained a f t e r a l l suc~-removals i s

c o f i n a l (mod A) , and any two d i s t i n c t elements o f D are never ordered i n

opposi te senses modulo A and modulo B . Moreover i f B i s a wel l -order ing, then A/D i s a well-founded p a r t i a l o rder ing which i s a co f i na l r e s t r i c t i o n o f A (POUZET 1979, unpublished). 0 F i r s t we see t h a t D i s c o f i n a l (mod A) . Indeed f o r each y which i s removed,

there e x i s t s an x o f l e a s t he igh t (mod B) among those s a t i s f y i n g x > y (mod A)

and x < y (mod B) . This x belongs t o D : i f i t were removed, then there would e x i s t an x ' > x > y (mod A) and x ' < x < y (mod B) , con t rad i c t i ng the

m in ima l i t y o f the he igh t o f x . Now consider the case t h a t B i s a wel l -order ing. For each non-empty subset X

o f D , l e t x be the minimum o f B/X : then x i s a minimal element (mod A/X) . For otherwise there would e x i s t a y i n X w i t h y < x (mod A) . Since x, y belong t o D , we do no t have y > x (mod 8) . Hence y < x (mod B) , contra-

d i c t i n g the m in ima l i t y o f x (mod B/X) . From an i n t u i t i v e p o i n t o f view, note t h a t when B i s a wel l -order ing, i f we

denote by bi the element w i t h he igh t i (mod B) , then D i s the s e t o f

elements c def ined as fo l lows among the b . Le t c,, = bo . Then c1 = bi(l)

= the bi o f l e a s t he igh t i ( 1 ) # 0 (mod B) , among the elements $ co (mod A) . Then c2 = bi(*) = the bi o f l e a s t he igh t i ( 2 ) > i ( 1 ) (mod B) , among the

elements .& co and 4 c1 t h a t the cu = b . (u < O( ) are defined. Then c o( - - bi(%) = the bi o f

L

(mod A) . I n general f o r each o rd ina l o( , assume

1 (u )

Chapter 2 45

l eas t height i ( % ) 7 i ( u ) f o r a l l u <o( , among the elements which are simultaneously 9. co and .$ c1 and . .. and + cu (mod A) f o r a l l u < O( . Corollarz. For each p a r t i a l o rder ing A , there e x i s t s a c o f i n a l s e t D such t h a t A/D i s a well-founded p a r t i a l o rder ing (uses axiom o f choice; ZF su f f i ces i f A i s countable o r has wel l -orderable base; ZF c l e a r l y s u f f i c e s a l s o i f A i s we l l -

founded). Take a we l l -o rder ing o f I A 1 and use the preceding propos i t ion .

5.2. STRATIFIED PARTIAL ORDERING A p a r t i a l o rder ing A i s sa id t o b e . s t r a t i f i e d i f f the union o f the re la t i ons o f incomparabi l i ty (mod A) and i d e n t i t y forms a t r a n s i t i v e r e l a t i o n , hence an equivalence re la t i on . Consequently x 4 y and y I z imp l ies t h a t x < z ; s i m i l a r l y wi th > i n the place o f < . The equivalence classes o f i ncomparab i l i t y - i den t i t y form a t o t a l ordering, by pu t t ing "c lass o f x " < "class o f y I' i f f x < y (mod A) . This t o t a l orde- r i ng o f the equivalence classes w i l l be c a l l e d the p r i n c i p a l t o t a l o r d e r i n g of the s t r a t i f i e d p a r t i a l o rder ing A . Every t o t a l l y ordered r e s t r i c t i o n o f A which i s maximal under i nc lus ion i s isomorphic w i t h the p r i n c i p a l t o t a l order ing.

5.3. Le t A be a well-founded p a r t i a l order ing. Then there e x i s t s a c o f i n a l r e s t r i c t i o n C - o f A s a t i s f y i n g H t C 6 Card H t A (POUZET 1979, unpublished). 0 To each ord ina l i < H t A associate the c lass Bi o f elements o f he igh t i (mod A) . Order the s e t o f the cardinal, which i s Card H t A . Denote by B the well-founded s t r a t i f i e d p a r t i a l ordering w i t h base I A I , def ined by p u t t i n g x < y (mod B) i f f "class o f x I'

< "class o f y I' according t o the preceding wel l -order ing. Then H t B =

Card H t A . By 5.1, there e x i s t s a c o f i n a l r e s t r i c t i o n C o f A such t h a t any two elements o f I C I are never ordered i n opposi te senses by A and B . Hence i f x, y belong t o I C l and x < y (mod A) , then x ( y (mod B) , as incomparabi l i ty (mod 8) i s impossible s ince x and y belong t o two d i s t i n c t classes Bi . Thus B / K I i s a reinforcement o f C = A/)CI . By 4.5, we have H t C 4 Ht(B/ICI) 6 H t B = Card H t A .

Bi by a we l l -o rder ing isomorphic w i t h i t s

5.4. COFINALITY, CO-INITIALITY Let A be a p a r t i a l order ing. If, among the c o f i n a l sets (mod A) , there ex i s t s one o f l e a s t card ina l , then t h i s card ina l i s c a l l e d t h e c o f i n a l i t y o f A , denoted by Cof A . Analogous d e f i n i t i o n o f c o - i n i t i a l i t y . With the axiom o f choice, every card ina l i s an aleph, hence the c o f i n a l i t y and co- i n i t i a l i t y e x i s t f o r each p a r t i a l order ing. With on l y the axioms o f ZF, these

o n l y e x i s t i n p a r t i c u l a r cases, f o r example when t h e base i s w e l l - o r d e r a b l e . T h e i r s t u d y i s ve ry d i f f e r e n t i n t h e case o f a t o t a l o r d e r i n g , t h e c l a s s i c a l case

cons ide red i n t h e p resen t and n e x t s e c t i o n s , f rom t h e genera l case o f a p a r t i a l o rde r ing , such a case i n t r o d u c e d and s t u d i e d by POUZET: see 5 7 below.

L e t subset U o f t h e base, w i t h A/U a w e l l - o r d e r i n g isomorphic t o Cof A ; same r e s u l t w i t h c o - i n i t i a l i t y .

0 Take a c o f i n a l subset D w i t h l e a s t c a r d i n a l , hence o f c a r d i n a l equal t o Cof A . T o t a l l y o r d e r D a c c o r d i n g t o i t s c a r d i n a l . Then a p p l y 5.1, r e p l a c i n g E by D

and A by A/D , and B by a w e l l - o r d e r i n g o f D i somorphic t o t h e c a r d i n a l o f D ( i . e . t h e l e a s t o r d i n a l based on D ) , which i s Cof A . 0

C o r o l l a r y . L e t o( be an o r d i n a l ; C o f d i s t h e l e a s t o r d i n a l u f o r which t h e r e e x i s t s a u-sequence of successor o r d i n a l s whose un ion i s

a s t r i c t l y i n c r e a s i n g u-sequence o f successor o r d i n a l s whose u n i o n i s 4 ) .

A be a t o t a l o r d e r i n g w i t h w e l l - o r d e r a b l e base. Then t h e r e e x i s t s a c o f i n a l

c( ( e q u i v a l e n t l y :

5 . 5 ~~t be an indecomposable ordinal. I a cof ina l subset of o( - w i t h w / I

o f l e a s t o r d e r t ype , hence equal t o Cof 4. To each element i o f I we asso-

c i a t e t h e o r d i n a l di , t h e i n i t i a l i n t e r v a l o f d formed by t h e e lements < i . Then we have N i ( i E I ) = (Sup d i ) = o(, where zcxi o r d i n a l sum o f t h e Mi a l o n g t h e o r d e r o f i n c r e a s i n g i : see 5 3.6.

0 Obvious ly s t r i c t i n e q u a l i t y . Suppose t h a t ment o f I f o r which z o( i(i < u) >/ o( . E i t h e r u has a predecessor v

( u = v + l ) , i n which case o( i s equal t o p l u s a non-zero o r d i n a l which i s < dV hence < d . I n t h i s case o( i s n o t indecomposable: con- t r a d i c t i o n .

O r u i s a l i m i t o r d i n a l , hence

o f t h e d i s j o i n t i n t e r v a l s N i ( i < u) y i e l d s an o r d i n a l isomorphic t o u , and

i s c o f i n a l f o r d . Hence u >, Cof o( , and so t h e o r d i n a l o f d/I i s s t r i c t l y g r e a t e r t han C o f M : c o n t r a d i c t i o n . 0

N o t i c e t h a t it i s necessary t o assume t h a t o( i s indecomposable: t a k e t h e counterexample o( = W.2 w i t h i i n t e g e r and di = 6~ + i . It i s a l s o necessary t o t a k e OC / I i somorphic t o Cof d , and n o t o n l y o f c a r d i n a l Cof o( : t a k e

2 t h e counterexample LX = Cr) w i t h i < d . 2 and o( = c3 + i f o r f i n i t e i , and o( = L J . 2 + ~ ( i - W ) f o r (,J 6 i < (2.2 .

des igna tes t h e

oCi >, (Sup Ni) = o( . It remains t o e l i m i n a t e t h e case o f

2 Hi > o( and l e t u be t h e l e a s t e l e -

o( i(i < v )

d i ( i < u ) = o( . Then t h e s e t o f minimums

Chapter 2 41

§ 6 - REGULAR AND SINGULAR ALEPH, ACCESSIBILITY

6.1. REGULAR AND SINGULAR ALEPH An a leph ( i . e . t h e c a r d i n a l of a w e l l - o r d e r a b l e s e t ) i s s a i d t o be r e g u l a r i f f , considered as an o r d i n a l , eve ry c o f i n a l subset i s e q u i p o t e n t w i t h i t . I n o t h e r

words, i t s c o f i n a l i t y i s equal t o i t . An a leph i s s a i d t o be s i n g u l a r i n t h e

oppos i te case where i t s c o f i n a l i t y i s s t r i c t l y sma l le r . For example 1 and w a r e r e g u l a r . Each i n t e g e r 3 2 has c o f i n a l i t y 1, hence i s s i n g u l a r . The c a r d i n a l ww o r has c o f i n a l i t y r i j , hence i s s i n g u l a r .

Modulo t h e c o u n t a b l e z i ' l m o f choice, U1 i s r e g u l a r . However t h e i n e q u a l i t y

0 Associate t o each coun tab le subset D o f ul t h e coun tab le un ion o f those

countable o r d i n a l s which a r e elements o f

countable o r d i n a l , hence cannot be t h e e n t i r e s e t a1 . 0

L e t A be a c h a i n w i t h w e l l - o r d e r a b l e base; t hen Cof A i s a r e g u l a r a leph. Consequence o f t h e f a c t t h a t a c o f i n a l r e s t r i c t i o n o f a c o f i n a l r e s t r i c t i o n i s i t s e l f c o f i n a l ( 5 . 1 above).

w 1 > W i s o b t a i n e d i n ch .1 5 6.3 u s i n g o n l y ZF.

D . By ch .1 5 2.5, t h i s un ion i s a

6.2. Every successor a l e p h i s - y e g - u x (uses axiom o f c h o i c e ) . 0 Our a leph i s o f t h e fo rm ONtl where o( i s an o r d i n a l . L e t u be i t s

c o f i n a l i t y . Take a u-sequence o f successor o r d i n a l s whose un ion i s c3d+1 : see c o r o l l a r y 5.4. From some p o i n t on, t hese o r d i n a l s a r e e q u i p o t e n t w i t h wq . Suppose u s t r i c t l y l e s s t h a n , hence Card u 6 ad . Then t h e un ion o f t h e o r d i n a l s i n o u r u-sequence has c a r d i n a l a t most equal t o fdH x Ld4 hence a t most equal t o fAd (ax iom o f cho ice g i v i n g a b i j e c t i o n o f each o r d i n a l onto do(). C o n t r a d i c t i o n p r o v i n g t h a t u = Wdtl . 0

6.3. L e t a be an i n f i n i t e a leph . For a t o be s i n g u l a r , i t i s necessary and

s u f f i c i e n t t h a t t h e r e e x i s t s a s e t u , s t r i c t l y subpotent w i t h a , o f elements s t r i c t l y subpotent w i t h a , whose u n i o n y i e l d s a (uses axiom o f cho ice ) . 0 I f a i s s i n g u l a r , t h e n o u r c o n c l u s i o n i s obv ious. Conversely i f a i s

r e g u l a r , t hen l e t u be a s e t o f subsets o f a whose un ion i s a . E i t h e r one o f t h e subsets i s c o f i n a l , hence o f c a r d i n a l a . O r each subset i s bounded above,

and t h e s e t o f l e a s t upper bounds i s c o f i n a l i n a , hence o f c a r d i n a l a . Replace each upper bound by one o f t h e co r respond ing subsets (ax iom o f c h o i c e ) : t he s e t u has a t l e a s t c a r d i n a l a . 0 6.4. The p reced ing p r o p o s i t i o n suggests t h e f o l l o w i n g g e n e r a l i z a t i o n . A c a r d i n a l a ( n o t n e c e s s a r i l y an a leph ) i s s a i d t o be s i n g u l a r i f f i t i s t h e un ion o f a s e t

s t r i c t l y subpotent w i t h a ; i t i s s a i d t o be r e g u l a r o the rw ise . I n t h e presence o f t h e axiom o f choice, eve ry c a r d i - n a l i s an aleph, and we have t h e c l a s s i c a l d e f i n i t i o n i n 6.1. I n t h e absence o f t h e axiom o f choice, we do n o t know whether t h i s g e n e r a l i z e d d e f i n i t i o n o f r e g u l a r and s i n g u l a r c a r d i n a l y i e l d s i n t e r e s t i n g r e s u l t s .

Wi th t h e axiom o f c h o i c e and t h e cont inuum hypo thes i s , we know t h a t t h e c a r d i n a l o f

t h e continuum equals CJ and so i s r e g u l a r . Wi th o n l y t h e axiom o f cho ice , t h e r e e x i s t models where t h e continuum i s a r e g u l a r a leph, and o t h e r s where t h e continuum

i s a s i n g u l a r a leph. I t can have any i n f i n i t e c o f i n a l i t y , except w : f o r example

i t cannot equal ww . T h i s r e s t r i c t i o n on t h e c o f i n a l i t y r e s u l t s f rom t h e f a c t t h a t , f o r any p a r t i t i o n o f t h e continuum i n t o a coun tab le number o f subsets , t h e r e i s a t l e a s t one which i s e q u i p o t e n t w i t h t h e continuum: see ch .1 5 4.3.

a , whose elements a r e s t r i c t l y subpotent w i t h

6.5. ( 1 ) L e t a be a r e g u l a r aleph; f o r e v e r y b (14 b < a) we have ba = a (TARSKI 1938; uses g e n e r a l i z e d continuum hypo thes i s ; ZF s u f f i c e s f o r a = W ;

f o r a = cJ1 , ZF p l u s cho ice p l u s cont inuum hypo thes i s ) . ( 2 ) L e t a be a l i m i t a leph; f o r e v e r y c, d c a we have ‘d < a ( I b i d . prop. 9 ; g e n e r a l i z e d cont inuum hypo thes i s i s used) .

0 (1) The s ta temen t i s t r u e f o r a = W . Suppose t h a t a i s an i n f i n i t e succes-

s o r a leph, hence o f t h e fo rm a = ‘2 ( g e n e r a l i z e d continuum hypo thes i s ) , and b s a t i s f i e s 1 b x c = c by 3.8. Now suppose t h a t a i s a l i m i t a leph which i s s t i l l r e g u l a r and

s t r i c t l y g r e a t e r t han a. There e x i s t s an i n c r e a s i n g a-sequence o f successor

c a r d i n a l s ai < a (i runs th rough a) w i t h a = Sup ai . From a c e r t a i n p o i n t on,

we have a i 7 b , hence b(ai) = ai by t h e p rev ious d i s c u s s i o n . S ince a i s

r e g u l a r , no b-sequence i s c o f i n a l i n a . Hence t h e s e t o f a l l b-sequences w i t h va lues i n a i s t h e union, as i v a r i e s , o f t h e s e t s o f b-sequences w i t h va lues i n each ai . Hence ba = Sup( (ai)) = Sup ai = a . 0

0 ( 2 ) L e t u be a successor a leph s a t i s f y i n g c < u < a and d < u < a . Then u i s r e g u l a r , hence ‘d \< ‘IJ = u by (1); hence ‘d < a . 0

Statement (1) does n o t h o l d f o r s i n g u l a r a lephs. L e t a = OU and b = U. Decompose a i n t o t h e un ion o f t h e Wi (i i n t e g e r )

and assoc ia te t o each Wi

t h e un ion o f t h e Oi

b $ c . Then we have ba = b(C2) = ( b x c ) 2 = ‘2 = a s i n c e

b

i t s successor

w

. By K O N I G ‘ S theorem (ch .1 5 1.8)

i s s t r i c t l y subpo ten t t o t h e C a r t e s i a n p r o d u c t o f t h e

cc, i+l . Hence we have < ( W ) . Even i n ZF (KONIG‘s theorem u s i n g t h e axiom of cho ice ) , we have t h e f o l l o w i n g

counterexample. L e t a. = (d , and f o r each i n t e g e r i l e t us d e f i n e :

Chapter 2 49

= ( 2 t o the power ai) , and then a = Sup ai . Then we have a2 5 @ a , ai+l hence a < Gta : t o get the f i r s t i nequa l i t y , associate t o each subset u o f a

the sequence o f the i n te rsec t i ons ui = a,.n u , and no t i ce t h a t ui 6 ai+l . 6.6. Le t a be a regu la r l i m i t aleph; then the aleph rank o f a & a i t s e l f ;

i n o the r words 0 Assume the contrary, t h a t a < ma , and l e t b be the aleph rank o f a , so a = W ( b < a) . Then e i t h e r b i s a successor o rd ina l , so a i s a succes-

sor aleph. Or the sequence o f the wi ( i < b) i s co f i na l i n a , hence a i s

not regu la r . 0

On the o the r hand, we can have ba = a where a i s a s ingu la r card ina l . Take the l i m i t aleph o f t he sequence a(0) = u, a(1) = Wu ,..., a ( i + l ) = W

f o r each in tege r i : the aleph obtained has c o f i n a l i t y w , hence i s s ingu la r .

wa = a .

a ( i )

6.7. ACCESSIBLE CARDINAL, A X I O M OF ACCESSIBILITY An i n f i n i t e cardinal

a se t b , s t r i c t l y subpotent w i t h a and whose elements a re s t r i c t l y subpotent w i t h a , which s a t i s f i e s u b = a ; o r f i n a l l y i f there e x i s t s a s e t c s t r i c t l y

subpotent w i t h a , and where a i s subpotent w i t h ( c ) . A card ina l i s sa id

t o be inaccessible otherwise. The axiom o f a c c e s s i b i l i t y asserts t h a t every se t has accessible c a r d i n a l i t y . Under the assumption t h a t ZF i s consistent, the theory ZF plus the axiom o f acces-

s i b i l i t y i s consistent: see SHEPHERDSON 1952.

Every accessible l i m i t a leph d i f f e r e n t from W i s s ingu la r (uses general ized con-

tinuum hypothesis). Consequently i n view o f 6.2, f o r an accessible aleph a # k~ , t h i s a i s regu la r i f f a i s a successor aleph.

0 Let c3, be our aleph, where 4 i s a non-zero l i m i t o rd ina l . Every s t r i c t l y smal ler card ina l i s an

?(mi) = our aleph i s assumed t o be accessible and d i s t i n c t from .LI , there e x i s t s a se t s t r i c t l y subpotent w i t h a d , whose elements are as we l l s t r i c t l y subpotent w i th

a i s sa id t o be accessible i f a = CJ , o r i f there ex i s t s

Ui ( i < & ) . There i s no se t s t r i c t l y subpotent w i th

(general ized continuum hypothesis) and i+l< d . F i n a l l y as

c+ , hence of c a r d i n a l i t y lrJi , w i t h wd subpotent w i t h ?( wi) : indeed

LJ4 , and whose union i s 1J, : our aleph i s thus s ingu la r (see 6 .3 ) . 0

An aleph i s sa id t o be weakly inaccess ib le i f i t i s d i f f e r e n t from 0 , regu la r

and a l i m i t aleph. The preceding p ropos i t i on i s equ iva len t t o saying tha t , w i t h

the axiom o f choice and the general ized continuum hypothesis, every weakly inacces- s i b l e aleph i s inaccessible. Using on ly the axiom o f choice, every inaccessible aleph i s weakly inaccessible.


Indeed, i t i s d i f f e r e n t from w , regu lar , and there i s no s t r i c t l y smal ler c

f o r which i t i s subpotent w i t h T ( c ) : hence i t i s not a successor aleph. Problem. Assume t h a t ZF p lus choice p lus the existence o f a weakly inaccess ib le aleph less than the continuum i s consistent. Then does t h i s theory remain cons is ten t i f we requ i re t h a t the continuum i t s e l f be weakly inaccessible. C lass i ca l l y we have an a f f i r m a t i v e answer i f we assume the consistency o f ZF p lus choice p lus the existence o f a weakly inaccess ib le aleph grea ter than the continuum.

6.8. Some unpublished r e s u l t s of BLASS, i n a l e t t e r t o HODGES, 1982. Consider the fo l l ow ing statements: (1) every countable union o f countable sets i s countable (see ch.1 5 2 .5 ) ; ( 2 ) i f a s e t a and i t s elements have card ina ls cdl , then the union o f a has card ina l $ u1 ; (3) given a func t i on f , if Rng f = dl , then U1 i s subpotent t o Dom f . Then i t i s proved t h a t ne i the r o f (1) and (2) imp l ies the other, and t h a t (2 ) i s equivalent t o (3) p l u s the r e g u l a r i t y o f w1 .

§ 7 - COFINALITY OF A PARTIAL ORDERING, COFINAL HEIGHT The c o f i n a l i t y of a p a r t i a l o rder ing A , denoted Cof A and introduced i n 5.4, was s tud ied above on ly i n the case o f a t o t a l order ing. I f A i s non- to ta l l y ordered, then Cof A can be a s ingu la r aleph. 0 Construct A by taking, f o r each in tege r i , a cha in of order type wi , and p u t t i n g them together mutual ly incomparably: then Cof A = W, . 0

Given a p a r t i a l o rder ing A and a c o f i n a l s e t D (mod A) , the re e x i s t s a co f i na l subset U of D with Card U = Cof A . Consequently Cof(A/D) = Cof A f o r each co f i na l s e t D (uses axiom o f choice; ZF s u f f i c i e n t i f A i s countable o r has wel l -orderable base). 0 Take a se t V co f ina l (mod A ) and w i t h l e a s t card ina l , hence Card V = Cof A ,

Then replace each element x o f V by an element i n D which i s g rea ter than x . 0

7.1. Le t A be a p a r t i a l o rder ing and u = Cof A . There ex i s t s a co f i na l subset U o f card ina l u , the r e s t r i c t i o n A/U being a well-founded p a r t i a l o rder ing w i t h he igh t Ht(A/U) 6 u . A refinement o f c o r o l l a r y 5.1, due t o PDUZET 1979, unpublished. Uses t he axiom o f choice; ZF s u f f i c e s i f A I s countable o r has we1 1-orderable base. 0 Le t D be a c o f i n a l subset w i t h l e a s t card ina l u = Cof A . Order D by

Chapter 2 5 1

i t s c a r d i n a l i t y , which we assume t o be an a leph, and c a l l quence (ax iom o f c h o i c e ) . E x t r a c t a sequence bi by removing, f o r each i , t h e

a such t h a t j >i and a . <a i (mod A) . The sequence o f t h e bi has l e n g t h j J

a t most equal t o u . By 5.1, t h e i n e q u a l i t y i <i' i m p l i e s bi < o r I bin (mod A) . Moreover, t h e s e t U o f va lues b i s c o f i n a l i n A and A / U i s a

wel l - founded p a r t i a l o r d e r i n g .

For each s u b s c r i p t i i (mod A/U) . Then t h e r e

e x i s t s a bk o f h e i g h t i , w i t h b k < bi (mod A) ; and hence k < i by 5.1, which c o n t r a d i c t s t h e m i n i m a l i t y o f i . F i n a l l y t h e h e i g h t o f A / U i s 6 u . 0

ai (i< u) t h i s se-

i , t h a t t h e h e i g h t o f bi i s & i . F i r s t o f a l l , a l l t h e bi a r e 2 o r \ b o (mod A) . Assume

7.2. For eve ry we l l - f ounded p a r t i a l o r d e r i n g A , we have Cof H t A ,< Cof A

(uses axiom o f choice; ZF s u f f i c e s i f A i s coun tab le o r w i t h w e l l - o r d e r a b l e base).

0 Take a c o f i n a l s e t D o f l e a s t c a r d i n a l i t y , so Card D = Cof A . The h e i g h t s (mod A) o f t h e elements o f D f o rm a s e t H , which i s c o f i n a l i n t h e o r d i n a l

H t A . By t h e axiom o f cho ice , H i s subpotent w i t h D , hence we have

Cof H t A ,< Card H 4 Card D = Cof A . 0

I f A i s a w e l l - o r d e r i n g , t hen H t A i s isomorphic w i t h A , hence Cof H t A =

Cof A . On t h e o t h e r hand, t a k e t h e we l l - f ounded p a r t i a l o r d e r i n g A formed o f a component c3 , a component w+1 , t h e e lements o f one c h a i n p u t incomparable w i t h those o f t h e o t h e r . Then Cof A = W , H t A = W+1 so Cof H t A = 1 .

h

7.3. COFINAL HEIGHT

Given a p a r t i a l o r d e r i n g A , i f t h e r e e x i s t s a we l l - f ounded c o f i n a l r e s t r i c t i o n of A , then t h e l e a s t h e i g h t o f such r e s t r i c t i o n s i s c a l l e d t h e c o f i n a l h e i g h t

o f A , and denoted Cofh A . With t h e axiom o f choice, t h e c o f i n a l h e i g h t e x i s t s

f o r eve ry p a r t i a l o r d e r i n g A : see c o r o l l a r y 5.1. Using o n l y t h e axioms o f ZF, the c o f i n a l h e i g h t e x i s t s a t l e a s t f o r a l l we l l - f ounded p a r t i a l o r d e r i n g s .

(1) For e v e r y p a r t i a l o r d e r i n g

quence o f 7 .1 and uses t h e axiom o f choice; ZF s u f f i c e s i f A i s coun tab le o r has w e l l - o r d e r a b l e base.

S t r i c t i n e q u a l i t y can happen: f o r example when A i s reduced t o t h e i d e n t i t y on i t s base E , then Cofh A = 1 and Cof A = Card E . (2 ) I,f A i s a t o t a l o r d e r i n g , t hen Cofh A = Cof A . Consequence o f (1) and o f t h e f a c t t h a t t h e c a r d i n a l o f a w e l l - o r d e r e d r e s t r i c t i o n o f A i s a t most equal

t o i t s h e i g h t ; same c o n d i t i o n s as i n (1).

A ,.we have Cofh A 6 Cof A . Th is i s a conse-


( 3 ) I f A i s a well-founded pa r t i a l ordering, then Cofh A 4 Card H t A : another form of 5.3.

7.4. Given a pa r t i a l ordering A , t he cof ina l height i s the l e a s t height of well- founded cofinal r e s t r i c t i o n s -. of A ,whose cardinal i s equal t o Cof A (uses axiom of choice; ZF su f f i ces i f A

S t a r t w i t h a cofinal subset D such t h a t A / D i s a well-founded pa r t i a l ordering of l e a s t height H t ( A / D ) = Cofh A . Take a subset D ' of D which is cofinal and of l e a s t cardinal Cof A (see beginning of present 5 7, using the axiom of choice) . Then Ht(A/D') $ H t ( A / D ) = Cofh A

i s countable o r has well-orderable base).

by 3.5. 0

7.5. I f B i s a cofinal r e s t r i c t i o n of t he pa r t i a l ordering A , then Cofh B i- Cofh A (assuming t h a t these cof ina l he ights e x i s t ) . Indeed each cofinal r e s t r i c t i o n of B i s a cofinal r e s t r i c t i o n of A . The following example, due t o POUZET 1979, shows t h a t s t r i c t inequal i ty i s poss ib le , contrary t o the s i t u a t i o n f o r c o f i n a l i t y (beginning of present 5 7 ) . 0 Take as base the Cartesian product w x with i i n t ege r , j countable o rd ina l , p u t ( i , j ) 6 ( i ' , j ) f o r every in t ege r i ' a i . For each even in teger i , p u t (i,j)s ( i , j ' ) f o r a l l j ' a j . For each odd in teger i , we l e t the ordered pa i r s w i t h f i r s t term i be mutually incomparable. Then we complete by t r a n s i t i v i t y . The pa r t i a l ordering A thus obtained has a cofinal subse t of a l l ordered pa i rs w i t h odd f i r s t term: thus we have Cofh A = a. The r e s t r i c t i o n t o ordered pa i r s with even f i r s t term has cofinal height w1 . 0

. For each ordered pa i r ( i , j )

7.6. For each p a r t i a l ordering A , the cofinal height of A i s a cardinal (uses axiom of choice; ZF su f f i ces i f A i s countable, o r has well-orderable base, o r i s well-founded; POUZET 1979, unpublished). 0 Take a well-founded cofinal r e s t r i c t i o n B of A w i t h l e a s t he ight , so H t B = Cofh A : see co ro l l a ry 5.1. By 7.5 we have we have Cofh B 6 Card H t B = Card Cofh A . 0

Cofh A $ Cofh B . By 7 .3 . (3 )

7.7. Let A be a well-founded pa r t i a l ordering: any of t he possible comparisons " s t r i c t l y l e s s than", " s t r i c t l y g rea t e r " , "equal" can be obtained f o r the cardina ls Cofh A and Cof H t A . (1) S t a r t from the chain of in tegers . To each in teger i a s soc ia t e an element i ' . P u t i ' 7 i ; p u t i ' incomparable with in tegers bit1 ; f i n a l l y f o r a l l in tegers i and j # i , p u t i ' incomparable with j ' . Denote by A th is pa r t i a l ordering: we have H t A = Cof H t A = c3 and Cofh A = 1 . (1') In the above example 7.5 we have Cofh A = W and Cof H t A = W1 .

Chapter 2 53

( 2 ) For each ordinal q , we have Cofh o( = Cof o( = Cof H t o( . ( 3 ) Construct A by s tar t ing with the ordinal c3,; for each integer i , add by a bifurcation another chain w i a f t e r the i n i t i a l interval c d i . Then we have H t A = aw so Cof H t A = W and Cofh A = w w . We can also have Cof H t A 4 Cofh A where Cofh A i s a regular aleph. See below in ch.7 5 3.10 where Cof A = Cofh A = W1 and H t A = d l . w thus Cof H t A = W .

§ 8 - SET OF SUBSETS, SET OF INJECTIVE FUNCTIONS, INCREASING FUNCTIONS

ON THE REALS, INTERMEDIACY, CONSECUTIVITY, CYCLE, C Y C L I C RELATION

8.1. Let E be an in f in i te s e t , I a s e t of subsets Ei ( i 6 I ) of E where I i s subpotent with E and each E i i s equipotent with E . (1) There exis t two dis joint subsets C , D of E such t h a t fo r each i t& intersections CnEi and DnEi are non-empty. ( 2 ) More strongly, there exis t dis joint C , D such that CnEi and DnEi are equipotent with E (uses axiom of choice; ZF suffices when E i s countable; see BERNSTEIN 1908 or KURATOWSKI 1966 p . 514. 0 (1) Well-order E according t o i t s cardinal a , and I according t o i t s cardi nal b ( l ess t h a n or equal t o a ) . P u t the f i r s t element xo of Eo into C , the second yo into 0 . I n general for each ordinal i < b , p u t the f i r s t xi of Ei which i s d i s t inc t from a l l x j and y j ( j < i ) into C , and the second yi d i s t inc t from a l l x and y into D . 0

0 ( 2 ) Replace the s e t I by a sequence of the terms E i with repetitions, as follows. If b = a so t h a t I i s equipotent with E , then take the f i r s t term

j j

Eo , then Eo followed by E l , and in general f o r each i , take the sequence of E j (0 6 j ,c i ) . I f b < a , then keep the above sequence of the E i , b u t keep repeating so as t o obtain a sequence with length a = Card E . In e i ther case, repeat the preceding proof: for each s e t Ei , the intersections CnEi and DnEi have cardinal a . 0

Consequently, i f E i s denumerable and 3 i s a non-trivial u l t r a f i l t e r on E , then there i s no countable se t of subsets E i f E which generates g , in the sense that the elements of are obtained by taking supersets of f i n i t e intersections of the E i . 0 Denote by E i the E i from the statement and also the i r f i n i t e intersections. Then the se t s C and D , which are dis joint , would b o t h belong t o the ultra- f i l t e r : contradiction. 0

More generally, l e t E be inf in i te and 3 be an u l t r a f i l t e r on E whose elements are equipotent with E . Then potent with E , of elements of g.

cannot be generated by a se t , equi-

8.2. (1) Let E be an i n f i n i t e s e t , I a s e t of injective functions f i ( i € I ) from E in to E , where I i s subpotent w i t h E , and each f i s a t i s f i e s f i ( x ) # x on a subset of E equipotent w i t h E . Then there - exis t s a subset C of E , equipotent w i t h E , w i t h the non-inclusion f i o ( C ) q C for each i (notation f " from ch.1 5 1 . 2 ) . ( 2 ) Under the same hypotheses, there ex is t two subsets C , D Df E which are dis joint and equipotent w i t h E , w i t h the-_?on-inclusion f i o ( C ) F C W D and similarly by interchanging C and D (uses axiom of choice; SIERPINSKI 1950; see also ROSENSTEIN 1982). 0 (1) Well-order E by i t s cardinal i ty a ; well-order I f i r s t by i t s cardinali- t y , then w i t h repeti t ions according to the ordinal a , in the case where Card I < Card E . P u t the f i r s t xo of E sa t isfying fo(xO) # xo into C , and p u t xb = fo(xO) into E - C . I n general, for each i < a , l e t x i be the f i r s t e lement of E such t h a t f i ( x i ) # xi , and where xi and x(i = f i ( x i ) are d is t inc t from a l l x and x! ( j < i ) . Such an x i ex is t s , because the number of j < i i s s t r i c t l y less t h a n the cardinal a , and there are (cardinal i ty a)-many x which are different from f i ( x ) , as f i i s inject ive. Finally p u t xi in to C and x i into E - C . 0

0 ( 2 ) The preceding proof modified as follows. F i r s t ly , p u t the f i r s t xo # fo(xO) into C and xb = fo(xO) into E - (Cu 0 ) . Secondly, p u t the f i r s t element

yo # fo(yo) , where yo and fo(yo)ar%oth d is t inc t from xo and xb , into D , and yb = fo(yo) into E - ( C u D) . I n general, for each i < a , f i r s t l y find the f i r s t element xi # f i ( x i ) where both x i and x; = f i ( x i ) are d is t inc t from a l l x j , x!, y . , y j ( j < i ) . Then p u t x i into C and x i in to E - ( C uD) . Secondly consider the f i r s t yi # f i ( y i ) with yi and f i ( y i ) d i s t i n c t from a l l

x j , x i , yj, y j ( j < i ) into D , and p u t

y! = f . ( y . ) into E - ( C u D) . 0

8.3. The reader i s familiar w i t h the notion o f increasing or decreasing ( s t r i c t l y or otherwise) function on the reals . Let E be the s e t of reals and R the i r total ordering, F a subset of E and Ft the smallest interval including F ( w i t h endpoints Inf F and Sup F modulo R ) . For each increasing (mod R ) function f from F into E , there exis ts a t l e a s t one increasing function f + which i s an extension o f f to the domain Ft . If F i s dense for the total ordering R/F+ , then the extension f' of f

is unique. Moreover, i f f i s s t r i c t l y increasing and F *for R/F' , then f' i s s t r i c t l y increasing and hence i s an isomorphism from R/F+ - into R . 0 For each element u of F+-F , l e t f t (u) be the infimum of the f ( x ) for a l l x in F such that x > u . 0

-

_______

j J

J J

as well as from x i , x i . Then p u t yi

1 1 1

Chapter 2 55

8.4. Let E be the s e t of r e a l s , R t h e i r t o t a l ordering, F a non-empty subset of E . The s e t of increasing (mod R ) functions from F into E i s equipotent with the continuum. 0 Assume f i r s t of a l l t h a t F i s an in t e rva l . An increasing function from F i n to E has countably many points of d i scont inui ty , s ince t o these points correspond non-singleton mutually d i s j o i n t i n t e rva l s ( s ee ch.1 5 4.5) . Hence an increasing function i s defined by i t s values on the r a t iona l s f o r example, plus i t s values on the points of d i scont inui ty : a l toge ther making a countable sequence of real values. I n the case of an a r b i t r a r y F , l e t Ft be the smal les t in te rva l including F . To each increasing function f from F i n t o E , assoc ia t e the function f + from F+ i n to E which i s increasing and an extension of f , obtained by the previous proposit ion. The set of i s the s e t of r e s t r i c t i o n s f of f t t o the domain F . 0

f f i s equipotent w i t h the continuum, hence so

8.5. Let R be the t o t a l ordering of the r e a l s , and F a s e t of r ea l s . I f f o r each automorphism h of R which i s d i f f e r e n t from the iden t i ty , t he re exists an element x f F with h(x) # x , then F is dense f o r R . 0 Let a < b be r e a l s . There e x i s t s a t l e a s t one automorphism, o r s t r i c t l y increasing function f from R onto R , sa t i s fy ing f ( x ) = x f o r a l l x 6 a and a l l x ) b , and f ( x ) # x f o r a l l x ( a 4 x G b ) . Hence F must have an e l e - ment between a and b . 0

8.6. INTERMEDIACY, CYCLIC RELATION, CONSECUTIVITY, BINARY CYCLE We f in i sh this chapter by introducing several r e l a t ions frequently associated with par t ia l and t o t a l o rder ings , and used i n l a t e r chapters . Given a pa r t i a l ordering A , we ca l l intermediacy associated with A , the ternary r e l a t ion “z i s between x and y (mod A ) ” ; in o the r words x & z 6 y o r y < z , < x . Given a to t a l ordering A , we ca l l cyc l i c r e l a t ion assoc ia ted w i t h A , the te rnary r e l a t ion taking the value + exac t ly when x s y,( z o r y~ z s x o r z , < x f y (mod A ) . Given a pa r t i a l ordering A , we ca l l consecut iv i ty assoc ia ted w i t h A , the binary re la t ion C such t h a t C(x,y) = t i f f y i s consecutive with x (mod A ) , i . e . y > x and the re does not e x i s t t sa t i s fy ing x < t < y . Given a to t a l ordering A , the binary cyc le , o r cycle o f consecutivity associated with A , i s the r e l a t ion of consecut iv i ty w i t h the following possible modifica- t ion: i f there e x i s t s a minimum u and a maximum v of A , then we p u t C ( v , u ) = + .


EXERCISE 1 - THE GENERALIZED CONTINUUM HYPOTHESIS IMPLIES THE AXIOM OF CHOICE (SIERPINSKI 1947; see also COHEN 1966) 1 - Let A , B be two se ts . If A u B i s equipotent with the s e t ? ( 2 x A ) , then T(A) i s subpotent with B .

0 Let f be a bijection from AuB onto y(2 x A ) = T(A) x ? ( A ) . Each element x of A i s transformed into the ordered pair f ( x ) = (y ,z) of elements of T(A) . To each x associate the f i r s t term y of t h i s ordered pair. By CANTOR'S lemma ch.1 5 1.5, there exis ts an element u of T(A) for which none of the pairs (u,z) i s the image of any x of A . Hence there exis ts a subset of B which i s bijectively transformed by f into the se t ? ( A ) of the second terms z of the ordered pairs (u,z) . 0

2 - For each s e t A , l e t To(A) = A ; for each integer i , l e t Ti+l(A) = ( 9 i ( A ) ) . If A has a denumerable subset, then Pi(A) = 2 x T i ( A ) for each positive integer i . 0 By hypothesis A i s equipotent with A augmented by an element. Hence we have

Tl(A) = T(A) , the s e t of subsets of A ;

P(A) = A2 equipotent with 2 x A2 . I t follows that ? ( A ) i s equipotent with i t s e l f augmented by an element: then we i te ra te . 0

3 - Let A be an inf in i te s e t , assumed t o be equipotent with 2 x A . I f for i = 0,1,2,3, the sets T i ( A ) ly intermediate with respect t o subpotence, then there exis ts a well-ordering of the s e t A . 0 The s e t A i s equipotent with a proper subset of i t s e l f , hence has a denumerable subset: ch.1 5 2 . 6 . Hence Ti(A) i s equipotent with 2 x Ti(A) for each integer i : see ( 2 ) above. Consider the well-orderings based on subsets of A , and the isomorphism cl.asses of these well-orderings. From t h i s point on , denote Ti(A) by Pi . The pairs of elements of A belong t o P1 . The ordered pairs belong t o P2 : see ch.1 5 1.2. The well-orderings based on subsets of A belong t o P3 . Finally the isomorphism classes belong t o P4 . The s e t H of isomorphism classes i s equipotent with the Hartogs aleph o f A : see ch.1 5 6 . 2 ; and i t i s included in P4 . The union HuP3 is intermediate, under subpotence, between P3 and P3 u P4 , the l a t t e r equipotent with P4 , since P3 i s equipotent with a subset of P4 as well as with 2 x P3 . By hypothesis, H u P3 i s equipotent e i ther with P3 . I n the f i r s t case, i t i s equipotent with 9 (2 x P3) , and by the previous (1) hence well-orderable, so A i s well-orderable as well. Consider the second case: H v P3 i s equipotent with P3 . Then H and P2 are bo th subpotent with P3 , hence H u P2 subpotent with 2 x P3 and hence with P3 . By hypothesis H u P 2 i s equipotent e i ther t o PJ or t o P2 . I n the f i r s t case, by the previous ( l ) , the s e t

and (j)i+l(A) do not have any s e t which is s t r i c t -

P4 or with P4 i s subpotent w i t h H ,

P3 i s subpotent with H ,

Chapter 2 51

hence well-orderable, so t h a t A i s wel l -orderable. I n the second case, H u P1 i s subpotent w i t h P2 . By i t e r a t i n g , we ob ta in t h a t H i s subpotent w i t h P1 , hence H IJ A i s equipotent w i t h A o r w i t h P1 . The f i r s t case i s excluded, since the Hartogs aleph o f A i s no t subpotent w i t h A . The previous (1) shows

tha t P1 i s subpotent w i t h H , hence wel l -orderable, and consequently A i t s e l f i s well-orderable. 0

4 - Le t A be an i n f i n i t e se t . Take the union o f A w i t h L.J and l e t B =

F ( A u w ) . Then by (2 ) , we have 2 x B equipotent w i t h B . Using the genera l i zed continuum hypothesis, the statement (3) shows t h a t B i s well-orderable,

so A u W and hence A as wel l .

EXERCISE 2 - THE CARDINAL OF THE SET OF FILTERS

L e t E be an i n f i n i t e se t w i t h card ina l a . We sha l l prove t h a t the cardinal o f the se t o f f i l t e r s , o r as we l l t he card ina l o f the se t o f u l t r a f i l t e r s on E

equals 2 t o the power (a2) (TARSKI; see BELL, SLOMSON 1969 p . 108; uses the

axiom o f choice). Since each f i l t e r i s a se t of subsets o f E , the se t o f f i l t e r s has a t most the

above s ta ted c a r d i n a l i t y . Hence i t su f f i ces t o cons t ruc t a se t o f u l t r a f i l t e r s

on E having t h i s card ina l . For each X we denote by (X) the se t o f f i n i t e subsets o f X . By the axiom o f choice, F ( X ) is equipotent w i t h X f o r each i n f i n i t e se t X . I n the fo l low ing , we ob ta in u l t r a f i l t e r s on F ( ( E ) ) which i s equipotent w i t h E . 1 - Divide E i n t o two complementary subsets A, B o f t he same card ina l a , and l e t f be a b i j e c t i v e mapping from A onto B . To each subset X o f A associate X+ = the union o f X and (B-fo(X)) (notat ion, O from ch.1 5 1.2).

Show t h a t f o r two d i s t i n c t subsets X, Y o f A , ne i the r o f the two images

X+, Y+ i s included i n the other. 2 - To the se t E associate the se t E' = F ( E ) o f a l l f i n i t e subsets o f E . To each subset X o f A associate X ' = E ' - F ( X + ) . Note t h a t the t rans for - mation from X i n t o X ' i s i n j e c t i v e , hence the card ina l o f the se t o f X ' i s a2 . Moreover, f o r X, Y d i s t i n c t subsets o f A , we have X ' , Y ' ne i the r included i n the other.

3 - Le t X be a subset o f A and Y1, ..., Yn ( n i n tege r ) be a f i n i t e se t o f

subsets Y o f A , a l l d i s t i n c t from X . Then the i n te rsec t i on o f the images Y ' = E ' - 3 ( Y ' ) o f the Y i s no t included i n the image X ' f X . Indeed t o each index i = 1, ..., n associate an element g ( i )

The f i n i t e se t o f the g ( i ) i s inc luded i n X+ y e t no t included i n any Y t . Hence i t does no t belong t o X ' = E ' - 3 ( X + ) y e t does belong t o each Y; = E ' - F ( Y f ) , hence belongs t o the i n t e r s e c t i o n o f the Y ' .

I-I_".

o f X+ - (YtnX') .


4 - L e t E " = F ( E ' ) = T( 3 (E ) ) , so t h a t E " has c a r d i n a l a . Denote by t h e s e t o f p reced ing images X ' o f subsets X o f A . Hence has c a r d i n a l a2 . L e t f denote t h e s e t o f a l l F ( X ' ) . Hence where X i s an a r b i t r a r y subset o f A . Thus i s a l s o o f c a r d i n a l a2 . More- over , each element o f i s a subset o f E" . L e t H, K be two f i n i t e non-empty d i s j o i n t subsets o f . Then t h e i n t e r s e c t i o n

A K i s n o t i n c l u d e d i n t h e un ion U H . Indeed H i s a s e t o f F(X;) ( j = 1, ..., n ) w i t h m, n i n t e g e r s . Each X; and Y ! i s t h e image o f a subset

Xi o r Y . o f A , a l l d i s t i n c t . F i x an i ndex i 6 m : by ( 3 ) above, t h e i n t e r s e c - t i o n o f t h e Y ! ( j = 1, ..., n ) i s n o t i n c l u d e d i n X; . Hence t h e r e e x i s t s an

element h ( i ) which belongs t o t h i s i n t e r s e c t i o n and n o t t o X i . The s e t of h ( i ) (i = 1, ..., m) i s a f i n i t e subset o f each Y ' , hence an element o f each

i n c l u d e d i n any o f t h e

ment o f t h e un ion uH . By t h e p rev ious s tatement , each f i n i t e i n t e r s e c t i o n o f elements of empty. I n p a r t i c u l a r , two complementary subsets o f E"

5 - L e t U be a subset o f . Assoc ia te t o i t U+ =, U p l u s a l l complements E" -S , where S belongs t o t h e d i f f e r e n c e s e t 't'- U . F o r two d i s t i n c t U , say U and V , t h e r e e x i s t s f o r example an element S o f which belongs t o U , hence t o U+ , and whose complement E"-S be longs t o V+ . It f o l l o w s t h a t U+ and V+ a r e d i s t i n c t . F o r o the rw ise , two complementary elements would be long t o U+ , these elements be ing o b t a i n e d f r o m two complementary e lements o f f . Thus t h e s e t o f a l l U+ , as t h e s e t o f a l l U , has c a r d i n a l i t y 2 t o t h e power (a2) . Beginning w i t h an a r b i t r a r y non-empty subse t U o f $ , cons ide r an a r b i t r a r y p o s i t i v e f i n i t e number o f elements o f

S1,. . . ,Sm o f U and t h e e lements E"-T1,. . . ,E"-Tn where these T belong t o t h e d i f f e rence s e t if - U . By t h e p rev ious ( 4 ) , t h e i n t e r s e c t i o n o f t h e S i s n o t i n c l u d e d i n t h e un ion o f t h e T , hence t h e i n t e r s e c t i o n o f t h e S and t h e E"-T i s non-empty. Thus t h e r e e x i s t s an u l t r a f i l t e r on E" ex tend ing U+ . TO two d i s t i n c t U correspond two d i s t i n c t u l t r a f i l t e r s , because o f t h e e x i s -

tence of complementary e lements ( w i t h r e s p e c t t o E" ) b e l o n g i n g t o t h e co r res - ponding s e t s U+ . Thus t h e s e t of u l t r a f i l t e r s on E" , j u s t as t h e s e t o f subsets o f f , has c a r d i n a l i t y 2 t o t h e power (a2) . 6 - Modulo t h e g e n e r a l i z e d cont inuum hypo thes i s , we have a much s i m p l e r p r o o f of t h e preceding, y i e l d i n g , f o r a s e t E o f c a r d i n a l a , t h a t t h e s e t o f u l t r a - f i l t e r s on E has c a r d i n a l i t y 2 t o t h e power (a2 ) .

i s t h e s e t o f a l l F ( E ' - F ( X + ) )

(i = 1, ..., m) and K i s a s e t o f T(Y!)

J

J

J

J

j ( Y j ) , hence an element o f t h e i n t e r s e c t i o n n K . However t h i s element i s n o t

X; , hence belongs t o no F ( X ; ) , and so i s n o t an e l e -

i s non-

cannot b o t h be long t o e . -

U+ , by d i s t i n g u i s h i n g t h e elements

Chapter 2 59

To see t h i s , consider a l l par t i t ions of E into two dis joint subsets of cardinal a , and total ly order th i s se t of par t i t ions by i t s cardinal Let uo be the f i r s t par t i t ion, whose two associated subsets shall be denoted E o ( + ) (associated t o the 1-sequence t ) and E o ( - ) . For each of these subsets, E o ( + ) fo r example, take the f i r s t par t i t ion ul(+) such t h a t the intersection of E o ( + ) with each of the subsets of E given by u l ( + ) has cardinal a . Denote by E l ( + + ) and E l ( + - ) these two intersections. Do the same thing for the 2-sequences (-+) and ( - - ) . I n general, f o r each ordinal i s t r i c t l y less t h a n b , note t h a t Card i 6 a by the generalized continuum hypothesis. Fix i and f i x an arbi t rary i-sequence x with values (+) and ( - ) . We have t o define the par t i t ion u i ( x ) . Assume that we have obtained, for each j < i , the sequence of the E.(x. ) where xj+l i s the i n i t i a l interval of x with length j+l . By 8.1, the se t of the E . ( x ) generates a f i l t e r which i s n o t an u l t r a f i l t e r . So there exis ts a par t i t ion u i ( x ) yielding for every intersection of f in i te ly many of the E.(x. ) a s e t of cardinal a . Finally, for each b-sequence of values (+) and ( - ) , we obtain a f i l t e r such t h a t two d i s t inc t b-sequences give two d is t inc t f i l t e r s . Hence there are b2 many such f i l t e r s .

b = a2 .

J J+1 J . j+l

J J+1

EXERCISE 3 - A CLASSICAL PROOF OF HAUSDORFF-ZORN WITHOUT ORDINALS With the notions of partial and total ordering, b u t without using ordinals, we outline as follows the classical proof, using the axiom of choice, of the existence of a maximal chain: HAUSDORFF-ZORN axiom. Let A be a partial ordering, C a to ta l ly ordered restr ic t ion of A . We denote by X any chain which i s simultaneously a res t r ic t ion of A and an extension of C . Suppose t h a t no X i s maximal under inclusion, and l e t f be a choice function which associates t o each X a s t r i c t extension f(X) of X , again a res t r ic t ion of A and an extension of C . We shall obtain a contradiction as follows. Denote by & every s e t of chains X satisfying the following conditions: (1) ‘6 i s to ta l ly ordered under res t r ic t ion (or extension);

( 2 ) i s closed with respect t o taking the union of the bases; hence e admits a maximum element, which i s the common extension of a l l the chains in t o the union of the i r bases; (3) for each X in 6 other than the maximum chain, f(X) belongs t o 8 , and no s t r i c t intermediate chain between X and f(X) belongs t o . There exis t such 6 : for instance the singleton of C and the pair {C,f(C)} . Given two such E , say val of the other ( for the ordering of extension defined between the chains X ) .

and E ’ , we shall prove t h a t one i s an i n i t i a l inter-


Le t 1 be t h e s e t o f X be long ing t o t h e i n t e r s e c t i o n elements o f & which a r e r e s t r i c t i o n s o f X and t h e elements o f E ' which a r e

r e s t r i c t i o n s o f X , a r e t h e same. 3 i s non-empty s i n c e t h e element C be longs t o i t .

De f ine t h e cha in

cha in U belongs t o t h e i n t e r s e c t i o n ,$ A I ' . Moreover, eve ry element X o f

U , e i t h e r i s i d e n t i c a l t o U , o r i s a r e s t r i c t i o n o f a cha in be long ing t o 9 : i n t h i s case X be longs t o 3 . Thus U belongs

E i t h e r U i s t h e maximum o f E , so t h a t & i s an i n i t i a l segment o f % ' ; o r U i s t h e maximum o f f', so t h a t & ' i s an i n i t i a l segment o f

c h a i n f ( U ) be longs t o t h e i n t e r s e c t i o n n E' . Moreover no s t r i c t l y i n t e r - mediate c h a i n between U and f ( U ) be longs t o € , n o r t o & ' ; thus f ( U ) belongs t o U . Hence o u r c l a i m i s proved. We s h a l l prove t h a t t h e r e e x i s t s a maximum

s h a l l t hen o b t a i n a c o n t r a d i c t i o n by c o n s i d e r i n g t h e maximum c h a i n V o f t h e maxi - mum & , and then add ing t h e s t r i c t e x t e n s i o n

maximum and l e t ment t h e c h a i n

It i s s u f f i c i e n t t o prove t h a t e k i s indeed an

t i s f i e s ( l ) , ( Z ) , ( 3 ) . C o n d i t i o n (1) i s obv ious, as w e l l as ( 2 ) s i n c e &' has as l e a s t element. To o b t a i n ( 3 ) : i f X and f ( X ) be long t o & * , and i f a s t r i c t - l y i n t e r m e d i a t e Y between X and f ( X ) be longs t o € * . then t h e r e e x i s t s a t

l e a s t one c o n t a i n i n g as e lements X, f ( X ) and Y ( s i n c e , f o r any two E , one i s an i n i t i a l segment o f t h e o t h e r ) . Then t h i s c o n t r a d i c t i o n .

E fi E' , f o r which t h e

U as t h e common e x t e n s i o n o f a l l elements o f 3 . By ( 2 ) , t h e

E which i s a r e s t r i c t i o n o f

t o 9 .

. O r by ( 3 ) , t h e

? , which c o n t r a d i c t s t h e m a x i m a l i t y o f ( w i t h r e s p e c t t o i n c l u s i o n ) ; we

f ( V ) . So suppose t h a t no &' denote t h e un ion o f a l l t h e & , t o which we add as l a s t e l e -

. I n o t h e r words, t h a t E" sa-

i s

V which i s t h e common e x t e n s i o n o f a l l t h e elements o f a l l t h e E .

V

i t s e l f does n o t s a t i s f y ( 3 ) :

61

CHAPTER 3

RAMSEY THEOREMS, PART I T I ONS , COMB1 NATORI A L P R I NC I PLES

5 1 - RAMSEY'S THEOREM

1.1. We s t a t e f i r s t t h e i n f i n i t a r y fo rm o f t h i s theorem (1926). P a r t i t i o n t h e (unordered) p a i r s o f n a t u r a l numbers i n t o k c lasses ( k f i n i t e ) which we c a l l c o l o r s . Then t h e r e e x i s t s an i n f i n i t e s e t E o f i n t e g e r s such t h a t p a i r s i n c l u d e d i n E have t h e same c o l o r .

T h i s genera l i zes t o t h e case o f m-element se ts , o r s e t s w i t h f i n i t e c a r d i n a l m , which a r e assumed t o be p a r t i t i o n e d i n t o k c o l o r s . There e x i s t s an i n f i n i t e s e t E of i n t e g e r s such t h a t a l l m-element subsets o f E have t h e same c o l o r . The s e t E i s c a l l e d monochromatic. 0 Case o f p a i r s . P a r t i t i o n t h e non-zero i n t e g e r s x i n t o k c lasses acco rd ing

t o t h e c o l o r of t h e p a i r { O,x{ : a t l e a s t one o f t hese c lasses i s i n f i n i t e . L e t

UO = 0 and l e t ul, u2, .. be t h e elements o f t h i s c l a s s . L e t u1 = u1 and pa r -

t i t i o n t h e i n t e g e r s x = up (i 2 2) i n t o k c lasses acco rd ing t o t h e c o l o r o f

t h e p a i r {ul,x} : a t l e a s t one o f t hese c lasses i s i n f i n i t e . L e t u2, u3, .. be

t h e elements o f t h i s c l a s s . L e t u2 = u2 and i t e r a t e . A t t h e end we o b t a i n t h e i n f i n i t e s e t w i t h elements

( i i n t e g e r ) , s a t i s f y i n g t h e f o l l o w i n g c o n d i t i o n . For eve ry i n t e g e r i , t h e p a i r s

\vi,vi+l\ , {vi,vi+?J, .. i n t e g e r i . Then a t l e a s t one c o l o r i s assoc ia ted w i t h i n f i n i t e l y many i n t e g e r s i : a l l p a i r s subsets o f t h e s e t o f vi have t h e same c o l o r . 0

0 Case o f m-element s e t s (m 2 2 ) . Assume t h a t t h e s tatement i s t r u e f o r m-1 . Put as ide t h e i n t e g e r s 0,1,2, ..., m-2 , and p a r t i t i o n t h e i n t e g e r s x 2 m-1 i n t o k c lasses , acco rd ing t o t h e c o l o r o f t h e m-element s e t { O , l ,..., m-2,x) : a t

- -

0 0 0 1 0

. 1 1 1 2 1

1 vo =.O, v1 = ul, ... , v . = u? 1 1

have t h e same c o l o r , which we say i s assoc ia te$ t o t h e

- --

l e a s t one o f these c lasses i s i n f i n i t e . L e t uo 0 = 0, u1 1 = 1, ... , Um-2 m-2 = m-2 ,

and l e t ulIf , u I - ~ , .. be t h e elements o f t h i s c l a s s . L e t

t i t i o n t h e i n t e g e r s x = t~:-~ (i >/ m) i n t o a f i n i t e number o f c lasses, by p u t -

t i n g x and y i n t o t h e same c l a s s i f f f o r each (m-1)-element s e t I w i t h l a s t

element , t h e s e t I augmented w i t h x and I augmented w i t h y y i e l d two m-element s e t s w i t h t h e same c o l o r (which depends on

f i n i t e l y many I , one o f t hese c lasses i s i n f i n i t e : l e t ul- ' , ul;:, .. be t h e

= and pa r -

m- 1

I ) . As t h e r e a r e o n l y

elements o f t h i s c l a s s . L e t u; = u i - ' and i t e r a t e t h i s w i t h t h e (m-1)-element

s e t s w i t h l a s t element u i . A t t h e end we o b t a i n t h e i n f i n i t e s e t w i t h elements

v ~ - ~ = m-l, .. and i n genera l vi = ui f o r each i n t e g e r i , s a t i s f y i n g t h e f o l - l o w i n g c o n d i t i o n . For each (in-1)-element s e t I formed o f elements v and whose

l a s t element i s vi , t h e m-element se ts o b t a i n e d by add ing a v j ( j >i) t o I , a l l have t h e same c o l o r , which we c a l l t h e c o l o r assoc ia ted w i t h t h e (m-1)-element s e t I . By t h e i n d u c t i o n hypothes is , a p p l y t h e theorem f o r m - 1 : we o b t a i n an

i n f i n i t e s e t o f elements v , a l l o f whose ( in-1)-element subsets have t h e same as- s o c i a t e d c o l o r . Hence an i n f i n i t e s e t , a l l o f whose m-element subsets have t h e same c o l o r . 0

vo = uo 0 = 0, v1 = u1 1 = 1, .. 1 i

1.2. Given a r e l a t i o n A , a sequence whose va lues a r e e lements of t h e base I A I s h a l l be c a l l e d a sequence i n A . I f A i s a p a r t i a l o r d e r i n g , t h e n i n c r e a s i n g

and dec reas ing sequences were i n t r o d u c e d i n ch.2 5 2.4; e x t r a c t e d sequence was

i n t r o d u c e d i n ch .1 5 2.2. L e t u be an a -sequence i n a p a r t i a l o r d e r i n g A . Then t h e r e e x i s t s an w-sequen-

ce e x t r a c t e d f rom u , which i s constant , o r s t r i c t l y . i n c r e a s i n q , o r s t r i c t l y de-

c reas ing , o r c o n s i s t i n g o f e lements which a r e p a i r w i s e incomparable (mod A ) . 0 P a r t i t i o n t h e s e t o f p a i r s {i,j) o f i n t e g e r s i n t o f o u r c lasses ( p u t i < j f i x t h i n g s f o r d i s c u s s i o n ) , by d e f i n i n g a f i r s t c l a s s by t h e e q u a l i t y ui = u

a second by ui u . (mod A) , and a f o u r t h by

ui I u j (mod A) ; then a p p l y RAMSEY's theorem. 0

t o

j '

J

1.3. FINITARY FORM OF RAMSEY'S THEOREM; MONOCHROMATIC SET

Given t h r e e i n t e g e r s m, k, p 5 m , t h e r e e x i s t s an i n t e g e r

f o r eve ry s e t o f c a r d i n a l

k c o l o r s , t h e r e e x i s t s a p-element subset, a l l of whose m-element subsets have t h e same c o l o r . It i s c a l l e d a monochromatic p-element subset .

Consider t h e case m = 2 . Repeat t h e p r o o f o f 1.1, b u t i n s t e a d o f "one of t h e

c lasses i s i n f i n i t e " , say "one o f t h e c lasses i s l a r g e " , meaning t h a t i t con ta ins

a t l e a s t

= p.kkp i n o r d e r t o o b t a i n , a f t e r kp -1 opera t i ons , a sequence o f l e n g t h >/ kp o f elements v , analogous t o those i n 1.1. Thus we have a l a r g e c l a s s o f v , o f

c a r d i n a l i t y >/ p . 0

p+>/ p such t h a t , t

>/ p , whose m-element subsets a r e p a r t i t i o n e d i n t o

l / k o f t h e o r i g i n a l e lements. 'It s u f f i c e s t o t a k e p+ = (kp).k(kp-l)

Chapter 3 63

1.4. RAMSEY NUMBERS

The l e a s t such p+ i n t h e p reced ing p r o p o s i t i o n i s c a l l e d a Ramsey number, denoted

(p): . p laced by = o r < o r > , e t c . We g i v e seve ra l va lues .

Case m = 1 . I f each o f t h e k c lasses had p -1 elements, t h e e n t i r e s e t would

have k . (p-1) e lements. Hence i t s u f f i c e s t o t a k e (p): = k (p -1 ) + 1 t o o b t a i n

a t l e a s t one c l a s s w i t h p e lements. T h i s argument i s c a l l e d t h e "p igeonhole

p r i n c i p l e " : i f k (p -1 ) + 1 o b j e c t s a r e p a r t i t i o n e d i n t o k p igeonholes, t hen

a t l e a s t one o f t h e p igeonholes has p o b j e c t s .

- Case k = 1 : a s i n g l e c l a s s , t hus

- Case p = m : a p-element s e t i s necessary monochromatic, t hus

Th is l ooks i k e t h e usual Erdos-Rado n o t a t i o n , where t h e arrow w i l l be r e -

-

(p); = p . (m): = m .

C a l c u l a t i o n o f 0 Consider t h e e lements 1,2, ..., 6 ; p a r t i t i o n t h e edges (1,2) t o (1,6) i n t o

two c o l o r s . A t l e a s t one c o n t a i n s t h r e e edges ( l , a ) , ( l , b ) , (1,c) . E i t h e r (a,b) o r (b,c) o r (c,a) has t h e same c o l o r , o r t hese t h r e e edges have t h e oppos i te

c o l o r : t h i s shows t h a t ( 3 ) 2 i 6 . To see t h a t ( 3 ) 2 > 5 , t a k e t h e usual pentagon w i t h one c o l o r , and t h e s t a r r e d

pentagon w i t h t h e o p p o s i t e c o l o r . 0

C a l c u l a t i o n o f (3); = 17 (GLEASON, GREENWOOD 1955). 0 Consider t h e elements 1,2,. ..,17 and p a r t i t i o n t h e 16 edges (1,2) t o (1,17)

i n t o t h r e e c o l o r s . A t l e a s t one c o n t a i n s 6 edges, say remains t o p a r t i t i o n t h e edges f a l l back t o t h e case ( 3 ) 2 = 6 ; t h i s shows t h a t

The f o l l o w i n g counterexample shows t h a t o f t h e i n t e g e r s 0 and 1 w i t h

the r i n g o f po l ynomia l s on t h i s f i e l d w i t h t h e i d e n t i t y x4 = x + l . Th is r i n g i s

composed o f 16 e lements These elements a r e e x a c t l y O,l,x,x , . . . ,x14 (we have x15 = 1). Every non-zero

element i s a power xi (i = O , l , ..., 14) , and has i n v e r s e x15-i . Hence t h i s r i n g i s a f i e l d . P a r t i t i o n t h e p a i r s o f po l ynomia l s i n t o t h r e e c o l o r s , acco rd ing t o whether t h e d i f f e r e n c e o f these two po lynomia l s i s a cube x3" ( u = O , l , ..., 4) o r

i s o f t h e fo rm x3'+' o r x3'+' . It s u f f i c e s t o see t h a t t h e sum o f two non-zero cubes i s n o t a cube. 0

(3); = 6 .

2 2

(l,al), ... , ( l ,a6 ) . It (ai,a.) ( i , j = 1 t o 6) i n t o two c o l o r s : hence we

2 J 2 ( 3 ) 3 6 17 .

( 3 ) 3 > 16 . Consider t h e f i e l d composed ( t h e f i e l d o f t h e i n t e g e r s modulo 2) , and

2

1+1 = 0

2 3 (0 o r 1) + ( 0 o r 1 ) . x + ( 0 o r 1 ) . x + (0 o r l ) . x . 2

1.5. L e t E be a f i n i t e se t ; p a r t i t i o n i t s m-element subsets i n t o k c o l o r s

u1 ,..., uk . Given k i n t e g e r s p1 ,..., pk 3 m , by (pl ,..., pk)m we denote the l e a s t c a r d i n a l o f E f o r which t h e r e e x i s t s e i t h e r a pl-element subset

w i t h c o l o r u l , ... , o r a pk-element subset w i t h c o l o r uk . The func - t i o n (p ,,..., pk)m i s symmetric. Moreover, t a k i n g p =

t h e ramsey number (p)! = (p,. . . ,p) .

C a l c u l a t i o n o f 0 We show t h a t t h i s number i s E i t h e r among t h e 8 edges (1,2) t o (1,9) t h e r e e x i s t 4 edges o f c o l o r ( + ) . T h i s then y i e l d s e i t h e r a 3-element s e t w i t h c o l o r (+) o r a 4-element s e t w i t h c o l o r ( - ) , O r t h e r e e x i s t 6 edges w i t h c o l o r ( - ) , which then y i e l d s e i t h e r a 3-element subset

(+) o r a 4-element subset ( - ) . Or f i n a l l y none o f t h e p reced ing cases i s r e a l i z e d

f o r any o f t h e p o i n t s 1 th rough 9 . Then f rom each p o i n t t h e r e emanate e x a c t l y

3 edges (+) and 5 edges ( - ) . Bu t t h i s i s imposs ib le , s i n c e we would then have

= pk = p, we o b t a i n 1 ..*

m

(3,412 = 9 . 9 . J o i n up t h e i n t e g e r s 1 th rough 9 by edges.

3 . (9 /2) = 27/2 edges (+) .

(3,4) > 8 . Take t h e i n t e g e r s 0 t o 7, and g i v e t h e c o l o r (+) 2 We now show t h a t

t o t h e edge (x,y) i f f t h e abso lu te va lue o f y -x i s 3, 4, or 5. 0

C a l c u l a t i o n o f ( 4 ) 2 = 18 . 0 Take t h e i n t e g e r s 1 t o 18. Among t h e edges emanating f rom 1, t h e r e a r e a t l e a s t

9 o f t h e same c o l o r which we des igna te (+ ) . They j o i n 1 t o t h e i n t e g e r s des igna ted

al,. . . , ag . By t h e preceding, i n t h e s e t o f al, ..., ag 3-element s e t w i t h c o l o r (+) , o r a 4-element s e t o f t h e o p p o s i t e c o l o r ( - ) . Hence t h i s Ramsey number i s a t most 18.

The f o l l o w i n g ewample w i l l p rove t h a t t h e Ramsey number i s n o t \< 17. Take t h e

i n t e g e r s modulo 17, so 0 t o 16. For any two d i s t i n c t x, y i n t h i s se t , we g i v e t o t h e edge (x,y) t h e c o l o r (+) i f f x i s congruent t o y modulo a q u a d r a t i c

res idue , so mod 1, +, 2, f 4, o r 28 ; t h e c o l o r ( - ) i n t h e o p p o s i t e cases.

Suppose t h a t t h e r e e x i s t 4 i n t e g e r s a,b,c,d such t h a t a l l t h e i r edges have t h e same c o l o r . We can r e p l a c e these i n t e g e r s by a-d, b-d, c-d and 0 , and thus

can cons ide r o n l y t h e case o f O,a,b,c . We can r e q u i r e t h a t t h e 6 i n t e g e r s a, b , c, b-a, c-b, a-c be non-zero and e i t h e r a l l r es idues o r a l l non-res idues. M u l t i - p l y i n g by t h e i n v e r s e o f a , we can reduce t h i s t o t h e case o f 0, 1, b, c w i t h t h e 5 i n t e g e r s b, c, b-1, c-1, c-b which a r e a l l non-zero and a l l q u a d r a t i c res idues . Then t h e p o s s i b i l i t i e s f o r b and c a r e reduced t o -1, +2, -8 . For b = -1 , we have c # -1 , c # 2 s i n c e c-b # 3 . Moreover c # -8 s i n c e

c - b # -7 . The same argument f o r c = -1 : imposs ib le . Fo r b = 2 and c = -8 we o b t a i n b-c = -7 : imposs ib le . 0

2

t h e r e e x i s t s e i t h e r a

C a l c u l a t i o n o f (3,5)L = 14 . 0 Take a s e t w i t h 14 p o i n t s and l e t a be i n t h i s s e t . E i t h e r f rom a t h e r e emana te a t l e a s t 5 edges w i t h c o l o r (+), which y i . e lds a 3-element s e t monochromatic

Chapter 3 65

w i t h c o l o r (+ ) , o r a 5-element monochromatic s e t w i t h c o l o r ( - ) . O r f r om a

= 9 , t h i s y i e l d s e i t h e r a (+)-monochromatic 3-element se t , o r a (-)-monochroma- t i c 5-element s e t . Thus (3,5)2 i s bounded above by 14.

To see t h a t i t equals 14, t ake t h e 13 i n t e g e r s 0 th rough 12. Give t h e c o l o r (t) t o

the p a i r {x,y) (where x and y a r e d i s t i n c t elements among O , l , ..., 12) i f f

the abso lu te va lue o f y -x equals 2, 3, 10 o r 11; c o l o r ( - ) i n o t h e r cases. 0

Other known ( o r a lmost known) va lues o f b i n a r y Ramsey numbers:

(3,6)2 = 18 : KALBFLEISCH 1964; i ndependen t l y KERY 1964;

t h e r e emanate a t most 4 edges (+) , hence a t l e a s t 9 edges ( - ) . Then s i n c e (3,4) 2

n

(3,7)L = 23 : GRAVER, YACKEL 1968;

(3,8) = 28 o r 29 ; a l s o (3,9) = 36 : GRINSTEAD, ROBERTS 1982. 2 2

1.6. Below we l i s t some i n e q u a l i t i e s f o r t h e s m a l l e s t Ramsey numbers whose exac t va lue i s n o t known.

B ina ry numbers. 25 ,< (4,5)[ 4 28 : lower bound o f 25 by KALBFLEISCH 1964; upper bound o f 28 by WALKER 1971.

42 6 (5); ,< 55 : lower bound o f 42 by I R V I N G 1973; a l s o G A R C I A 1975; a l s o HANSON

1976; upper bound o f 55 by WALKER 1971.

128 4 ( 4 ) 3 d 254 : lower bound o f 128 by HILL, I R V I N G 1982; upper bound o f 254 e a s i l y ob ta ined v i a (2,4,4) = (4,4) = 1 8 and (2,3,4) = (3,4) = 9 ; then

( 3 , 3 , 4 ) 2 6 34 and

51 6 (3); 6 65 : lower bound o f 51 by CHUNG 1973; upper bound of 65 by FOLKMAN

1974, a l r e a d y announced by WHITEHEAD 1973. 159 6 ( 3 ) 5 < 322 : lower bound 159 by FREDRIKSON 1979; t h e upper bound i s easy. S t i l l c o n s i d e r i n g b i n a r y Ramsey numbers, we have t h e easy i n e q u a l i t y :

(3 )k 5 k ( (3 )k -1 - 1) + 2 , which y i e l d s

I n t h e o t h e r d i r e c t i o n , we e a s i l y o b t a i n

on Schur ’s numbers. Improved by 2 2 (3 )4 >/ 3. (3)3 = 3.17 = 51 ); see CHUNG 1973.

2 Moreover ( ~ + 1 ) ~

2

2 2 2 2

(3,4,4)2 6 85 .

2

2 2 2 ( 3 ) k 4 k !e + 1 2 k

( 3 ) k 3 ( 3

( 3 ) k + 1 3 3 . (3 )k + (3)k-2 - 3

( c l a s s i c a l number e ) .

+ 3 ) / 2 : see e x e r c i s e 1 2 2 2 ( f o r example

i s bounded above, up t o a c o n s t a n t f a c t o r , by t h e f o l l o w i n g

r a t i o :

Ternary number. 13 4 (4); 6 15 : lower bound o f 13 by ISBELL 1969; upper

bound o f 15 by GIRAUD 1969.

(2p) ! (Log Log p ) / (p! ) [Log p : see YACKEL 1972.

§ 2 - LEXICOGRAPHICALLY ORDERED SET, GALVIN'S INITIAL INTERVAL THEOREM, NASH-W I LLI AMS ' THEOREM

2 . 1 . LEXICOGRAPHICALLY ORDERED SET, LEXICOGRAPHIC RANK Totally order the s e t of f i n i t e sets of integers lexicographically, by f i r s t difference: ( s e t a ) ,< ( s e t b ) i f f the leas t integer in a i s s t r i c t l y less than the leas t integer in b ; or in the case of equality, compare the second l e a s t integer of t o precede a l l other se t s in th i s ordering. Finally i f a i s a proper i n i t i a l interval o f b , we p u t a < b . A s e t i f the lexicographic ordering of elements o f i s a well-ordering. The corresponding order type i s called the lexicographic rank of F . I t i s a countable ordinal. For example, the s e t of singletons of integers i s lexicographically well-ordered with rank w, the s e t of pairs with rank U2 . The s e t o f a l l f i n i t e sets of integers i s n o t lexicographically well-ordered. Indeed we obtain a lexicographically decreasing W-sequence, by taking the singleton i l } , then the pair { 0 , 2 ) ,

then {0,1,3} , then {0,1,2,4}

Let (1) the elements of ( 2 ) every inf in i te s e t of integers includes a subset which belongs t o 3; then Notice tha t ( 2 ) alone i s not suff ic ient for t o be lexicographically well-ordered: take the s e t of a l l the f i n i t e se t s of integers; (1) alone i s not suff ic ient : take the above decreasing W-sequence. 0 Consider a non-empty subset 5 of F , and show that there ex is t s a minimum element in 5 for the lexicographic ordering. Let a. be the leas t integer such t h a t there exis ts an element of 5 belongs t o 3 , then i t i s the minimum of 5 . Otherwise, take the elements of

which begin with a. , and l e t al be the leas t integer such t h a t there exis ts an element of 5 beginning with ao, al . If the pair {ao,al} belongs t o

then we obtain an in f in i te increasing sequence a. < al < . . . < ai < . . . ( i integer). By our hypothesis ( 2 ) , there exis ts a f i n i t e s e t composed of certain of the a i and belonging t o : denote by ah the l a s t among these. There exis ts also a f i n i t e set beginning with ao, a l , ... , ah and belonging t o 5 , hence t o F. B u t th i s contradicts our hypothesis ( I ) of incomparability. 0

a with the second leas t integer of b , e tc . The empty s e t i s defined

of f i n i t e se t s of integers i s said t o be lexicographically well-ordered,

, e tc . be a s e t of f i n i t e se t s of integers, such that :

are mutually incomparable under inclusion;

i s lexicographically well-ordered (communicated by POUZET in 1980).

beginning with a. . I f the singleton 4 ao)

9 , then i t i s the minimum of 5 . Otherwise, i f th i s procedure never terminates,

Chapter 3 67

2 . 2 . INITIAL INTERVAL THEOREM Let p be a s e t of f i n i t e se t s of integers, such that every inf in i te s e t of integers includes as a subset a t l eas t one element of F . Then there exis ts an inf ini te se t E of integers such that every inf in i te subset of E has an ele- .- ment of the axioms of ZF , i s due t o POUZET 1980, unpublished). Note f i r s t that RAMSEY's theorem easi ly follows from the preceding statement. Indeed, par t i t ion the pairs of integers into two colors (+) and ( - ) . Then either there ex is t s an in f in i te s e t of integers a l l of whose pairs belong t o (+), or every inf in i te s e t includes an element of color ( - ) . Then by the above statement, there exis ts an inf in i te se t such that every inf in i te subset of E begins with a pair belonging to ( - ) : in other words every pair belongs t o ( - ) . To simplify the proof, note that i t i s always possible t o assume that the elements of are mutually incomparable with respect t o inclusion. Indeed, starting with an arbi t rary 3, we obtain the subset To by taking those elements of which are minimal with respect t o inclusion. Every inf in i te se t of integers includes a t l eas t one element of F' . The i n i t i a l interval theorem, when restr ic ted by the preceding condition, says that there exis ts an inf ini te set E , such that every inf in i te subset of E has an element of 3" , hence of F , as an i n i t i a l interval. By the preceding 2 . 1 , we see t h a t i t suffices t o prove the i n i t i a l interval theorem for an arbitrary lexicographically well-ordered s e t (whose elements will no longer necessarily be incomparable under inclusion). Thus we are led t o prove the following statement.

a s a n i n i t i a l interval ( G A L V I N 1968; the following proof, using only

E

2.3. Let (3' could contain the empty s e t as an element); then: (1) ei ther there ex is t s an in f in i te s e t of integers which includes no element

( 2 ) or there exis ts an in f in i te s e t of integers, each of whose inf in i te subset has an element of 3- as i n i t i a l interval (the empty s e t i s considered as an in i t ia l interval of every s e t ) . 0 We argue by induction on the lexicographic rank of F . Suppose f i r s t that the rank i s equal to 1 , so that 3 i s the singleton of a f i n i t e se t F of integers. Then the inf in i te s e t of a l l integers not belonging t o F sa t i s f ies our conclusion (1) i f F i s non-empty, our conclusion ( 2 ) i f F i s empty. Let o( be a countable ordinal. Suppose the statement i s true for every s e t of lexicographic rank < o(. We shall prove i t f o r every s e t of rank n( .

be a lexicographically well-ordered s e t of f i n i t e subsets of integers

of F-;

More strongly, in order to avoid use of the axiom of choice, or even a weakened form of choice, suppose t h a t there exists a function h which, to each ordered p a i r (E,F) where E i s an infinite set of integers, If a set of f in i te subsets of E of lexicographic ranks < I% , associates an infinite set h ( E , F ) s E satisfying one of the conclusions (1) or ( 2 ) . More precisely, either h ( E , y ) includes as a subset no element of 5 , or every infinite subset of h ( E , F ) begins with an init ial interval which belongs t o F . We will prove that there exists an analogous function for the ordered pairs whose second term lexicographic rank d . T h u s h will be progressively extended t o al l countable lexicographic ranks, so t o a l l ordered pairs S t a r t with an infinite se t E of integers and a set of f in i te subsets of E with lexicographic rank O C . For each integer i of E , denote by Fi the subset of those elements of which begin w i t h i . Hence i s the union of the Fi and the lexicographic ordering o( of 3;- i s the sum along w of the lexicographic orderings of the Fi . If there exist infinitely many integers i of E with Fi empty, then the set of these i does not include as a subset any element of F . This will be, by definition, h ( E , F ) which then verifies our conclusion (1). Consider the other case, and l e t m(0) be the least integer of E after which

Fi and by Mi the set Mo with i t s minimum m(0) lexicographic ranks t d , hence the function h i s already defined for these. We p u t $ = Fm(o) and then 5 0 = the set of the elements of go , each of whose m i n i m u m m(0) s t i l l < e( . P u t El = h(Mg, 5 0) 2 Mi . Either there exist infinitely many integers i of El for which El includes no element of Ti . Then the set o f these i i s by definition h ( E , g ) , which satisfies our conclusion (1). Or i n the opposite case, l e t m(1) point on every Ti restricted t o i t s elements which are subsets of El i s never empty. Denote by M1 the set of integers of El which are >/ m(1) , and Mi the set M1 with i t s minimum m(1) removed. Let zl = Fm(l) res-

tricted to elements which are subsets of M1 . Then 5 i = the set of the ele- mnts of

and of 5 Let E2 = h(M;, 5 i) 5 Mi , and iterate this procedure. Then either, a t the end of a f in i te number of steps, we obta in an Er ( r integer) with infinitely many integers i of E r f o r which no element of Fi i s a subset of E r . Then by definition, we p u t h ( E , $ ) t o be the set of these i which satisfies (1). Or this described process continues indefinitely: we must consider two subcases.

has

(E,F) .

i s never empty. Denote by Mo the set of integers of E which are >,m(O) removed. The Ti all have

has been removed: the lexicographic r a n k of this 5 ; i s

be the least integer of El , from which

$ 1 each w i t h i t s minimum removed: the lexicographic rank of

i s s t r ic t ly less than o( .

Chapter 3 69

First subcase. There ex is t in f in i te ly many integers k for which the set Ek+l = h(Mi , 9 k ) contains as a subset no element of 9 corresponding se t of minimums m ( k ) . Then m ( k ' ) tZ Ek+l Mk for a l l k , k ' > k . So the se t of minimums m(k) element of F m ( k ) for any k . Finally our se t of m ( k ) contains as a subset no element of 8 : we take i t as our definition of

In the second subcase, because of the definition of the function h , there exis ts an integer ko from which point on, every inf in i te subset of Ek+l begins by a possibly empty element of % m ( k ) for k >/ kg , then every inf in i te subset of K begins by an element of a

sat isf ies our conclusion ( 2 ) . 0

. Take the inf in i te

contains as a subset no element of % K , hence no

h ( E , 7 ), which sa t i s f ies (1) .

. A f o r t i o r i , i f K denotes the se t of minimums

k , hence by an element of F . Thus the s e t K , which we take for h(E, 3 ) ,

2.4. NASH-WILLIAMS' THEOREM (1965) Consider two dis joint se t s 5 k of f i n i t e se t s o f integers. Suppose t h a t no element of F i s an i n i t i a l interval o f an element of

of F , orwhich contains as a s u b T t no element of t h a t , for two d i s t inc t elements of of the other. The present stronger statement resul ts from a remark by HODGES.

Motice t h a t RAMSEY's theorem follows. Indeed, given two d is t inc t pairs of integers, or in general, for p a fixed integer, given two d is t inc t p-element sets of integers, one i s never an i n i t i a l interval of the other.

0 Either there exis ts an inf in i te s e t o f integers having no subset which i s an element of 3 . Or every inf in i te se t of integers has a sirbset which i s an element of F . I n the l a t t e r case, by GALVIN's theorem 2 .2 , there exis ts an inf in i te se t

i n i t i a l interval . Then E has no subset which i s an element of . Indeed, i f i t contained as a subset an element G of , then take an in f in i te subset X of E with i n i t i a l interval G . There exis ts an element F of which i s an i n i t i a l interval of X : thus F i s an i n i t i a l interval of G , or G an i n i t i a l interval of F : contradiction. 0

, and vice-versa. Then there exis ts an inf in i te se t E of integers, which con 9 ains a s a subset no element pl_l___

. NASH-WILLIAMS only assumes - 9 F w 5 , one i s never an i n i t i a l interval

E of integers, such t h a t every inf in i te subset of E has an element of T as

9 5

5 3 - UNCOUNTABLE CASE, PARTITION THEOREMS: DUSHNIK, MILLER, E R D ~ S , RADO 3.1. SIERPINSKI 'S COUNTEREXAMPLE (1933) There ex is t s a partition of the pairs of reals into two colors, such t h a t every monochromatic s e t i s countable (uses axiom of choice).

0 Take a w e l l - o r d e r i n g A o f t h e s e t o f r e a l s . Then t o each p a i r o f r e a l s x, y

w i t h x c y i n t h e usual o r d e r i n g , g i v e t h e c o l o r (+) i f x < y (mod A) and

t h e c o l o r ( - ) i f x > y (mod A) , The p r o p o s i t i o n f o l l o w s f rom t h e f a c t t h a t eve ry s t r i c t l y i n c r e a s i n g ( o r s t r i c t l y decreas ing) sequence o f r e a l s i s coun tab le

(see ch.1 5 4.5) . 0

As a p a r t i c u l a r case of 3.4 below, we ment ion he re t h a t , f o r eve ry p a r t i t i o n o f

t h e p a i r s o f elements of a s e t o f c a r d i n a l

monochromatic subset o f c a r d i n a l a1 (uses axiom o f cho ice p l u s t h e cont inuum hypo thes i s ) . We can summarize t h i s s i t u a t i o n by u s i n g t h e n o t a t i o n f o r Ramsey numbers w i t h

f i n i t e o r i n f i n i t e c a r d i n a l va lues. Then t h e usual Ramsey theorem i s w r i t t e n

( w ); = W w i t h 3.4 y i e l d s

i n t o two c o l o r s , t h e r e i s a

f o r a l l i n t e g e r s m, k . The p reced ing p r o p o s i t i o n complemented 2

( w 1 ) 2 = Ld2 .

3.2. PARTITION LEMMA (DUSHNIK, MILLER 1941) L e t A = C d N be an i n f i n i t e r e g u l a r a leph. P a r t i t i o n t h e p a i r s o f e lements o f A i n t o two c o l o r s which we des igna te by (+) and ( - ) . Then e i t h e r , f o r eve ry subset B which i s e q u i p o t e n t w i t h A , t h e r e e x i s t s an

element a of B and a s e t o f elements x f B which i s e q u i p o t e n t w i t h A , where a l l t h e p a i r s {a,x) have c o l o r ( - ) . O r t h e r e e x i s t s a subset o f - A equ ipo ten t w i t h A , a l l o f whose p a i r s have c o l o r (+) . 0 Assume t h e r e e x i s t s a subset B of A e q u i p o t e n t w i t h A , which negates t h e

f i r s t conc lus ion . Consider t h e elements o f B orde red by t h e usual o r d e r i n g on t h e o r d i n a l s . L e t a. be t h e minimum o f B ; by hypo thes i s t h e r e e x i s t s an

al > a,, i n B , such t h a t f o r eve ry x >, al t h e p a i r { aO,x 1 has c o l o r (+). By i n d u c t i o n , g i v e n i C w* , assume t h a t f o r t h e j < i we have a s t r i c t l y i n c r e a s i n g sequence o f e lements a o f B , such t h a t a l l p a i r s o f t h e a . have

t h e c o l o r (+) . For eve ry j < i , t h e r e a r e l e s s than wd many x i n B such t h a t t h e p a i r { a j , x ) has c o l o r ( - ) . S ince u4 i s r e g u l a r , t h e s e t o f a l l such x f o r a l l j , ai and a l l

j J

j j c i , t h e p a i r (a j , x ) has t h e c o l o r (+) . F i n a l l y we o b t a i n an

C30(-sequence o f elements, a l l o f whose.pai rs have c o l o r (+) . 0

N o t i c e t h a t t h e p r o p o s i t i o n i s f a l s e f o r e v e r y s i n g u l a r c a r d i n a l r~~ . Indeed

l e t T < W , be t h e c o f i n a l i t y of . For eve ry i < , t a k e a subset

Ai o f A = W 4 , such t h a t t h e un ion o f t h e Ai i s A , b u t eve ry Ai i s s t r i c t l y subpotent w i t h Ai , g i v e t h e c o l o r

( - ) , and f o r p a i r s o f e lements be long ing t o d i s t i n c t Ai ' s , t h e c o l o r (+ ) .

A . For eve ry p a i r con ta ined i n an

Chapter 3 71

3.3. PARTITION THEOREM (DUSHNIK, MILLER 1941) Let A be an arbi t rary inf in i te s e t ; partition the pairs of elements of A into two colors (+) and ( - ) . Then ei ther there exis ts a denumerable subset of A which i s (-)-monochromatic, or there exis ts a subset of A which i s equipotent w i t h A fi (+)-monochromatic (uses axiom o f choice). DUSHNIK and M I L L E R mention the influence of ERDOS. A different proof of the theorem i s given by ERDOS, RADO 1956. Using Ramsey numbers notation and replacing A by an aleph w, , we have

Replace A by an aleph which we designate by &>,;,(axiom of choice), and assume f i r s t t h a t th i s aleph i s regular. By the preceding lemna, i f our second conclusion is fa l se , then there exis ts an a. in A for which the s e t A. o f

x 7 a. (mod A ) such that {ao ,x] has color ( - ) i s equipotent with A . Take th i s a. minimum (mod A) . Then replace A by A. , t h u s yielding an element al of A. satisfying the same condition and taken minimum. By i te ra t ion , we obtain an w-sequence o f elements elements have color ( - ) . Assume now that A = ud i s singular. Then c4 i s a l imit ordinal (ch.2 9 6.2, axiom of choice). Let x+ o(. Thus M, i s the ordinal l imit of the r-sequence w , where i < 2( and g ( i ) < o( . Moreover, we can choose the LX ( i ) t o be s t r i c t l y increasing with i , and every o \ ( i ) > 8 . Finally every w g ( i ) can be assumed t o be regular, replacing i f need be d ( i ) Suppose t h a t the f i r s t conclusion f a i l s : there i s no denumerable subset o f A a l l of whose pairs have color ( - ) . Then by the preceding, there ex is t s a subset B of A , equipotent with A , such that for every x in B , there are s t r i c t l y less t h a n Q,, many y in B with { x,y) having color ( - ) . For every subset X of B , denote by M ( X ) the se t of elements of B-X which, together w i t h a t l eas t one element of X , have color ( - ) . Let U be any subset of B equipotent with A , and l e t i be an ordinal s t r i c t l y less than 8 . We shall show that there ex is t s a subset W of U with cardinal IA

satisfying the two following properties: every pair of elements of W has color (+) ; the se t M ( W ) has cardinal s t r i c t l y less t h a n ma. Indeed, by our f i r s t paragraph and because for every i s regular, there exis ts a subset V of U with cardinal wNc(i) , a l l of whose pairs have color (+) . For every j < d , denote by V elements B - i x ] which together with x have color ( - ) . Then V = \I V . ( j < b') since, by our third paragraph , no x together with wd many elements of B , has color ( - ) , and i s the limit of the

( ~ 3 , w W ) ' = L&, .

ai ( i integer) , a l l of whose pairs of

be the cofinal i ty of un , SO r< UUc and

m ( i )

by i t s successor.

M ( i ) '

i < 8 the cardinal L3 % ( i )

the se t of j

x of V , such t h a t there ex is t a t most woc(j) elements in

J

(j C I f ) . Recall t h a t the

card ina l wec(i) o f V i s regu la r and s t r i c t l y g rea ter than . It fo l lows t h a t there e x i s t s a t l e a s t one o rd ina l k 6 8 w i t h Vk equipotent w i t h V : pu t W = Vk . Then Card M(W) ,< W,c(i). L3 o ( ( k ) < do(. Thus the two proper t ies s ta ted above f o r W are obtained. It remains t o cons t ruc t a subset o f B which i s equipotent w i t h B and thus w i t h A , a l l o f whose p a i r s have co lo r (+) . L e t W1 be a subset o f B w i t h card ina l w less than ~3~ . I t e r a t e by tak ing W2 a subset o f B - (W1 u M(W1)) w i t h car- d ina l G) o((2) , a l l o f whose p a i r s have c o l o r (+) , w i t h Card M(W2) less than ucA . Note tha t , i n the union W1u W2 , a l l t he p a i r s have co lo r (+). Le t i < 8 and assume t h a t the Wi are def ined f o r j , Wd , w i t h Cdo((j) < and Card M(Wj) < c 3 ~ f o r each j < i . Hence the c o f i n a l i t y o f wOc would be 6 i < 8 , con t rad ic t i ng the f a c t t h a t 8 i s the c o f i n a l i t y o f a l l j < i) : t h i s has c a r d i n a l i t y ad . Hence there e x i s t s a subset pe r t i es : Card Wi = CJ Card M(Wi)

Wi ( i < 8 ) Example. For u1 we ob ta in e i t h e r a denumerable a (+)-monochromatic subset having card ina l W l . With the no ta t i on o f Ramsey

a1 - numbers: ( a, =

3.4. PARTITION LEMMA (ERDOS 1942) L e t oa be an i n f i n i t e aleph. Set A = WM+2 continuum hypothesis i n the form (

P a r t i t i o n the p a i r s o f elements o f A i n t o two co lo rs (+) and (-) . Then e i t h e r there e x i s t s a e x i s t s a (-)-monochromatic subset w i t h card ina l w . See E R D b 1942 o r ERDOS, RADO 1956. With Ramsey numbers: 0 We prove f i r s t t h a t there e x i s t s a monochromatic subset o f A w i t h card ina l

w i t h card ina l CC,

We say t h a t a sequence of terms ai i n A (i ord ina l ) i s pre-monochromatic i f f for every index i , the c o l o r o f the p a i r s {ai,aj) remains the same f o r a l l j > i : we say t h a t the c o l o r i s associated t o the index i . Construct as fo l lows

, a l l o f whose p a i r s have co lo r (+) , w i t h Card M(W1) s t r i c t l y o((1)

s t r i c t l y

j < i )

d ( j ) J

. For U take the di f ference B - u ( W . u M(Wj) f o r J

Wi w i t h a l l p a i r s o f Wi having c o l o r (+) , and

o f t h i s d i f fe rence which s a t i s f i e s the two pro-

M ( i ) s t r i c t l y less than L,J~ . It remains t o note t h a t the union o f the has card ina l W, and t h a t a l l i t s p a i r s have co lo r ( t ) . 0

(-)-monochromatic subset, o r

and assume the general ized

i ) 2 = c3 i+l f o r every i .4< d .

(+)-monochromatic subset o f A w i t h card ina l ~ 3 ~ + ~ , o r there

0(+2 2 %+2) = c30(+2 . ( w o(+l,

. Take ‘4 = 0 , so t h a t A = W 2 : we s h a l l ob ta in a monochromatic se t o(+l . The proo f w i l l e a s i l y extend t o the general case.

Chapter 3 73

such a pre-monochromatic ul-sequence. We then immediately extract a monochroma- t i c GJ l-sequence, by taking a l l those indices i with the same color, provided t h a t th i s se t i s cofinal in CJ . Let a. denote the minimum of A ( in the usual well-ordering of A = "2 ; thus a. = 0 , the value i s unimportant). Partition the elements x # a. of A into two classes: the class d f of those x such that { a o , x j has color (+), and the class 1 i

class similarly defined with ( - ) . Similarly, l e t

similarly defined with ( - ) . Let a; be the minimum in the class 8 1 , and then partition the elements x # a; of r i into two subclasses: the

t' of those x for which { a l , x ) has color (+) , and the class i- a; be the minimum of the class

, and then par t i t ion the elements of xi dis t inc t from th is minimum, into

two subclasses 8 ;+ and r i- , defined as previously. I n general, l e t u be a n ordinal s t r i c t l y less than w . If u has a prede-

cessor u-1 , assume t h a t the classes d :-1 are already defined, each characterized by a sequence s of + and - , with length u-1 , hence defined on the indices s t r i c t l y less t h a n u . Denote by a:-1 the minimum of r:-l , provided th i s class i s non-empty. the minimum (assuming of course that there are such), into two subclasses: the

class $ :+ , characterized by the sequence s completed by the ( ~ - 1 ) ' ~ term + ,

hence a sequence of length u , of x for which {a:-l,x) has color (+) . Analo-

gous definition for the class 1( :- . Suppose now that u i s a l imit ordinal. Given a sequence s of length u , hence of indices s t r i c t l y less t h a n u , consider for each i < u the rest r ic ted sequence s / i taking the same values as s , b u t defined only for indices < i . Then we define the class r : as the intersection of the classes 8 :Ii for a l l

ordinals i < u . Finally, whether u i s a l imit ordinal or n o t , we define the

element a: as the minimum of f: , provided t h a t th i s c lass i s non-empty. Since the ordinals considered are a t most countable, there are continuum many sequences s , hence u1 many, since we assume the continuum hypothesis. Hence there are cjl many classes 4 and the i r minimums a , for a l l indices u and a l l sequences s . Since A has cardinality &J2 , there ex is t other elements besides the minimums a . Let r be one such. Beginning with a. , pick ei ther a; o r a; , depending on whether the pair \ a o , r ) has color (+) or ( - ) . I f we have chosen a: , then choose

ei ther a? o r a;- u < W , i f we have already chosen the sequence s with length u-1 and values

Partition the elements in th i s class which are d is t inc t from

by the same consideration. Continue thusly: given an ordinal

(+) and ( - ) , choose a:' or a:- , depending on whether the pair {a:-l,r} has the color (+) or ( - ) . For every l imit ordinal u , take the sequence s with length u which i s the l imit of the sequences already obtained, and take the corresponding a s . By the definition of the classes ce defined by r , a l l following terms belong t o the same class , i . e . t h a t which contains r . Hence for every v > u a l l pairs {au ,av} have the same color as {a" , r ) . Moreover, the class y through which we pass i s never empty, since r belongs t o i t . We thus obtain a pre-monochromatic W1-sequence from which we extract , as already explained, a monochromatic wl-sequence. 0

0 Let us now take u p the proof of our stated lemma. Assume t h a t the second conclusion does not hold, so t h a t every (-)-monochromatic subset of a t most W1 . Take a subset of A , which i s maximal with respect t o inclusion among the (-)-monochromatic subsets; hence Do has cardinality ,< W . For every element x of A-Do , there ex is t s a t least one element y of Do such t h a t the pair ( x , y ) has color ( t ) . Associate t o each Thus we par t i t ion the elements of A-Do a. of Do . We denote by 'd (a,) the class thus associated with a, , with the color (+) for {ag,x) for a l l x in th i s c lass . I f If(a,) i s non-empty, then take a subset Dl(ao) among (-)-monochromatic subsets; hence D1 has cardinality U l . As previous-

ly , par t i t ion the elements of (a,) - Dl(ao) by an element a l of Dl(aO) . We denote by (a,,al) the class thus associated with the sequence (a,,al) , with the color (+) for { ao,x) and for { al,x) for every x in t h i s c lass . The i terat ipn can be continued in an obvious manner for a l l successor ordinals u s t r i c t l y less t h a n cJ1 . For a l imit ordinal u and a sequence s with length u , having values a i ( i < u) , we define the class 8 ( s ) of the g ( s / i ) for a l l i < u ( recal l t h a t s / i i s the rest r ic t ion o f s t o indices s t r i c t l y less than i ) . Of course, each such sequence s = ao,al, ..., a must sa t i s fy the following conditions. For every successor ordinal i < u , the term ai belongs to Di(ao,al,.. . , a i - l ) , which i s a maximal (-)-monochromatic subset of i < u , the term ai belongs t o Di(ao,a l , . . . , a j , . . ) ( j < i ) , which i s a maximal (-)-monochromatic

subset of r i ( a o , a l , . . . , a j , . . ) , the l a t t e r being the intersection of the

i s used t o define subsets of A .

U , whenever we reach a term a: in the sequen-

8

A has cardinality

Do

x of A - D o the minimum such y . into classes, each defined by an element

of th i s class which i s maximal with respect t o inclusion

into classes which are defined

as the intersection

i ' "

r i ( a D , a l ,..., a i - l ) . For every l imit ordinal

8 j (ao,al , . . . ,a j - l ) (j successor ordinal < i ) . Notice that the axiom of choice D : we do not have i n i t i a l l y a well-ordering of the s e t o f

Chapter 3 75

For eve ry o r d i n a l u < w ao,al, ... hypo thes i s . Hence t h e s e t o f a l l t h e a has c a r d i n a l i t y W1 and l i k e w i s e f o r t he un ion o f t h e D . Since A has c a r d i n a l i t y W 2 , t h e r e e x i s t elements which belong t o no D . Le t r be one such. P i c k a. such t h a t r belongs t o r ( a o ) . Then as r does

n o t be long t o Do , p i c k al such t h a t r belongs t o 8 (ao,al) ; and i n genera l ,

f o r eve ry u < a p i c k au such t h a t r belongs t o Z f ( a ,,...,au) . The sequence o f these au

considered c lasses a l l c o n t a i n r as an element, and so a r e a l l non-empty. 0

, t h e r e a r e a t most & ( U 1) many sequences w i t h l e n g t h u , thus a t most W 1 many, s i n c e we assume t h e continuum

i s (+)-monochromatic and has l e n g t h (,d1 , s i n c e t h e above

3.5. PARTITION THEOREM (ERDOS, RADO 1953) (1) L e t A be a r e g u l a r l i m i t a leph, B an a leph < A . P a r t i t i o n t h e p a i r s o f elements o f A i n t o two c o l o r s (+) and ( - ) . Then e i t h e r t h e r e e x i s t s a subset of A e q u i p o t e n t w i t h A and ( + ) - m o n o c h r o m a t i c , z t h e r e e x i s t s a subset o f A w- po ten t w i t h B (-)-monochromatic (uses g e n e r a l i z e d continuum hypo thes i s ) .

( 2 ) L e t a be an i n f i n i t e a leph, b t h e l e a s t a leph s a t i s f y i n g ba > a . L e t A be t h e a leph a+ , t h e successor o f a . P a r t i t i o n t h e p a i r s o f elements o f A i n t o two c o l o r s (+) and ( - ) . Then e i t h e r t h e r e e x i s t s a subset o f A equ ipo ten t

- w i t h A and (+)-monochromatic, t h e r e exl’sts a subset of A equ ipo ten t w i t h b and (-)-monochromatic (uses axiom o f cho ice ) .

0 (1) Assume t h a t eve ry (+)-monochromatic subset o f A has c a r d i n a l i t y < A . We s h a l l c o n s t r u c t a subset o f A e q u i p o t e n t w i t h B and (-)-monochromatic. To eve ry non-empty subset X o f A , a s s o c i a t e a subset T(X) o f X which i s (+)-monochromatic and maximal w i t h r e s p e c t t o i n c l u s i o n , among (+)-monochromatic

subsets o f X (uses axiom o f c h o i c e ) . N o t i c e t h a t T(X) i s non-empty: a t t h e worst T(X) c o u l d be a s i n g l e t o n .

For eve ry element x o f A and eve ry o r d i n a l i , we d e f i n e as f o l l o w s t h e e l e - ment f i ( x ) o f A . F i x x ; t a k e f o ( x ) = x i f x belongs t o T(A) . Otherwise

s e t f o ( x ) t o be t h e l e a s t element i n T(A) ( w i t h respec t t o t h e w e l l - o r d e r i n g o f A ) , f o r which t h e p a i r { x , fo (x ) ) has c o l o r ( - ) (uses m a x i m a l i t y o f T(A) ) . Now l e t i be a non-zero o r d i n a l , and assume t h a t f . ( x ) i s d e f i n e d f o r each j < i . E i t h e r t h e r e a l r e a d y e x i s t s a j < i f o r which f . ( x ) = x and i n t h i s

case, s e t f i ( x ) = x , O r a l l t h e f . ( x ) a r e d i s t i n c t f rom x and d i s t i n c t among

themselves, and t h e p a i r s they fo rm among themselves o r w i t h x a l l have t h e c o l o r ( - ) . Then l e t U be t h e s e t o f t h e y i n A f o r which ( y , f j ( x ) ] has c o l o r ( - ) f o r a l l j < i . I n p a r t i c u l a r x be longs t o U . I f x belongs t o

T(U) , then s e t f i ( x ) = x . Otherwise, i f x be longs t o U-T(U) , then by t h e

max ima l i t y o f T(U) , t h e r e e x i s t elements z o f T(U) wi th {x,z} hav ing

- -

J J

J

c o l o r ( - ) . Take f i ( x ) t o be t h e l e a s t such z i n t h e w e l l - o r d e r i n g o f A . By t h e p reced ing c o n s t r u c t i o n , f o r eve ry x o f A , t h e r e e x i s t s an o r d i n a l ix such t h a t , f o r a l l i < ix , t h e f i ( x ) a r e d i s t i n c t f rom x and d i s t i n c t f rom each o t h e r , and t h e p a i r s t h e y fo rm among themselves o r w i t h x have c o l o r ( - ) ; and f i ( x ) = x f o r i 3 i For eve ry o r d i n a l i , l e t Mi be t h e s e t o f these f i ( x ) , w i t h i f i x e d and x r u n n i n g through A . We s h a l l p rove t h a t Mi has c a r d i n a l i t y < A f o r eve ry

i < A . F i r s t o f a l l , Mo i s con ta ined i n T(A) which i s (+)-monochromatic, hence by hypo thes i s has c a r d i n a l i t y < A . Consider an o r d i n a l C < A and assu-

me t h a t f o r eve ry i -= C we have Card Mi < A . Since A i s r e g u l a r by hypo- t h e s i s , t h e maximum c a r d i n a l o r t h e supremum c a r d i n a l o f t h e

s t r i c t l y less t han A . Otherwise A would be t h e un ion o f s t r i c t l y < A many se ts , each s t r i c t l y subpotent w i th A . L e t D denote t h e maximum o r supremum c a r d i n a l o f t h e Mi . To e v e r y element x o f A , a s s o c i a t e t h e sequence o f t h e

f i ( x ) ( i 4 C ) . The number o f d i s t i n c t such sequences i s l e s s than o r equal t o

(Card . Now by t h e f a c t t h a t A i s a l i m i t a leph, and by ch.2 5 6.5 ( 2 ) (gene-

r a l i z e d cont inuum hypo thes i s ) , t h i s c a r d i n a l i s s t r i c t l y l e s s than A . Two i d e n t i - c a l sequences g i v e t h e same s e t U , hence t h e same T(U) o f c a r d i n a l i t y < A . For any x , t h e new element f C ( x ) be longs t o T(U) . Hence as x v a r i e s , t h e

p o s s i b l e sequences f i ( x ) ( i d C) g i v e < A many s e t s T(U) . Because o f t h e r e g u l a r i t y o f A , t h e s e t MC o f a l l p o s s i b l e f C ( x ) has c a r d i n a l i t y C A . By hypo thes i s B i s a c a r d i n a l < A . Thus t h e r e e x i s t s an r i n A such t h a t

ir >/ B . For o the rw ise , MB would be i d e n t i c a l w i t h A , c o n t r a d i c t i n g t h e p re -

ceding. Hence t h e fi(r) a r e d i s t i n c t f o r i < B , and t h e i r s e t i s (-)-monochro- ch romat i c and o f c a r d i n a l i t y B . 0

0 (2 ) F o r A o f c a r d i n a l i t y a+ , suppose t h a t eve ry (+)-monochromatic subset of A has c a r d i n a l i t y < a . We s h a l l c o n s t r u c t a (-)-monochromatic subset o f A w i t h c a r d i n a l i t y b . For eve ry x o f A , c o n s t r u c t as p r e v i o u s l y t h e f i ( x ) and t h e Mi f o r a l l o r - d i n a l s i (axiom o f c h o i c e ) . We s h a l l p rove t h a t , f o r eve ry i o f c a r d i n a l i t y a . The s e t Mo i s con ta ined i n T(A) , which

i s (+)-monochromatic and hence has c a r d i n a l i t y 6 a . Take an o r d i n a l c < b , hence c < a , and assume t h a t Card Mi 6 a f o r eve ry i < c . The maximum

o r t h e supremum o f t h e Card Mi ( i < c ) i s a . The sequences fi ( x ) (i < c )

have c a r d i n a l i t y a t most

p r o o f o f (1) above, i t f o l l o w s t h a t t h e c a r d i n a l i t y o f t h e s e t Mc o f f c ( x ) f o r x r u n n i n g th rough A , i s ,< a.a = a . Since

x '

Mi ( i < C ) i s

(Card ')a , hence 6 a s i n c e Card c < b . As i n t h e

b ,< a , t h e un ion o f t h e Mi (i < b) has c a r d i n a l i t y 6 b.a = a . L e t

Chapter 3 77

r .be an element of A not in this union. Then r # fi(x) for every x of A and every i < b . In particular r # fi(r) for every i < b : thus i, 3 b . It follows that the set of the fi(r) (i < b) is (-)-monochromatic and has cardinality b . 0

Note that, in the proof of (2 ) , the cardinal condition, and also the use of generalized continuum hypothesis, is only necessary for (1) in order to apply ch.2 5 6.5.(2). The example of w1 , given at the end of 5 3.3, can be expressed here by taking a = d , b = w and A = bJ1. Recall that with generalized continuum hypothesis, the only regular limit alephs are CJ and the inaccessible alephs (ch.2 5 6.7). The statement (1) holds only for these. On the other hand, with only the axiom of choice, the continuum can be regular or not, a limit or a successor aleph, with cofinality >/a

A = a, is not a limit aleph. This

(ch.2 § 6.4).

§ 4 - COMBINATORIAL LEMMAS, COLOR AND INCLUSION

4.1. (1) Let E be a set, p, q two integers such that ptq 4 Card E . Take a set of p-element subsets included in E and call this the color u . If every (p+q)-element subset includes the same number k f p-element subsets with color & , then every k(p+qtl)/(qtl) of p-e&%n_t_Aybsets with color u . (2) Given two not necessarily disjoint sets of p-element subsets o f E , call these the colors ?d, and 2'. If every (p+q)-element subset includes as many p-element subsets with color u as p-element subsets with color ?f, then the same is true fmeve_ry element subset of E .

(1) Let F be a (p+q+l)-element set. The cardinality of the set o f (p+q)- element subsets of F is ptq+l . Each includes k many p-element subsets with color k , which yields k(p+qtl) ordered pairs, each formed with a p-element set having color ?,(, and with a (ptq)-element set which includes it. For every p-element subset with color % in F , there are qtl many (ptq)- element subsets which include it. This yields k(p+q+l)/(q+l) of p-element subsets included in F and having color & . 0 (2) Let F be a (p+q+l)-element set. Every (p+q)-element subset o f F includes as many p-element subsets with color % as with color r . Thus we have that in F , there are the same number of ordered pairs, each with first term a p-element set with color 21 and second term a (p+q)-element set including the first term, as of ordered pairs, each with first term a p-element

(ptqt1)-element subset includes the same number

~

(p+qtl)-

as the number


set having color 'It and second term a (p+q)-element set including the first term. We obtain the number of p-element subsets with color % by dividing the preceding number by q+l . Similarly for the color 'v . 0

4.2. (1) Let E be a finite set, p. q two integers such that p+q 6 Card E . Let 'LL be a color of certain p-element subsets of E . Let ss p and s 6 (Card E) - p - q . If every with color u , then every s-element subset is included in the same number of_ p-element subsets with color /d~ . (2) Let %, be two colors of p-element sets, not necessarily disjoint. If every (p+q)-element set includes as many p-element subsets with color % as s-element subset is included in as many (communicated by POUZET in 1975). 0 First we prove (2). For s = 0 , this follows from 4.1. (2) iterated from p+q to Card E . Assume that s 31 and assume that the statement is true for s - 1

and p+q . In other words, for every E with finite cardinality >/ s+p+q-1 and every (s-1)-element subset of E . We shall prove this for s and p+q , hence for E with finite cardinality 3 s+p+q and a s-element subset H G E . Let u be an element of H . By the induction hyoothesis, there exists a same number k of p-element subsets with color % as with color 2 / , included in E and including H - {u) . Similarly, there exists a same number 1 of p-element subsets with color u as with color r, included in E - { u) and including and equal to s-1 ). By subtraction, there existsthesame number k-1 of p-element subsets with color '?A!, as with color , included in E and including H .O

0 Statement (1) follows from (2). Indeed, let H and H' be two s-element subsets of E . Take a permutation f of E which transforms H into H ' . Take the p-element subsets with color u , and let 21' be the color of their images via f . Then every (p+q)-element subset X includes as many p-element subsets with color u (p+q)element subset (f-l)"(X) (notation from ch.1 !j 1.2). By ( 2 ) , the s-element subset H ' is included in as many p-element subsets with color %& as with color u' . But the latter are the images via color u and including H . Thus H and H ' are included in the same number of p-element subsets with color u . 0

(ptq)-element subset includes the same number of p-element subsets

p-element subsets with color 1/, then every - p-element sets with color u 2 p-element sets with color

H - {u} (the cardinality of these sets being respectively s+p+q-1

as with color a ' , as one can see by taking the

f of p-element subsets with

Chapter 3 79

4.3. (1) Let E be a set, p an integer less than or equal to Card E , and let u be a non-empty set, called color, of p-element subsets.

I f there exists an integer q such that 2p + q Card E , and if every (p+q)- element set includes the same number of p-cgment subsets with color u , then every p-element subset has color ld . (2) Given E and p less than or equal to Card E , let I d , 3‘ be two sets, called colors, of p-element subsets. If there exists an integer q such that 2p + q 4 Card E , and for which every (p+q)-element subset includes as many p-element subsets with color 1;’ , then the colors

p-element subsets with color _ _ ~ _ ^ _ _

and 2/ are identical.

0 We can assume that E is finite, by replacing E if necessary by a finite subset of cardinality at least equal to 2p + q . Now take the preceding statements with s = p . By statement (l), every p-element subset is included in the same number of p-element subsets with color 16. In other words, every p-element subset has color 11 (since it is assumed that 2 is non-empty). By statement (2), every subsets with color 2 as p-element subsets with color 17. In other words, the colors ‘lG and are identical. 0

p-element subset is included in as many p-element

If Card E < 2p + q , then by taking s 6 (Card E)-p-q , it is easy to give an example in which the color sets. Thus with E = {a,b,c,dj hence of cardinality 4 , with p = 2 , q = 1 , and only the edges ab and cd with color % , every 3-element subset contains such an edge, and every element belongs to such an edge. Adding ac and bd with color v , every 3-element subset contains an edge with each color, and all elements belong to an edge of each color.

Calculation. If e designates the cardinality of E , and k the number of p-ele-

does not extend to the entire set of p-element sub-

ment subsets of color u? contained in every (p+q)-element subset, then by 4.1.(1 the number of all p-element subsefis with color a is k.e!q!/(p+q)!(e-p)! . The number of those which contain a given element u , hence are not contained in the (e-1)-element subset E - (u) , is obtained by subtraction. It equals k.(e-l)!q!p / (p+q)!(e-p)! , by assuming e >/ p+q+l . The number of those which contain a given pair (u,v) , hence which contain u and are not contained in the (e-1)-element subset E - { v ) , i s obtained by subtraction. It equals

,

k.(e-2)!q!p.(p-1) / (p+q)!(e-p)! , in assuming e >/ p+q+2 . In general, the number of those containing a given s-element subset is: k.(e-s)!q!p! / (p+q)!(e-p)!(p-s)! , in assuming s & e-p-q .

§ 5 - I N C I D E N C E M A T R I X , K A N T O R ' S L I N E A R I N D E P E N D E N C E LEMMA, MULTICOLOR THEOREM

5.1. INCIDENCE MATRIX Let p. q be two integers, E a s e t of f i n i t e cardinal h . Represent as "ordinate values" the s e t of p-element subsets of E , of cardinality h!/p!(h-p)!, and as "abscissa values" the s e t of (p+q)-element subsets, of cardinality h!/(p+q)!(h-p-q)! . To each couple ( x , y ) where x i s a (p+q)-element s e t and y i s a p-element s e t , a t t r ibu te the value 1 i f f y s x and the value 0 otherwise. The rectangular table thus obtained shall be called the incidence matrix of E for p and q . Note t h a t i f h = Card E 2 2p + q , then each row of the matrix, corresponding t o a column, corresponding t o a (p+q)-element se t : indeed p!(h-p)! 3 ( p + q ) ! ( h - p - q ) ! . The reader i s assumed t o be familiar with the elementary theory of determinants and with the notion of l inear dependence. If Card E < 2p + q , then i t i s possible that a row of the incidence matrix depends linearly on one or several other rows. For example, for p = q = 1 and Card E = 2 , the matrix reduces t o two rows and one column, with value 1 . B u t for Card E 3 2p + q , we have the following resul t . L I N E A R INDEPENDENCE LEMMA I f Card E 3 2p + q , the rows of the incidence matrix are linearly independent: no row i s a l inear combination of other rows. Equivalently, every non-zero determinant extracted from the matrix and depending on a f i n i t e number r of rows can be extended t o a non-zero determinant based on the previous rows together with an arbi t rary (r+l)st row.

p-element s e t , i s a t l eas t as long as each -

I n the case t h a t E i s f i n i t e , i t follows t h a t there exis ts a non-zero determi- n a n t depending on a l l the rows. Hence there exis ts an injection which t o each p-element s e t y associates a (p+q)-element s e t including y as a subset (KANTOR 1972).

~ _ _ _ ~

0 To each permutation f of E associate the corresponding permutation f " of p-element subsets of E . Hence f" permutes the se t of rows. There corresponds t o f as well a permutation of the se t of (p+q)-element subsets, hence of the se t of columns, b u t i t i s unnecessary t o consider t h i s , since we are working with l inear combinations of rows and reasoning by the coefficients a t t r ibuted t o each row in a given l inear combination. Assume t h a t E has f i n i t e cardinal h ; we argue ad absurdum. Assume t h a t there exis ts a p-element subset, hence a row which i s a l inear combination of a l l the other rows, with positive, negative or zero rational coefficients, since these are quotients of determinants with values 0 or 1 .

Chapter 3 81

Le t us c a l l b t h i s p-element se t and the corresponding row. Given an a r b i t r a r y permutation f o f E which preserves the se t b (bu t not necessar i l y each element o f b ), then fo preserves the row b and permutes the se t o f the other rows; two rows which are transformed one i n t o the o ther represent two p-element sets y, y ' such t h a t b n y and b A y ' are equipotent. Transform the given l i n e a r combination by a l l poss ib le fa , the number o f such

being (h-p)!p! , then take the combination which i s the a r i t hmet i c average o f the combinations thus transformed. By s y n e t r y , a l l t he rows which represent p-element sets d i s j o i n t from b w i l l have the same c o e f f i c i e n t . S i m i l a r l y f o r a l l rows which represent p-element se ts having a s i n g l e element i n common w i t h b , and i n general f o r a l l rows which represent p-element sets having equipotent i n te rsec t i ons w i t h b . Consider a column a represent ing a (p+q)-element s e t d i s j o i n t from b : t h i s a ex i s t s s ince h >/ 2p + q . I n the column a , the p-element sets included i n a a re a l l d i s j o i n t from b , and so a l l have the same c o e f f i c i e n t i n our combination. Moreover, i f we denote these p-element sets by y , these are the on ly ones y i e l d i n g the value 1 i n p o s i t i o n (a,y) i n the incidence matr ix, wh i le the ma t r i x has the value 0 i n p o s i t i o n (a,b) . It fo l lows t h a t t h e i r c o e f f i c i e n t i s zero, hence each row which represents a p-element se t d i s j o i n t from b has c o e f f i c i e n t zero. The problem i s thus answered negat ive ly f o r p = 1 , since i n t h i s case the p-element sets d i s t i n c t from b are d i s j o i n t w i t h b , hence the above assumed l i n e a r combination does no t ex i s t . Assume t h a t p 2 , and consider a column al represent ing a (ptq)-element se t which i n te rsec ts b i n a unique element. Then the rows y f o r which the ma t r i x has value 1 i n (a,,y) a re those which represen! e i t h e r a p-element se t d i s j o i n t from b , hence w i t h c o e f f i c i e n t zero, o r a p-element se t i n t e r - sec t ing b i n a s ing le po in t . By the preceding discussion, the l a t t e r have the same coe f f i c i en t i n the combination. Since the ma t r i x has the value 0 i n (al,b) , t h i s coef f i c ien t i s zero. The problem i s thus answered negat ive ly f o r p = 2 , since i n t h i s case the p-element se ts d i s t i n c t from b have a t most one element i n common w i t h b . I n the general case, by i t e r a t i n g the preceding argument, we prove t h a t a l l t he coe f f i c i en ts are zero, .hence t h a t t he above assumed l i n e a r combination does no t e x i s t . The r e s u l t fo l lows imnediately i n the case o f E i n f i n i t e . F ina l l y , f o r t he cnnclusion concerning the e x t e n d i b i l i t y o f a non-zero determinant, assume on the cont ra ry t h a t there e x i s t s a non-zero determinant which i s no t extendible, and deduce t h a t an a r b i t r a r y row o f t he ma t r i x i s a l i n e a r combination o f rows o f the submatrix which corresponds t o t h i s determinant,

5 .2 . In the "degenerate case" where h = Card E < 2p+q , the number of columns i s s t r i c t l y l e s s than the number of rows. In this case the columns of the incidence matrix a r e l i nea r ly independent; i n o ther words, t he re e x i s t s a non-zero determinant based on the col umns. 0 Interchange each p-element s e t y w i t h t he (h-p)-element s e t E-y , and each (p+q)-element s e t x with the (h-p-q)-element s e t E-x . Then the inclusion y c x i s equivalent t o E-xc E-y . The r o l e of p i s played by p ' = h-p-q ;

t he ro l e of p+q i s played by p '+q ' = h-p , so t h a t q ' = q . We have 2p' + q ' = 2 h - 2 p - q < h : hence we can apply the l i n e a r independence lemna with rows and columns interchanged. 0

5.3. MULTICOLOR Let E be a f i n i t e s e t , h i t s cardinal and p , q two in tegers . P a r t i t i o n the p-element subsets of E i n t o a f i n i t e number k of c lasses which a r e ca l led colors uo , ul, ..., u ~ - ~ . For each (p+q)-element subset a of E , we ca l l the multicolor of a the function which t o each co lor u i ( i k ) associates the number of p-element s e t s of co lor u i which a r e included i n a . When this number i s non-zero, we say t h a t the co lor MULTICOLOR THEOREM I f - of E i s a t l e a s t equal t o the number of co lors of p-element subsets. More prec ise ly , t he re e x i s t s an in j ec t ion which t o each co lor l e a s t one p-element s e t belongs) assoc ia tes a multicolor i n which u f i gu res , and t o which a t l e a s t one

Assume f i r s t t h a t E has f i n i t e cardinal h > , 2p+q . Hence the number of (p+q)-element s e t s i s a t l e a s t equal t o t h a t of t he p-element s e t s , and the rows of the incidence matrix a r e l i n e a r l y independent. To each co lor t he re corresponds a f i n i t e s e t of rows of t h a t co lor . Replace these by a un ique row which i s t h e i r sum, obtained by adding the values 0 o r 1 i n each column. T h u s each new row represents a co lor u . Each column continues t o represent a (p+q)-element s e t , and ind ica t e s the number of p-element sets of co lor u which a r e included i n th i s (p+q)-element set . Note t h a t , i n the new matrix t h u s obtained, the rows a r e l i nea r ly independent. I t su f f i ces t o see t h a t , given a matrix w i t h k independent rows ( k 3 2 ) , the replacement of two rows b and b ' by t h e i r sum y ie lds a matrix w i t h k-I independent rows. Indeed, t he re e x i s t s a non-zero determinant based on the k-2 i n t a c t rows. So t h a t the only o the r poss ib i l i t y would be t h a t t he row sum o f b and b' i s a l i n e a r combination of the k-2 i n t a c t rows. B u t then the row b , f o r example, would be a l i n e a r combination o f the k-2 i n t a c t rows plus

u i f i gu res in the multicolor.

Card E 2, 2p+q , then the number of mul t ico lors 'o f (p+q)-element subsets

u ( t o which a t

(p+q)-element s e t belongs (POUZET 1976).

Chapter 3 83

the row b' , cont rad ic t ing the hypothesis. T h u s , i f k i s now the number of co lo r s , hence of rows, we have a non-zero determinant of order k . Take i n th is determinant a sequence of k ordered pa i rs (x ,y) where x i s a column and y a row, w i t h non-zero value of the new matrix in each considered ordered pa i r . We thus obta in the in j ec t ive function i n the theorem. This i n j ec t ion a s soc ia t e s , t o two d i s t i n c t co lors y. y ' two ( p + q ) - element s e t s x , x ' whose mul t ico lors a r e d i s t i n c t . Otherwise we would have two ident ica l columns i n the determinant. T h u s th i s is an in j ec t ion from the s e t of colors i n t o the s e t of mul t ico lors . I t remains t o consider the case when E i s countably i n f i n i t e . I f we only have a f i n i t e number of co lors , then we r e s t r i c t E t o a s e t of f i n i t e ca rd ina l i t y a t l e a s t equal t o 2p+q and including p-element subse ts of each co lor . The rows, which represent the co lo r s , a r e l i nea r ly independent, and remain so when one takes up the e n t i r e i n f i n i t e set E . I f there a re i n f i n i t e l y many co lo r s , then we s t i l l have l i n e a r independence. Then as mentioned f o r the l i n e a r independence lennna, every non-zero determinant i s extendible t o a non-zero determinant over one more row, hence one more co lor . The ex is tence of the in j ec t ive function i n the theorem follows. 0

5 6 - RAMSEY SEQUENCE: ANOTHER PROOF OF GALVIN'S THEOREM

The following notion of Ramsey sequence of conditions is a form of the c l a s s i ca l Ramsey s e t : see E R D O S , RADO 1952. The connected proof of GALVIN 's i n i t i a l in te rva l theorem i s due t o LOPEZ 1983'. As opposed w i t h POUZET's proof in 5 2 , here we need ne i ther lexicographic rank nor t r a n s f i n i t e induction. As well a s i n 5 2 , the axioms of ZF w i l l be s u f f i c i e n t : see 5 6.5 below.

6 .1 . Given two s e t s A, B of in t ege r s , p u t A A i f f every element of B i s s t r i c t l y g rea t e r than every element of A . We adopt the convention t h a t the empty s e t i s < and > any s e t ; so t h a t < i s i r r e f l e x i v e and t r ans i - t i v e only f o r non-empty s e t s . L e t H be a f i n i t e s e t , Z an i n f i n i t e set of in tegers . A f i n i t e sequence of conditions g i ( H , Z ) the following:

( i = 1, ..., r ) i s sa id t o be a Ramsey sequence i f f we have

V H f i n V X i n f X > H 93, inf YCX A [ ( b z i n f Z c y 3rl(H,Z)) V ... ... v ( d z i n f Z s Y +Vr(H,Z))]

(nota t ions : f i n = f i n i t e , i n f = i n f i n i t e set of in tegers ; obvious log ica l symbols).


Example. P a r t i t i o n the pa i r s o f in tegers i n t o two co lo rs (+) and (-). Take f o r

where z belongs t o Z , have same c o l o r (depending on h )" . Then alone cons t i tu tes a Ramsey sequence. Another example. Take a cond i t i on f and de f ine i g as the negation o f d . Then the sequence ( ff, -I e ) i s o f t e n a Ramsey sequence. I t i s the case, for instance, i f yf (H,Z) means t h a t the preceding p a i r s { h,z) have co lo r (t). I n the case o f two such opposi te condi t ions, the above formula means tha t , g iven H , the se t o f a l l i n f i n i t e Z s a t i s f y i n g C(H,Z) i s Ramsey i n the sense o f ERDOS, RADO 1952. I n other words, there ex i s t s an i n f i n i t e se t Y o f in tegers such t h a t e i t h e r each i n f i n i t e Z C_ Y belongs t o the se t def ined by and H o r each i n f i n i t e Z c Y belongs t o the complement. Among sets o f i n f i n i t e sets o f integers, i . e . among sets o f rea ls , i t i s known t h a t t he fo l l ow ing are Ramsey: a l l open sets (see the topology def ined i n ch.1 exerc. 4) ; Bore1 sets (see GALVIN, PRIKRY 1973); a n a l y t i c se ts (SILVER 1970). See a lso ELLENTUCK 1974, who character izes the "completely Ramsey sets" by the Bai r e property.

f(H,Z) the fo l l ow ing statement: " f o r each in tege r h i n H , a l l p a i r s 1h.z) ,

6.2. Given a Ramsey sequence el ,..., er , we have the fo l l ow ing statement, modulo the axiom o f dependent choice:

J ~ i ~ f f l ~ f i , , (vzinf ( H C A A Z S A ~ Z > H ) ~ V 1 ( H J ) ) V - - *

... ( V z inf ( H c A h Z E A A Z > H ) 3 f,(H,Z))

0 The proo f general izes the f i r s t p a r t o f RAMSEY's p roo f 1 . 1 , i n ob ta in ing e le - ments vi . S t a r t from uo = 0 , Ho = { D l and Xo = se t o f in tegers # 0 . We get an i n f i n i t e Ys Xo , c a l l e d Yo and s a t i s f y i n g the above cond i t i on i n brackets, where H = Ho and A = {O)uYa . Then l e t u1 be the f i r s t element of Yo . S t a r t from HI = { uo,ul) and XI = Yo -{ul) . We get an i n f i n i t e

Y1s X1 which s a t i s f i e s our above cond i t i on i n brackets, where H = Ho o r H1 and A = \uo,uI) u Y1 . Then s t a r t from H i = {ul) and X i = Y1 . We ge t an i n f i n i t e Y i Y1 which s a t i s f i e s our condi t ion, where H = Ho o r HI o r H i and A = {uo,ul)u Y i . Then l e t u2 be the f i r s t element o f Y i . S t a r t from H2 = {uo,u1,u2) and X2 = Y i - {u2 ) . We ge t an i n f i n i t e Y 2 5 Y i which sa t i s f i es our condi t ion, where H = Ho o r HI o r H i o r H2 and where A = {uo,u1,u2\ u Y2 . I t e r a t e , t ak ing f o r H i , H;,... a l l the sets w i t h l a s t element u2 , and so g e t t i n g Y h y Y i , ... before de f i n ing u3, Hg and Y3 ; and so on. F i n a l l y take f o r A the s e t o f ui ( i i n tege r ) . The axiom o f dependent choice i s used f o r choosing se ts Y . 0

6.3. L e t H, F be f i n i t e sets o f in tegers and b (H,F) Then the p a i r o f t he fo l low ing cond i t i on e(H,Z) w i t h i t s negat ion i s Ramsec

c (H ,Z ) : 3 F fin F c Z A a ( H J ) . be an a r b i t r a r y cond i t ion .

Chapter 3 85

0 Suppose the contrary. There exis t a f i n i t e se t such tha t , for every inf in i te s e t Y 5 X , there exis t two inf in i te subsets and Z2 with (H,Z1) and the negation iff (H,Z2) . Then each f i n i t e subset F of Z2 sa t i s f ies (H,F) . Now replace Y by Z2 : there exis ts an in f in i te subset Z i of Z2 such that 'f: ( H , Z i ) . Thus there exis ts a f i n i t e subset F of

H and an in f in i te s e t X 7 H

Z1

Z i 5 Z2 which sa t i s f ies a (H,F) : contradiction. 0

6 .4 . A PROOF OF G A L V I N ' S INITIAL INTERVAL THEOREM Let under inclusion, and t o sa t i s fy GALVIN's hypothesis: every inf in i te s e t of integers includes a t least one element of 3 as a subset. Take 3 (H,F) interval belonging t o " ; more br ief ly " H,J F has i . i . " . Then by 6 . 2 there exis ts an inf in i te s e t A such t h a t , f o r every f i n i t e subset H of A :

ei ther ( 1 )

or (2 )

Firs t ly we prove tha t , assuming GALVIN's hypothesis, there exis ts an in f in i te - se t A such the above ( 2 ) i s fa lse: so only (1) i s true. 0 For H empty, the above conclusion ( 2 ) i s fa lse . Indeed ( 2 ) reduces t o saying t h a t , fo r every inf in i te s e t Z , there does n o t exis t any f i n i t e subset of Z which belongs t o . Now l e t G be a f i n i t e se t of integers. Assume that the above ( 2 ) i s fa lse for every subset H of G . Then i t suffices t o prove that there exis ts an integer g 7 M a x G such that every H 5 G u{g> f a l s i f i e s ( 2 ) . For th i s purpose, i t suffices t o prove t h a t , for each subset H of G , there ex is t only f in i te ly many integers suffice t o choose g s t r i c t l y greater t h a n a l l such h . Arguing ad absurdurn, assume the existence of an i n f i n i t e sequence h l , h 2 , ..., h i , ... with Hu{hi} satisfying ( 2 ) . Take Z to be the inf in i te s e t of these h i : by hypothesis H f a l s i f i e s ( 2 ) , so H verif ies (1). T h u s there exis ts a f i n i t e subset F of Z such that H u F has an i n i t i a l interval which belongs to 3-. Let h (p integer) be the f i r s t element of F ; then H w{h ) f a l s i f i e s (2) : contradiction. 0

be a se t of f i n i t e sets of integers, assumed t o be mutually incomparable

t o be the following condition: "the union H u F admits an in i t ia l

v, in f ( Z g A A Z > H ) =) 3, f i n ( F c Z A H u F has F i . i . )

tJz i n f ( Z 5 A A Z 7 H) =) vF f i n ( F c Z + H u F has no '3 i . i . ) .

h Z Max G such t h a t H u { h ) sa t i s f ies ( 2 ) : indeed i t will

P P

Secondly we obtain GALVIN's conclusion. 0 Consider the inf in i te se t A of 6.2, now assumed to sat isfy only the above (1). Let B be an arbi t rary inf in i te subset of A . Let K be a f i n i t e subset of B belonging t o , and denote by H the i n i t i a l interval of B which ends with Max K . Then by ( 1 ) above, there exis ts a f i n i t e subset F of 6-H such


t h a t HuF cannot surpass Max K , since elements of inclusion. Consequently our i n i t i a l interval of H u F interval of H , thus of B . 0

has an i n i t i a l interval which belongs t o 3 . This i n i t i a l interval are mutually incomparable under

reduces t o an i n i t i a l

6.5. To end th i s section, l e t us prove t h a t the axiom of dependent choice, used in 6.2, i s avoidable in view of obtaining GALVIN's theorem.

0 Come back t o our hypotheses in 6.1. Given a f i n i t e se t se t X > H :

ei ther (1) 3 i n f ys x A 'z i n f ( Z C y

or ( 2 )

Either ( 2 ) i s fa lse; in other words:

H and a n in f in i te

3~ f i n (Fc Z A H u F has $-i.i .))

3, i n f Y c _ X A vz inf(ZcY => 'dF f i n ( F c Z => H u F has n 0 F i . i . ) ) .

vy inf Y c _ X => 2z i n f ( Z S Y

I n such a case, take Y = X ; then take any Yc X b u t change the notation, writing Z instead of Y : we get the following

3, f i n ( F c Z A H U F has 5 i . i . ) ) .

3, i n f Y = X A \dz in f ( Z S X => 3, f i n (Fc Z A HuF has F i . i . ) ) . So we obtain (1) strengthened by the unambiguous definition Or ( 2 ) i s t rue, with (1) true or fa l se , which i s immaterial. In t h i s case, take a l l the inf in i te sets Y which sa t i s fy ( 2 ) , and note that each inf in i te subset of a Y i s s t i l l a Y satisfying ( 2 ) . Let us proceed lexicographically: we take the least integer uo fo r which there exis ts a Y beginning with uo . Then the leas t u l , uo for which there exis ts a Y beginning with u o , u1 ;

and so on. Finally we adopt the unambiguous definition Y = i u o , u l , . . . ) . 0

Y = X .

EXERCISE 1 - SCHUR NUMBERS 1 - Given a par t i t ion of the s t r i c t l y positive integers into a f i n i t e number of classes called columns, show t h a t a t l eas t one of the columns contains three dis t inct integers a,b,c with c = a+b (SCHUR 1916). Hint. To each column U associate the class of pairs of integers x , y such that the absolute value I x-yl belongs t o U ; then apply RAMSEY's theorem. 2 - A s e t U of integers i s said t o be additively free i f f the sum of any two integers belonging to U does not belong t o U . Given an integer k , show that there exis ts an integer k+, k such tha t , for each par t i t ion of the integers 1,2, ..., k+ i n t o k classes called columns, there is a t l eas t one non- additively free column. The smallest k+ will be denoted by s (k) and called the Schur number re lat ive t o k . Show t h a t s ( 1 ) = 2 , s (2) = 5 , s(3) = 14 ( s t a r t with the column 5,6,7,8,9).

Chapter 3 87

3 - I n 1961, Leonard D. BAUMERT (see p u b l i c a t i o n 1965) e s t a b l i s h e d , w i t h t h e a i d

o f a computer, t h a t s ( 4 ) = 45. Here i s t h e example he gave o f a p a r t i t i o n o f t h e

f i r s t 44 s t r i c t l y p o s i t i v e i n t e g e r s i n t o 4 a d d i t i v e l y f r e e columns: 1 2 4 9 3 7 6 1 0 5 8 13 11 15 18 20 12 17 21 22 14 19 24 23 16 26 27 25 29 28 33 30 3 1 40 37 32 34 42 38 39 35 44 43 41 36

2 Show t h a t t h e Ramsey number : a s s o c i a t e t o each s e t o f p a i r s o f i n t e g e r s f rom 1 t o s ( k ) f o r which t h e abso lu te

d i f f e r e n c e belongs t o U ; thus ( 3 ) 4 >/ 46 .

( 3 ) k >/ s ( k ) + l

2

column U t h e va lue o f t h e

4 - Show t h a t s ( k + l ) >/ 3 . s ( k ) - 1 . Begin w i t h t h e p a r t i t i o n o f t h e i n t e - gers 1 t o p = s ( k ) - 1 i n t o k columns. Add a ( k + l ) s t column o f t h e i n t e -

gers p + l t o 2p + 1 . Then complete each column formed o f i n t e g e r s u by t h e i n t e g e r s 2p+l+u . Hence s ( 5 ) 3 134 and (3); >, 135 . FREDRICKSON 1979 o b t a i -

ned s ( 5 ) & 158 , thus ( 3 ) 5 + 159 . I n genera l

i n e q u a l i t y i s improved by ABBOTT, HANSON 1972 who o b t a i n , i f one r e c t i f i e s t h e i r

numer ica l e r r o r ,

2

k s ( k ) 3 ( 3 +1) /2 and even 3 (3k-4.89/2) + 1/2 f o r k >/ 4 . Th is

s ( k ) 3 89(k-7) /4 .1201 + 1 f o r k >/ 4 .

EXERCISE 2 - RAMSEY'S THEOREM WITH m-TUPLES

1 - L e t m, k be two i n t e g e r s ; t a k e an m-sequence o f i n t e g e r s ai (i = 1, ..., m ) .

Pa t te rned a f t e r t h e f i n i t e Ramsey theorem, p rove t h a t t h e r e e x i s t s an m-sequence of i n t e g e r s bi + ai s a t i s f y i n g t h e f o l l o w i n g . Given an m-sequence of s e t s

Fi o f c a r d i n a l i t y bi , f o r e v e r y p a r t i t i o n o f t h e s e t o f m- tup les ( x ,.... , Xm) where xi be longs t o Fi , into k c o l o r s , t h e r e e x i s t s a sequence o f subsets Gi Fi w i t h Gi o f c a r d i n a l i t y ai , such t h a t a l l t h e m-tup les i n t h e Car tes ian p r o d u c t G1 x ... x Gm have t h e same c o l o r . Note t h a t n o t h i n g i s changed i n assuming t h e p a i r w i s e d i s j o i n t , which i s o f t e n convenient . Example: m = 2 w i t h al = a2 = p . Beg inn ing w i t h a s e t F1 o f c a r d i n a l i t y

(kp) .kkp and F2 o f c a r d i n a l i t y (p) .kkp , we o b t a i n G2 5 F2 o f c a r d i n a l i t y p and an " i n t e r m e d i a t e " s e t H1 C, F1 o f c a r d i n a l kp , such t h a t , f o r eve ry

element x o f H1 , a l l t h e o rde red p a i r s w i t h f i r s t t e r m x and second term

- - Fi

an element of G2 have the same co lo r . F ina l ly we obtain G1 c H1 of cardinal p , w i t h the monochromatic Cartesian product G1 x G2 . 2 - For m = 2 and a l = a2 = 2 , we can take bl = 3 and b2 = 9 , o r conver- s e ly . We search f o r a symmetric so lu t ion bl = b2 ; show t h a t bl = b2 = 4 in su f f i c i en t . O n the o the r hand, f o r the common value 5 , we always have a monochromatic Cartesian product with Card G1 = Card G2 = 2 . One could assume f i r s t t h a t there e x i s t s an element x of F1 joined t o a t l e a s t 4 elements of F2 by the same co lo r , denoted (+ ) . Call F i th is subset of F2 of cardinal 4. Then e i t h e r there e x i s t s another element of y ie ld ing the co lor (+) w i t h two elements of F i , o r two elements of F1 d i s t i n c t from x y i e l d the opposite co lor ( - ) w i t h two elements of Next, we assume t h a t each element of F1 t o 3 elements of F2 f o r the co lors . Such an element x i s ca l l ed (+)-majo- r i t y o r (-)-majority, according t o whether t he re a re 3 edges ( o r pa i r s ) emana- t i n g from x with co lor (+), o r 3 w i t h co lo r ( - ) . Then there e x i s t 3 elements of F1 of the same majority: this y i e lds our conclusion. 3 - Attempt t o extend the statement in sec t ion 1 above t o the case of i n f i n i t e card ina ls a ( i = 1,. . . , m l . Note t h a t for m = 2 and al = a? = W , the values bl = b2 = ~3 do not hold: we do not have a simple i n f i n i t a r y theorem analogous t o RAMSEY's theorem. Indeed, taking i t su f f i ces t o co lor (+) the ordered pa i r s ( x , , ~ , ) i f f x l \ < x2 . However, with the axiom of choice, ERDOS, RADO 1956 proved t h a t one can take

and b2

i s

F1

F; . y ie lds a p a r t i t i o n of 2 as opposed

i F1 = F2 = is , the s e t of i n t ege r s ,

bl countable equipotent w i t h the continuum, o r conversely: see exerc ise 3 below.

EXERCISE 3 - A PARTITION THEOREM WITH POINTS, i . e . ORDERED PAIRS Take a denumerable s e t D , a s e t C equipotent w i t h t he continuum, and the Cartesian product D x C whose elements (x ,y) sha l l be ca l l ed points with the abscissa x in D and the ord ina te y i n C . P a r t i t i o n the poin ts i n t o two c lasses ca l led the co lors (+) and ( - ) . We sha l l show t h a t there e x i s t s either a denumerable subse t subset Co of C equipotent w i t h the continuum, w i t h Do x Co e n t i r e l y of - color ( - ) , o r two denumerable subse ts D1 0-f D and C1 C with D1 x C1 e n t i r e l y of co lor (+) (uses axiom of choice; ERDOS, RADO 1956 p . 482).

1 - Assume t h a t t he re does not e x i s t any (-)-monochromatic s e t which i s t he Cartesian product of a denumerable s e t w i t h a continuum-equipotent set. Take an a rb i t r a ry denumerable subset X of D and an a r b i t r a r y subse t Y of C which is continuum-equipotent: by hypothesis X x Y i s never (-)-monochromatic. Call Y o the projection onto Y of the s e t of points of X x Y w i t h co lor (+), and note t h a t , by our hypothesis, Card(Y-Yo)

Do of D

i s s t r i c t l y l e s s than the

Chapter 3 89

continuum. Every pa r t i t i on of a continuum-equipotent s e t i n to countably many c lasses y i e lds a t l e a s t one c l a s s which i s continuum-equipotent (ch.1 5 4.3 , axiom of choice) . T h u s there e x i s t s an element xo i n X yie ld ing continuum many points w i t h abscissa x o and ord ina tes i n Y with the co lor (+). 2 - W i t h the s e t s X, Y as above, f o r each point y in Y , denote by f ( y ) the s e t of absc issas x i n X such t h a t the point (x ,y) has co lor (t). Note t h a t f ( y ) equipotent s e t of ord ina tes y w i t h the same f ( y ) , hence the same complement X - f ( y ) and f i n a l l y a (-)-monochromatic s e t which i s t he Cartesian product of a denumerable and continuum-equipotent s e t . T h u s t he re exists an element y o of Y yie ld ing continuum many poin ts w i t h o rd ina te yo and absc issas i n X , having the co lor (t). 3 - Beginning w i t h Xo = D and Yo = C , take x1 i n Xo yie ld ing continuum many points w i t h absc issa x1 and co lor (+) . Denote by Y1 the continuum- equipotent set of ord ina tes . Take y1 in Y1 y ie ld ing a denumerable s e t of points with ord ina te y1 and co lor (+) . Denote by XI the denumerable set of absc issas . I t e r a t e th i s , beginning v i t h X1 - {xl} and Y1 - {y l \ , thus obtaining an element x2 of t he f i r s t se t , and y2 an element of the second, w i t h continuum many poin ts with absc issa and co lo r (+), and denumerably many points w i t h o rd ina te y2 and co lor (+). The four points w i t h absc issas x l , x2 and ord ina tes y l , y2 have co lor (+) . By i t e r a t i o n , obtain a denumerable s e t of absc issas x i and a denumerable s e t o f ordina tes y i y ie ld ing the color (t) f o r a l l po in ts ( x i , y . ) ( i , j pos i t ive in t ege r s ) . 4 - I t follows t h a t i f the plane, which i s the Cartesian product of two sets both equipotent w i t h the continuum, is par t i t ioned i n t o two co lo r s , then e i t h e r there exists a denumerable "gr id" of the f i r s t co lo r , or there e x i s t both a g r id obtained as the product of denumerably many absc issas w i t h continuum many ord ina tes , and a g r id obtained as the product of continuum many abscissas with denumerably many ord ina tes , both of the second co lor . On the o ther hand, using SIERPINSKI's counterexample i n 3.1, we can pa r t i t i on the plane i n t o four co lors so t h a t every monochromatic g r id i s a t most the product of a denumerable s e t of absc issas w i t h a denumerable s e t of ord ina tes .

i s not always f i n i t e , f o r otherwise the re would e x i s t a continuum-

x2

J

Problem. Does there ex is t a par t i t ion of the plane into two colors, or into three colors, such that every monochromatic grid i s a t most the product o f a denumerable s e t with a denumerable set.

EXERCISE 4 - SPERNER'S LEMMA Let E be a s e t with f i n i t e cardinality 2h or 2h+ l . Then every s e t o f subse ts of E which are mutually incomparable with respect t o inclusion has cardinality a t most ( 2 h ) ! / ( h ! ) (even case) or ( 2 h + l ) ! / h ! ( h + l ) ! (odd case) . In other words, the largest possible cardinality i s obtained by taking the set of a l l h-element subsets. Beginning with a s e t Jf o f subsets of E , none of which i s included in another, by replacing i f necessary each se t by i t s complement, we can always assume that the smallest cardinality of the elements of 4 i s p,< h . We shall prove that i f p < h (even case) or p 4 h (odd case) , then we can injectively substitute for every subset A of minimum cardinality p a superset B of A of cardinality p + l . Indeed, i f t h i s i s possible, then the B will be d is t inc t and of the same cardinal p + l , hence will be mutually incomparable with respect t o inclusion. Moreover, no B can be included in an element of 4 of larger cardinality, since then the p-element se t A from which B was obtained would i t s e l f be included in tha t se t . Now note that SPERNER's lemma easi ly follows from the l inear independence lemma 5.1 in the case of E f i n i t e and q = 1 . For further developments on th i s subject, see POUZET, ROSENBERG 1982.

-. -

2

91

CHAPTER 4

GOOD AND BAD SEQUENCE, FINITELY FREE PARTIAL ORDERING, WELL PARTIAL ORDERING, IDEAL, TREE, DIMENSION

§ 1 - PARTIAL ORDERING OF THE INITIAL INTERVALS, PARTITION IN SLICES

1.1. INITIAL INTERVALS Let A of language, we confuse the notion of interval as subset of the base with that o f rest r ic t ion of A , and speak of the union and intersection of &! . The union of u i s an i n i t i a l interval o f A and i s the supremum or least upper bound of with respect t o inclusion; similarly the intersection of 4 i s an i n i t i a l interval and i s the infimum of u . I n the case of a chain, o r total 01 jering A , we get a total ordering of the in i t ia l intervals of A . To each i n i t i a l interval X of A , associate the final interval X ' , the complement of X ; the ordered pair ( X , X ' ) i s said t o be a cut of A . The comparison ordering between cuts i s defined by inclusion of their f i r s t terms, which are i n i t i a l intervals.

Separate the se t of cuts of A into two complementary subsets '& and v , such that every cut in u i s less t h a n every cut in v ; we have DEDEKIND's resul t :

u has a maximum cut , or ?f has a minimum cut. 0 Take the union C of the i n i t i a l intervals in the cuts in u , and the final

be a par t ia l ordering, k a se t of i n i t i a l intervals of A ; by an abuse

interval C ' , the complement of C . Then (C,C') this cut belongs t o then i t i s the maximum; i f the minimum. 0

l e t A be a chain, B a res t r ic t ion of A . If the common restr ic t ion B ' of B and A ' t o the

i s the supremum of . I f i t belongs t o t' then i t i s

A ' , A " ) i s a cut of A , then intersection of the bases, and

the common restr ic t ion B" of B and A " constitutes a cut ( B ' , B " ) of B , called the restriction, of the cut ( A ' " " ) t o B . We also say t h a t ( A ' " " ) i s a cut induced by ( B ' , B " ) on A . Obviously there may ex is t many such cuts.

1 . 2 . Let E be a s e t , f a se t of subsets of E . Two d is t inc t elements of E

are said t o be separable by f , i f there exis ts an element of ff which contains one and n o t the other.

Let be a s e t of subsets of E , which we assume t o be to ta l ly ordered under inclusion. For t o be a maximal chain with respect t o inclusion, i t i s necessary and suff ic ient t h a t f? be closed under unions and intersections

( f i n i t e or otherwise), and that any two d is t inc t elements of

I n general, l e t L& be a s e t of subsets of tion. Let f? be a subset of & which i s to ta l ly ordered by inclusion. - For and suff ic ient that f dist inct elements of

0 Assume that i s a chain which i s maximal among the restr ic t ions of &? . Then i s closed under union and intersection, for otherwise we could add the unions and intersections of elements of ‘f . Moreover, i f x , y are two d is t inc t elements of E which are separated by an element X o f ‘4 with x belonging t o X yet n o t y , then denote by U the union of the elements of which contain neither x nor y , and by V the intersection of the elements of which contain both x and y . By the preceding discussion, U and V are elements of . Since (e i s to ta l ly ordered by inclusion, then U i s s t r i c t l y included in V . Either ‘Cr contains an element separating x and y , in which case our conclusion holds. Or every element of f i s a subset o f U

Hence U and V are consecutive with respect t o inclusion in the chain . So ( U u X ) n V i s an element of (R , situated between U and V with respect t o inclusion, and d i s t inc t from U and V as i t contains x b u t n o t y . This contradicts the maximality o f . Conversely, assume that the chain i s not maximal among the to ta l ly ordered restr ic t ions of & . We can thus add t o f? a subset W of E which i s an element of L & , and e i ther including Or included in every s e t which belongs t o

and by V the intersection of those elements o f which include W . Then ei ther i t i s the case that U or V does not belong t o : so that i s e i ther n o t closed under union or under intersection. Or i t i s the case that U and V belong t o ‘t? : so t h a t W i s d i s t inc t from U and V , and hence i s properly situated between U and V . Let u be an element of W-U and v an element of V-W : the elements u and v are separated by the element W of dz , and yet are n o t separated by any element o f

E be separable t x v .

E , closed under union and intersec-

t o be maximal among to ta l ly ordered restr ic t ions of LR , i t i s necessary be closed under union and intersection, and that any t w q

E which are separable by ‘4 are also separable by f? . .-___

or a superset of V .

. Denote by U the union of those elements of which are included in W ,

. 0

1.3. For any par t ia l ordering A , denote by 3 ( A ) i n i t i a l intervals o f A (with respect t o inclusion). We know that 9 ( A ) i s closed under union and intersection. Moreover any two elements x , y of I A I are separable by 3 ( A ) : indeed we can assume t h a t x < y or x l y (mod A ) , and i t suffices t o take the interval 5 x (mod A) t o separate x and y .

the partial ordering o f the

Chapter 4 93

To every to ta l ly ordered reinforcement B of A (see ch.2 § 4 ) there corresponds the t o t a l ordering 3 ( 6 ) chains (mod B ) , and although n o t necessarily chains (mod A ) , they remain i n i t i a l intervals (mod A ) . I n other words, every element of J ( B )

Let A be a par t ia l ordering. If B i s a to ta l ly ordered reinforcement of A , then 3 ( B ) res t r ic t ions of 'j ( A ) . Conversely, every maximal to ta l ly ordered restr ic t ion of

moreover this B i s unique (BONNET, POUZET 1969). 0 Starting with a to ta l ly ordered reinforcement B of A , we already know t h a t 3(B) i s closed under union and intersection. Moreover, any two d is t inc t elements x , y of I A I are separable by J (B) : suppose for example that x c y (mod B ) x (mod 6) . By the preceding 1 . 2 , the total ordering 3(B) i s maximal among those to ta l ly ordered restr ic t ions of 3 ( A ) . Conversely, l e t f be a total ordering which i s maximal among those total ly ordered restr ic t ions of 3 ( A ) . By 1 . 2 , t h i s kf i s closed under union and intersection, and any two d is t inc t elements in the base I A I are separable by an element of . To obtain = j ( B ) , define the to ta l ly ordered reinforcement B of A by the condition tha t , when given two elements x , y of A , we p u t xd y (mod 6 ) i f f every element of e which contains y as an element also contains x as an element. The antisymmetry of B follows from the fact tha t , for d i s t inc t x , y , there exis ts an element of Moreover, since f i s a total ordering, B i s also. Finally, the uniqueness of B follows from the fac t that two d is t inc t total orderings B and B ' yield two d is t inc t 3 ( B ) and Y ( B ' ) . 0

of the i n i t i a l intervals of B . These intervals are

i s an element of 3(A) .

i s a total ordering which i s maximal among those total ly ordered

3 ( A ) i s of the form g(B) , where B i s a to ta l ly ordered reinforcement of A ;

and take the interval

which separates them.

1 .4 . I n par t icular , l e t E be an arbi t rary se t . For every chain B based on E , the i n i t i a l intervals of B form a maximal to ta l ly ordered restr ic t ion 3 ( 6 ) of the par t ia l ordering of inclusion. Conversely, every chain which i s a maximal res t r ic t ion of the par t ia l ordering of inclusion of subsets of E i s the s e t of i n i t i a l intervals 3(B) for a certain chain B on E . AXIOM OF MAXIMAL CHAIN OF INCLUSION This axiom asserts tha t , for every s e t E , the par t ia l ordering of inclusion among subsets o f E admits a maximal total ordering among i t s res t r ic t ions. I t i s considerably weaker than the general maximal chain axiom, as stated in ch.2 5 2 .7 : the HAUSDORFF-ZORN axiom, equivalent t o the axiom of choice. By the preceding proposition, the axiom of maximal chain o f inclusion i s equivalent t o the ordering axiom, ch.2 5 2.3.

AXIOM OF MAXIMAL C H A I N O N THE SET OF INITIAL INTERVALS This axiom asser ts t h a t , given a partial ordering A , the partial ordering of inclusion on the i n i t i a l intervals of A admits a maximal chain among i t s res t r ic- t-. I t i s equivalent t o the reinforcement axiom, ch.2 5 4.2.

1.5. Starting with an arbi t rary partial ordering A , there ex is t s t r a t i f i e d partial ordered reinforcements of A (see ch.2 5 5 . 2 ) : for examole the chains which are reinforcements of A . We obtain as follows a minimal s t r a t i f i e d reinforcement, in the sense that i f x ( y (modulo the reinforcement), then e i ther x c y (mod A) , or there ex is t x ' , y ' w i t h x -= x ' I y ' + y or with x & X I I y ' < y (mod A ) (however the converse can be fa l se) .

LEMMA FOR PARTITION IN SLICES Let A be a par t ia l ordering. There exis ts an equivalence relation R on I A I for which the equivalence classes are free subsets (mod A ) , and a c& H the s e t of equivalence classes, such that for any two elements x , y of I A I : (equivalence class of x ) 6 (equivalence class of y ) modulo H

i f f there ex is t two elements X I , y ' which are equivalent (mod R ) and sa t i s fy x + x ' and y ' d y (mod A ) (uses axiom o f choice; BONNET, POUZET 1969).

0 Firs t of a l l , denote by R any equivalence relation on the base 1 A l , and by H following condition: (equivalence class of x ) 6 (equivalence class of y ) (mod H ) i f f e i ther x i s equivalent t o y (mod R ) , or there exis ts an equivalence class U of R which i s a f ree subset of I A l and two elements x ' , y ' of U with x & x ' and y ' b y (mod A ) . Such ordered pairs ( R , H ) ex is t : i t suffices t o take R t o be the t r iv ia l equivalence relation with one equivalence class I A I , and H the chain on the singleton of I A I . An ordered pair ( R ' , H ' ) i s said t o be finer t h a n ( R , H ) i f every equivalence class (mod R ) which i s f ree (mod A) remains an equivalence class (mod R ' ) , and moreover every equivalence class (mod R ) i s a union of equivalence classes (mod R ' ) constituting an interval of H ' , and f inal ly for any two of these intervals , the total ordering induced by H ' i s identical t o the total ordering (mod H ) of the corresponding equivalence classes. The comparison "f iner than" defines a par t ia l ordering o n the ordered pairs We shall prove t h a t an ordered pair ( R , H ) are n o t a l l free (mod A ) admits an ordered pair (R' ,HI) which i s s t r i c t l y f iner . For thisn take an equivalence class U of R which i s not free. Take a subset V of IJ which i s maximal free (mod A) (ch.2 5 2.9) , and par t i t ion U into three dis joint subsets: V , the se t of elements having a greater element (mod A ) in V , and the s e t of elements having a smaller element (mod A ) in V .

on

any chain on the se t of the equivalence classes of R , which sa t i s fy the

( R , H ) . for which the equivalence classes

Chapter 4 95

Define the equivalence classes (mod R ' ) as those (mod R ) , except ing U which i s p a r t i t i o n e d i n t o three equivalence classes (mod R ' ) . The chain H ' i s the same

as H , except f o r the s u b s t i t u t i o n o f U by the th ree classes w i t h the obvious order ing . Using HAUSDORFF-ZORN axiom, consider a maximal ordered p a i r obtained from a maximal chain o f ordered pa i r s (R,H) , t o t a l l y ordered by the comparison " f i n e r

than". For t h i s maximal (R,H) , the equivalence classes def ined by R are a l l f r ee (mod A) . 0

1.6. There e x i s t s a p a r t i a l o rder ing w i t h c a r d i n a l i t y W 1 i n which every chain and every an t icha in i s countable (uses axiom o f choice; compare w i t h ch.7 exerc. 4 ) .

0 To every countable o rd ina l u i n j e c t i v e l y associate a r e a l r ( u ) (axiom o f choice). Define a p a r t i a l o rder ing based on the countable o rd ina ls

s e t t i n g u < v i f f simultaneously u < v modulo the usual order ing o f the ord ina ls and r ( u ) < r ( v ) modulo the usual o rder ing o f the rea ls . To ob ta in a

t o t a l l y ordered r e s t r i c t i o n o f t h i s p a r t i a l order ing, one must take a s t r i c t l y

inc reas ing sequence o f rea l s : such a sequence i s countable: ch.1 fj 4.5. To obtain a f ree set, one must take a s t r i c t l y decreasing sequence o f rea l s . 0

u, v ... by

1.7. The existence o f a tree (as def ined i n fj 6 below) o f card ina l

which every chain and every an t icha in i s countable, i s no t provable i n ZF and even no t provable using the axiom o f choice and the continuum hypothesis. This

problem i s r e l a t e d t o SUSLIN's hypothesis: see ch.5 f j 5 below.

cJ1 i n

1.8. GENERATED INITIAL INTERVAL, FINITELY GENERATED INTERVAL

Given a p a r t i a l o rder ing A and a subset 0 o f i t s base, the i n i t i a l i n t e r v a l

generated by D (mod A) i s the se t o f elements having an upper bound i n D . An i n i t i a l i n t e r v a l o f A i s sa id t o be f i n i t e l y generated i f i t i s generated

by a f i n i t e subset o f the base. We denote by r ( A ) i nc lus ion ) o f f i n i t e l y generated i n i t i a l i n t e r v a l s o f A . - I f A i s a well-founded p a r t i a l order ing, then so i s I f ( A ) (uses axiom o f dependent choice; see BIRKHOFF 1948)..

0 To each f i n i t e l y generated i n i t i a l i n t e r v a l X associate the se t o f i t s maxi-

mal elements (mod A) , which i s f i n i t e and non-empty (unless X i s i t s e l f

empty), and which i s a f ree set, i .e . a se t o f mutual ly incomparable elements. Le t US say t h a t a s e t Y i s smal ler than Y ' i f f f o r each element y i n Y

the p a r t i a l o rder ing (under

t h e r e e x i s t s a t l e a s t one upper bound o f y i n Y ' (mod A) . Then i t s u f f i c e s t o prove t h a t t h e r e does n o t e x i s t any s t r i c t l y dec reas ing u - s e q u e n c e o f f i n i t e f r e e

se ts , w i t h r e s p e c t t o t h e p reced ing comparison (dependent cho ice ) . Given such a sequence Yi ( i i n t e g e r ) , e i t h e r each e lement o f Y o be longs t o f i n i t e l y many

s e t s Yi : then remove these Yi so t h a t Yo i s d i s j o i n t f rom a l l o t h e r s e t s Y . O r t h e r e e x i s t elements o f Y o which be long t o i n f i n i t e l y many s e t s Yi . Then remove one o f these elements and keep o n l y t h e t i n g t h i s procedure, we e l i m i n a t e a l l such e lements f rom

o f Y o . F i n a l l y we o b t a i n a new, non-empty which i s d i s j o i n t f rom a l l o t h e r

s e t s Yi . I t e r a t i n g t h i s , we o b t a i n a s t r i c t l y dec reas ing w-sequence o f f i n i t e

f r e e , m u t u a l l y d i s j o i n t s e t s .

L e t Zi be these d i s j o i n t se ts . There e x i s t s a t l e a s t one e lement zo i n Zo w i t h s m a l l e r elements i n a l l o t h e r s e t s Zi . Then a t l e a s t one z1 < zo i n Z1 w i th s m a l l e r elements i n a l l s e t s Zi (i > 1) ; and so on. Thus we have a s t r i c t - l y dec reas ing w-sequence o f elements (mod A) : c o n t r a d i c t i o n . 0

Yi t o which i t belongs. By i t e r a - b u t n o t a l l elements Yo

Yo

5 2 - LESS THAN RELATION AMONG SEQUENCES, EMBEDDING BETWEEN SEQUENCES, GOOD, BAD, MINIMAL BAD, STRONGLY MINIMAL BAD SEQUENCE

Given a sequence u i n t h e p a r t i a l o r d e r i n g A , t h e i n i t i a l i n t e r v a l generated by u i s t h a t generated by t h e s e t o f va lues o f u : see 1.8 above.

LESS THAN RELATION, EMBEDDING BETWEEN SEQUENCES L e t A be a p a r t i a l o r d e r i n g and u, v be two sequences w i t h va lues i n A , b o t h w i t h t h e same l e n g t h g . The sequence v i s s a i d t o be l e s s than u (mod A ) o r s m a l l e r t han u , o r u i s s a i d t o be q r e a t e r t han v (mod A) , i f vi< ui (mod A) f o r each index i < o( thus d e f i n e s a p a r t i a l o r d e r i n g on e v e r y s e t o f sequences i n A o f a g i v e n l e n g t h .

Given a p a r t i a l o r d e r i n g A and two sequences u, v i n A , we say t h a t v i s embeddable i n u o r t h a t u admits an embedding o f v (mod A) , i f t h e r e e x i s t s an e x t r a c t e d sequence o f u ( ch .1 5 2.2) which i s g r e a t e r t han v (mod A) . Consequently ( l e n g t h o f v)s ( l e n g t h o f u ) . Embeddab i l i t y i s r e f l e x i v e , t r a n s i t i v e b u t n o t an t i symmet r i c , s i n c e t h e n o t i o n o f e x t r a c t e d sequence i s n o t i t s e l f a n t i -

symmetric: i t d e f i n e s a q u a s i - o r d e r i n g on e v e r y s e t o f sequences i n A . Two sequences which a r e each embeddable i n t h e o t h e r , a re n o t n e c e s s a r i l y e x t r a c - t e d one f rom t h e o t h e r : i n t h e t o t a l o r d e r i n g o f t h e p o s i t i v e i n t e g e r s , t ake t h e i n c r e a s i n g sequences o f even i n t e g e r s , r e s p e c t i v e l y odd i n t e g e r s .

I f a sequence v i s embeddable i n u , then t h e i n i t i a l i n t e r v a l generated b y v i s i n c l u d e d i n t h e i n i t i a l i n t e r v a l generated by u . Note a l s o t h a t , modulo t h e axiom o f choice, s t a r t i n g w i t h an i n i t i a l i n t e r v a l A of a p a r t i a l o r d e r i n g , i t

s u f f i c e s t o w e l l - o r d e r I A I i n o r d e r t o o b t a i n a sequence which generates A .

. T h i s comparison i s r e f l e x i v e , an t i symmet r i c , t r a n s i t i v e ,

Chapter 4 97

PARTIAL ORDERING OF WORDS S t a r t i n g wi th a p a r t i a l o r d e r i n g A , t a k e t h e s e t o f words, o r f i n i t e sequences o f elements o f t h e base I A l . Embeddab i l i t y between words i s an t i symmet r i c and so de f i nes a p a r t i a l o r d e r i n g o f words. I f A i s a we l l - f ounded p a r t i a l o r d e r i n g , t hen t h e p a r t i a l o r d e r i n g o f words i n A i s wel l - founded.

0 S t a r t i n g w i t h a non-empty s e t o f words i n A , o b t a i n a min imal word i n t h i s

set . F i r s t l y t a k e t h e subset o f words o f minimum l e n g t h

words beginning w i t h a te rm words beginning w i t h uo, u1 where u1 has min imal value; and so on. 0

-

n , and among such, those

u,, o f min imal va lue. Then among t h e l a t t e r , those

2.1. INITIAL INTERVAL OF A SEQUENCE

Given a sequence u w i t h l e n g t h o( , an i n i t i a l i n t e r v a l o f u i s any r e s t r i c -

t i o n o f u t o an o r d i n a l domain s m a l l e r t han o r equal t o o( . The i n i t i a l i n t e r - val i s c a l l e d s t r i c t o r p roper i f f i t s l e n g t h i s s t r i c t l y l e s s than o( . Let A be a p a r t i a l o r d e r i n g , u, v be two o rd ina l - i ndexed sequences w i t h va lues

i n A . I f t h e l e n g t h o f u i s a l i m i t o r d i n a l and i f each p roper i n i t i a l

i n t e r v a l o f u i s embeddable i n v (mod A ) ,then u i s embeddable i n v . 0 Le t h(0) be t h e l e a s t o r d i n a l such t h a t vh(o) uo (mod A) . By i n d u c t i o n ,

given an o r d i n a l i s t r i c t l y l e s s than , assume t h a t t h e s t r i c t l y i n c r e a s i n g sequence o f t h e h ( j ) (j < i) i s de f i ned , and l e t h ( i ) be t h e l e a s t o r d i n a l s t r i c t l y g r e a t e r t han t h e h ( j ) and such t h a t v ~ ( ~ ) > ' ui (mod A) . C l e a r l y t h i s

h ( i ) i s t h e s m a l l e s t poss ib le , and by hypo thes i s f o r each i s t r i c t l y l e s s than 7 , our h ( i ) i s s t r i c t l y l e s s than t h e l e n g t h o f v : o u r p r o p o s i t i o n f o l l o w s . 0

2.2. GOOD SEQUENCE, BAD SEQUENCE

Let A be a p a r t i a l o rde r ing . A sequence u i n A w i t h l e n g t h ( o r domain)

o( i s c a l l e d &(mod A) i f f t h e r e e x i s t a t l e a s t two i n d i c e s i, j with i < j .C a i f f every te rm i s < o r I (mod A) w i t h t h e terms o f s t r i c t l y s m a l l e r i ndex . Every sequence e x t r a c t e d f rom a bad sequence i s bad. Given a bad cs -sequence u , f o r eve ry uo o b t a i n e d f rom u by a pe rmuta t i on

o f the s e t o f i n d i c e s , t h e r e e x i s t s a bad w -sequence e x t r a c t e d f rom u o . We can even r e q u i r e t h a t t h e bad e x t r a c t e d sequence beg in w i t h , f o r i ns tance ,

the f i r s t t e rm uo o f u .

and ui Q u. (mod A ) . Otherwise a sequence i s c a l l e d bad, i . e . J

2.3. MINIMAL BAD SEQUENCE

Let A be a p a r t i a l o r d e r i n g . A bad a -sequence u i n A i s s a i d t o be

minimal bad i f , f o r any &-sequence v ex t rac ted from u , every bad sequence l e s s than v (mod A) i s equal t o v . Every a-sequence ex t rac ted from a minimal bad sequence i s minimal bad. I n a minimal bad sequence, a l l the terms are mutua l l y incomparable. 0 Le t u be a minimal bad sequence i n the p a r t i a l o rder ing A , and l e t p C q be two in tegers w i t h u > u (mod A) . Def ine the ex t rac ted sequence v by

P 9 p-l , v = uq< u , and vpti = uq+i f o r every i 3 1

Then v i s bad since i t i s ex t rac ted from u . Moreover v i s a sequence less

-

P P vo = uo ,..., v = u P: 1

,... which i s ex t rac ted from u , and p-1 ' up ' Uq+l ' uq+2 than uo ,..., u as v i s d i s t i n c t from t h i s ex t rac ted sequence, t h i s cont rad ic ts the m in ima l i t y o f u . 0

Le t A be a p a r t i a l order ing, u a minimal bad sequence i n A , and l e t p be an integer, and x < u (mod A) . Then x < a l l terms o f u (mod A) , except

P poss ib ly f i n i t e l y mani. 0 Assume on the cont ra ry t h a t there e x i s t s a s t r i c t l y inc reas ing sequence o f integers p < f ( 1 ) < f ( 2 ) < ... C f ( i ) < ... w i t h u < o r 1 x (mod A) . Then the sequence x, uf(l) ,..., uf(i) ,... i s a bad sequence less than up, uf(l),...

u ~ ( ~ ) , . . Not ice t h a t a sequence w i t h incomparable terms which s a t i s f i e s the cond i t i on o f the preceding p ropos i t i on i s no t necessar i l y minimal. 0 Take A t o be a well-founded p a r t i a l o rde r ing formed from denumerably many minimal elements which we denote by the in tegers 0, 1, 2 , , . and w i t h O'> 0 , 1'> 0 and 1 '> 1 , and f o r every i n tege r i an element i ' 7 0, 1, ..., i and f i n a l l y make the i' mutua l ly incomparable. Then the sequence 0 ' , 1' , 2 ' ,.. . has incomparable terms and s a t i s f i e s the cond i t ion o f the preceding propos i t ion , but the sequence 0, 1, 2, . . . i s a bad sequence which i s l ess than the sequence considered. 0

-

f ( i )

which con t rad i c t s the m in ima l i t y o f u . 0

2.4. Let A be a p a r t i a l order ing. I f u i s a minimal bad w-sequence (mod A) , then the image of u by an a r b i t r a r y permutation o f t he s e t o f ind ices i s minimal bad. 0 Denote by uo the image o f u under a permutation o f the ind ices . By 2.3, the terms of u , hence o f uo , are mutua l l y incomparable, so t h a t uo i s bad. Suppose t h a t uo i s no t minimal bad. Then there e x i s t s an W -sequence v ex t rac ted from u0 and a smal le r w -sequence w , which i s bad and d i s t i n c t from v . We can assume t h a t the f i r s t terms o f v and w are d i s t i n c t , so wo < vo (mod A) . Retransform uo, v and w by the inverse o f our i n i t i a l permutation. Then we have again u and we obta in the images o f v and w . L e t w' denote the l a t t e r image. By 2.2 there e x i s t s a bad m-sequence

Chapter 4 99

extracted from w ' , which begins by wo (more exactly, by the term of w ' whose value i s wo ). This contradicts the minimality of u ( t h i s proof using only ZF i s communicated by H O D G E S ) . 0

2.5. Given a partial ordering A and a minimal bad sequence u in A , the in i t ia l interval of A generated by u i s a well-founded partial ordering (uses dependent choice; ZF suffices i f A i s countable, or obviously i f A i s well-founded). 0 Assume on the contrary that there exis ts a term u of u with a s t r i c t l y decreasing choice). Then

P -sequence up > a. > a l 7 ... > ai > ... (mod A) (dependent

a. 6 a l l except f in i te ly many of the terms of u : see 2.3.

Hence there exists an integer q such that a. 6 uqti for every integer i . Finally we have ai < uq+i fo r every integer i , thus contradicting the minimality of the sequence u . 0

2.6 . Let A be a par t ia l ordering, u an (.J -sequence in A with mutually incomparable terms. Then u i s minimal bad i f f every bad a-sequence embeddable in u i s extracted from u . 0 If u i s minimal bad, our conclusion follows immediately from the definition. Conversely, suppose that u minimal bad. Then there ex is t s a s t r i c t l y increasing a-sequence of integers f (0) < f ( l ) < ... < f ( i ) < ... with a bad sequence i s a dis t inct sequence less than uf(o) ,uf( l ) ,..., u f t i ) ,... . T h u s there exis ts an integer p such t h a t x q u (mod A ) . Consider the sequence o f the x (j = 0,1,2, ...) . This sequence i s embeddable in u and moreover i s bad since x i s bad. By hypothesis, t h i s sequence i s extracted from u , hence there exis ts a term u = x 4 u (mod A ) , which contradicts the hypothesis t h a t the terms o f u are mutually incomparable. 0

The hypothesis o f incomparability of the terms of u Take A t o be a well-founded partial ordering with O < 0 ' and denumerably

many integers 1,2,3, ... which are s e t t o be mutually incomparable, and also incomparable with 0 and 0 ' . Then the sequence 0',0,1,2,3, ... i s bad yet not minimal bad (since 0 ' > 0 ) , and every bad w-sequence embeddable in i t i s extracted from i t . 0

Moreover, a bad non-minimal bad w -sequence u can sat isfy the condition t h a t every bad sequence less than u i s equal to u . nTake an element a s e t t o be minimal, and the even integers s e t t o be mutually incomparable, with 0 > a , 2 > a , . . . . Take the odd integers 1 ,3 , . . . a l l to be minimal and incomparable w i t h the preceding elements. Then the sequence of a l l

has incomparable terms, so i s bad, and yet not

x = (xo ,x l ,..., xi ,...) which

P f ( P ) p+j

q P f ( p )

i s necessary.

i n t e g e r s 0,1,2,. . . i s n o t min imal s i n c e a i s l e s s t h a n eve ry even i n t e g e r . Yet

eve ry d i s t i n c t sequence l e s s than i t must t a k e t h e va lue an even i n t e g e r , and hence i s good. 0

Problem. L e t A be a p a r t i a l o r d e r i n g , u an w - s e q u e n c e i n A w i t h incompa-

r a b l e terms. F o r u t o be min imal bad, does i t s u f f i c e t h a t eve ry w -sequence w i t h incomparable terms which i s embeddable i n u be e x t r a c t e d f rom u .

a i n t h e p o s i t i o n o f

2.7. THEOREM ON THE MINIMAL BAD SEQUENCE L e t A be a wel l - founded p a r t i a l o r d e r i n g . F o r eve ry bad w-sequence u & A , t h e r e e x i s t s a min imal bad sequence which i s embeddable i n u (uses dependent

choice; ZF s u f f i c e s i f A i s denumerable). -

0 Replace t h e f i r s t t e rm uo o f u by voc< uo w i t h vo min imal among those elements x & uo f o r which t h e r e e x i s t s a bad w-sequence which begins by x and i s embeddable i n u . Denote by vo, w1 t h e f i r s t two terms i n such a bad m-sequence embeddable i n u . Replace w1 by vl& w1 w i t h v1 min imal

among those vo, x and i s embeddable i n u . Denote by vo, vl, w2 t h e f i r s t t h r e e terms i n

such a bad w-sequence embeddable i n u . By i t e r a t i o n , we o b t a i n t h e sequence

v = vo, vl, v2, ... which i s bad and embeddable i n (dependent c h o i c e ) . More- over , f o r eve ry i n t e g e r i and e v e r y x < vi (mod A ) , no w - s e q u e n c e beg inn ing

by vo, vl, ..., vi-l, x i s s imu l taneous ly bad and embeddable i n u . To see t h a t i n c r e a s i n g u - s e q u e n c e o f i n t e g e r s f ( O ) < f ( l )< ... < f ( i ) < ... and a bad

W -sequence x = (xo, xl, ..., xi,...) which i s d i s t i n c t and l e s s than v

vf(l) ,..., vf( i) ,... . Then t h e r e e x i s t s an i n t e g e r p w i t h x < v

Consider t h e sequence vo, vl, ..., v ~ ( ~ ) - ~ , xp, x ~ + ~ , . . . . Th is sequence i s embeddable i n v and so embeddable i n u . By t h e p reced ing i n e q u a l i t y , t h i s

sequence i s good. As t h e sequences x and v a r e bad, t h e r e e x i s t s an i n t e g e r

j C f ( p ) and an i n t e g e r k >/ p , hence j <f(p),< f ( k ) w i t h v j 6 xk 6 v (mod A ) , so t h a t v i s good: c o n t r a d i c t i o n . 0

x B w l f o r which t h e r e e x i s t s a bad a - s e q u e n c e which begins by

u

v i s min imal , suppose on t h e c o n t r a r y t h a t t h e r e e x i s t s a s t r i c t l y

f(O) ’ (mod A)

P f (P)

f ( k )

2.8. STRONGLY MINIMAL BAD SEQUENCE

Given a p a r t i a l o r d e r i n g A , a bad w -sequence u i n A i s c a l l e d s t r o n g l y

min imal bad i f , f o r each i n t e g e r i

w -sequence i n A beg inn ing by uo, ul, .... ui-l, x i s good.

Every s t r o n g l y min imal bad sequence i s min imal bad.

0 Suppose t h a t u i s s t r o n g l y min imal bad b u t n o t min imal bad. There e x i s t s a

s t r i c t l y i n c r e a s i n g W-sequence o f i n t e g e r s f ( O ) < f ( l )< ... < f ( i ) < ... and a d i s t i n c t sequence x l e s s than u ~ ( ~ ) , u ~ ( ~ ) , . . . , u ~ ( ~ ) , . . . . Hence t h e r e e x i s t s an i n t e g e r p wi th x < u ~ ( ~ ) (mod A) . The w -sequence uo, ul, ...

and each element x < ui (mod A) , every

P

Chapter 4 101

u ~ ( ~ ) - ~ , xp , x ~ + ~ , . . . i s good, since u i s strongly minimal. As the sequences

u and x are bad, there ex is t s an integer j < f ( p ) and also ka p w i t h j 4 f ( p ) 5 f ( k ) and u . d x k ,< u f ( k ) (mod A) , hence u i s good: contradiction. 0

There exis t minimal, non-strongly minimal sequences. More precisely, there exis ts a n w-sequence extracted from a strongly minimal bad sequence, which i s not strongly minimal; however by 2.3 i t i s minimal bad. In addition, there exis ts a strongly minimal bad sequence and a sequence obtained from i t by permutation of the s e t of indices, which i s not strongly minimal bad; however by 2.4 i t i s minimal bad.

J

0 Take two minimal elements a , b and p u t a l l even numbers 0 , 2 , 4 , . . . t o be mutually incomparable and incomparable with b , b u t a l l 7 a . P u t a l l odd numbers 1, 3 , 5 , ... t o be mutually incomparable, a lso incomparable with the even numbers and with a , yet a l l 7 b . Then the sequence b , 0 , 2 , 4, ... i s strongly minimal; b u t 0 , 2, 4 , ... i s n o t , since, for each integer i , the sequence 0 , 2 , 4 ,..., 2 i , a , 1, 3 , 5 ,... i s bad. 0

0 Take a minimum element a , then b , c mutually incomparable and > a . P u t the even numbers 0 , 2 , 4 , ... to be mutually incomparable and incomparable with c , yet a l l 7 b . P u t the odd numbers t o be mutually incomparable, also incomparable with the even numbers and with b , yet a l l 7 c . Then the sequence c , 0 , 2 , 4 , ... indeed every strongly minimal bad sequence begins with b or c . 0

i s strongly minimal bad; b u t some of i t s permutations are not:

2.9. There exis ts a minimal bad sequence which i s n o t extracted from any strongly minimal bad sequence. 0 Let A be the well-founded partial ordering constructed in the following manner. Begin with the sequence of integers i in increasing order. To each i

associate an element i ' 7 i , the i ' being mutually incomparable, each i ' incomparable with a l l integers > i . Finally add inf ini te ly many minimal elements ao, a l , ... which are incomparable with the i and i ' . Let u be the sequence of the i ' . This sequence i s minimal bad, since for each extracted W -sequence, every d is t inc t smaller sequence i s good. Suppose t h a t u i s extracted from a strongly minimal bad GJ -sequence v . Then v i s obtained from u by inserting elements a i . Let p be an integer such t h a t the pth term of v i s a term of u , in other words an i ' ( i integer .\< p ). Hence v begins by O', l ' , Z', ..., i ' between which can be inserted elements a j . I t suffices to replace i ' by i , which i s incomparable with 0 ' , 1' , . . . , ( i -1) ' and which is < i' , and t o add a f t e r i an a-sequence of elements a . which are not already inserted, in order t o obtain a bad sequence. J

T h i s c o n t r a d i c t s t h e s t r o n g m i n i m a l i t y o f v . 0

2.10. Theorem 2.7 as s t a t e d does n o t ex tend t o s t r o n g l y min imal bad sequences. 0 Take t h e we l l - f ounded p a r t i a l o r d e r i n g o f t h e second counterexample 2.6, w i t h

t h e min imal element a f o l l o w e d by t h e even i n t e g e r s 0,2, ... t o g e t h e r w i t h

t h e odd i n t e g e r s taken as min imal and incomparable w i t h t h e even i n t e g e r s . Then

t h e sequence o f even i n t e g e r s sequences as embedded sequences. I t i s n o t s t r o n g l y min imal bad, s i n c e f o r each

i n t e g e r i , t h e sequence 0,2,4 ,..., 2i,a,1,3 ,... i s bad. 0

However, g i v e n a wel l - founded p a r t i a l o r d e r i n g A , i f t h e r e e x i s t s a bad

(uses dependent choice; ZF s u f f i c e s i f A i s coun tab le ) .

0,2, ... i s bad and admi ts o n l y i t s e l f and good

w - s e q u e n c e i n A , then t h e r e e x i s t s as w e l l a s t r o n g l y min imal bad sequence

0 Take an element uo min imal among t h e f i r s t terms o f bad a -sequences , t hen an element min imal among those x f o r which t h e r e e x i s t s a bad w-sequence be-

g i n n i n g by uo, x e t c . 0

F i n a l l y , by t h e method i n t h e f i r s t paragraph o f t h e p r o o f o f 2.7, we see t h a t : g i ven a we l l - f ounded p a r t i a l o r d e r i n g A and a bad w-sequence u i n A , t h e r e

e x i s t s a bad w -sequence v embeddable i n u , such t h a t f o r each i n t e g e r p

and each x < v (mod A) , every w -sequence beg inn ing by vo, v1 ,... , vp-l, x and which i s embeddable i n u i s good. We can c a l l t h i s sequence s t r o n g l y min imal w i t h r e s p e c t t o e m b e d d a b i l i t y z

t h e sequence u (as i n 2.7, we use dependent choice; ZF s u f f i c e s i f A c o u n t a b l e ) .

P

v

§ 3 - FINITELY FREE PARTIAL ORDERING, WELL PARTIAL ORDERING

FINITELY FREE PARTIAL ORDERING

Th is i s a p a r t i a l o r d e r i n g such t h a t any f r e e subset o f i t s base, o r any a n t i -

chain, i s f i n i t e (see ch.2 § 2.9) . A p a r t i a l o r d e r i n g can be we l l - f ounded w i t h o u t b e i n g f i n i t e l y f r e e : t a k e an i n f i - n i t e base and t h e p a r t i a l o r d e r i n g o f i d e n t i t y , t hen a l l elements a r e m u t u a l l y incomparable. A p a r t i a l o r d e r i n g can be f i n i t e l y f r e e w i t h o u t b e i n g wel l - founded:

take t h e n a t u r a l o r d e r i n g on t h e n e g a t i v e i n t e g e r s .

Every f i n i t e p a r t i a l o r d e r i n g i s f i n i t e l y f r e e .

Every r e s t r i c t i o n o f a f i n i t e l y f r e e p a r t i a l o r d e r i n g i s f i n i t e l y f r e e . Every p a r t i a l o r d e r i n g which i s a re in fo rcemen t o f a f i n i t e l y f r e e p a r t i a l orde-

r i n g i s f i n i t e l y f r e e (see ch.2 5 4 .1 ) .

3.1. Given a f i n i t e l y f r e e p a r t i a l o r d e r i n g A w i t h i n f i n i t e base, t h e r e e x i s t s a t o t a l l y o rde red r e s t r i c t i o n o f A which i s e q u i p o t e n t w i t h A (uses axiom o f

Chapter 4 103

choice; ZF s u f f i c e s i f A i s countable o r i f Card A i s a regu la r aleph!. 0 P a r t i t i o n the pa i r s o f elements x, y o f the base I A l i n t o two co lo rs , according t o whether x and y are comparable o r incomparable (mod A). Assuming the axiom o f choice, take an aleph equipotent t o the base, and apply the p a r t i t i o n theorem ch.3 5 3.3 (DUSHNIK, M I L L E R ) . Then e i t h e r there e x i s t s a denumerable subse t w i t h a l l i t s elements incomparable: con t rad ic t ion . O r there e x i s t s a subset w i th same c a r d i n a l i t y as the base, a l l o f whose elements a re comparable. 0

3.2. WELL PARTIAL ORDERING

This i s a p a r t i a l o rder ing which i s well-founded and f i n i t e l y f ree . For example, every f i n i t e p a r t i a l o rder ing i s a we l l p a r t i a l order ing. S i m i l a r l y f o r any wel l -order ing, o r even a we l l -o rder ing where each element i s replaced by a f ree f i n i t e set .

A quasi-order ing ( r e f l e x i v e and t r a n s i t i v e b inary r e l a t i o n ) i s sa id t o be a we l l quas i -o rder ing iff the p a r t i a l o rder ing o f t he equivalence classes i s a we l l p a r t i a l o rder ing (each c lass i s formed by elements each o f which i s both g rea ter and smal le r than other elements).

(1) A i s a we l l p a r t i a l o rder ing i f f f o r any non-empty subset X o f the base, the s e t o f minimal elements (mod A/X) i s f i n i t e and non-empty.

0 Le t A be a we l l p a r t i a l o rder ing and X a non-empty subset o f the base. Since A i s well-founded, there e x i s t minimal elements (mod A/X): ch.2 5 2.4. These elements are mutual ly incomparable (mod A), hence there are on ly f i n i t e l y many such, s ince A i s f i n i t e l y f ree . Conversely i f A i s no t a we l l p a r t i a l ordering, then e i t h e r there e x i s t s a subset X o f the base w i thout any minimal element, o r there e x i s t s an i n f i n i t e f r e e subset X . I n the l a t t e r case a l l the elements o f X are minimal (mod A/X). 0

( 2 ) A necessary and s u f f i c i e n t cond i t i on f o r A t o be a we l l p a r t i a l order ing, i s t h a t every Cd-sequence i n A be good ( the su f f i c i ency uses dependent choice; ZF s u f f i c e s i f A i s countable o r has wel l -orderable base). 0 Le t A be a we l l p a r t i a l order ing. If there e x i s t s a bad w-sequence i n A , then there i s an ex t rac ted w -sequence which i s e i t h e r s t r i c t l y decreasing o r with a l l i t s terms mutua l l y incomparable: see ch.3 5 1.2 (RAMSEY). Conversely, i f A i s no t a we l l p a r t i a l order ing, then e i t h e r A i s no t we l l - founded, so there e x i s t s a s t r i c t l y decreasing w-sequence (ch.2 § 2.4, dependent choice). O r there e x i s t s an i n f i n i t e f r e e set, hence an w-sequence o f mutua l l y incomparable elements: i n both cases, a bad sequence. 0

(3) A necessary and s u f f i c i e n t cond i t i on f o r A t o be a we l l p a r t i a l o rder ing i s t ha t , f o r every w-sequence i n A w i t h d i s t i n c t values, there e x i s t s an

E j g ,

extracted same proof as for ( 2 ) .

w -sequence which i s s t r i c t l y increasing (mod A ) . Same conditions and

3.3. A necessary and suff ic ient condition for A t o be a well partial ordering i s tha t , for each element u of the base I A I , the res t r ic t ion of A elements < 1 u i s a well partial orderinq. 0 I f A i s a well partial ordering, then every restr ic t ion of A i s a well partial ordering: hence the necessity of our condition. Conversely, assume that the condition holds. Let be a non-empty free subset of the base. Take an element u of X . By hypothesis X - { u) i s f i n i t e , so X i s f i n i t e : thus A i s f ini te ly f ree . Suppose now tha t X base. Take an element u of X and l e t Y be the s e t of those elements n o t greater than o r equal t o u (mod A) . Then ei ther Y i s empty, so that u i s minimal in X . Or Y i s non-empty and so by hypothesis has a minimal element which i s also minimal in X . Thus A i s well-founded (proof of POIZAT). 0

X

i s an arbi t rary non-empty subset of the

3.4. THE EXTRACTION THEOREM FOR WELL PARTIAL O R D E R I N G S Let A be an in f in i te well partial ordering and C J ~ a given aleph. Let u be a sequence in A with d is t inc t values a n d length ad . Then there exis ts a sequence extracted from u which i s s t r i c t l y i n c r e a g x (mod A ) and has same length as u i s singular; ZF suffices i f ad i s regular) .

0 Partition the pairs of indices i , j ( i < j < ad ) into two colors: the color (t) i f 5 3.3 (OUSHNIK, M I L L E R ) , e i ther there exis ts a se t of indices equipotent to U 4 ,

a l l of whose pairs have color (+ ) , and hence an and s t r i c t l y increasing. Or there ex is t s a denumerable set of indices a l l of whose pairs have color ( - ) , and hence a bad 0 -sequence (mod A ) , thus contradicting 3.2.(2) . 0

Conversely by 3 .2 . (3) , only well partial ordering5 sat isfy our conclusion. However, the ordinal product 0 l.( W - ) ; and the partial ordering with denumerably many mutually incomparable components, each isomorphic with W

sat isfy our conclusion, when restr ic ted t o sequences (communicated by POUZET; see also 9.3 below).

3.5.(1) Every par t ia l ly ordered reinforcement of a well partial ordering i s a well partial ordering.

Let A be a well partial ordering, and B a par t ia l ly ordered reinforcement of A . If X

(consequence of DUSHNIK, M I L L E R 1941; uses axiom of choice i f wd

u i or I u . (mod A ) . Using ch.3 J J

C+-sequence extracted from u

, u of length Wac( o( # 0)

i s a free se t (mod B ) , i . e . a s e t whose elements are incomparable

Chapter 4 105

(mod B), then X i s f r e e (mod A ) and hence f i n i t e ; thus B i s f i n i t e l y f ree . Now l e t X be a non-empty subset o f the c o m n base IAI = I B I . Le t Y be the se t o f those elements o f X which are minimal (mod A) . Since these elements are incomparable (mod A), there are on ly f i n i t e l y many. Hence there e x i s t s i n Y an

element y which i s minimal (mod B) . We sha l l show t h a t y i s minimal (mod B) f o r the se t X , which then imp l ies t h a t B i s well-founded. I f there e x i s t s an

element x of X such t h a t x < y (mod B) , then the non-empty se t o f elements of X which are 6 x (mod A) ha5 a minimal element z ; so t h a t 2 6 x < y

(mod B) and z i s an element of Y . But t h i s cont rad ic ts the m in ima l i t y o f y

(mod B) i n Y (p roo f communicated by POIZAT 1976, using on ly ZF). 0

(2) A necessary and s u f f i c i e n t cond i t ion fo r A t o be a we l l p a r t i a l order ing, i s t h a t every t o t a l l y ordered reinforcement o f A be a well-ordering(sufficiency uses the reinforcement axiom p lus dependent choice; ZF su f f i ces i f A i s coun- tab le ) . 0 If A i s a we l l p a r t i a l order ing, then by the previous propos i t ion every t o - t a l l y ordered reinforcement o f A i s a we l l -o rder ing . Conversely, suppose t h a t A i s no t a we l l p a r t i a l order ing. Then e i t h e r A i s not well-founded, so t h a t there e x i s t s a s t r i c t l y decreasing w-sequence (mod A ) : see ch.2 5 2.4, dependent choice. O r A has an i n f i n i t e f r e e subset, hence a denumerable f r e e subset. I n the l a t t e r case, take a t o t a l o rder ing 0 - (converse o f I% , see ch.2 5 1.7) on t h i s f r e e subset. By the reinforcement axiom, and i t s immediate consequence i n ch.2 0 4.2, there e x i s t s a t o t a l o rder ing which extends fd and i s a reinforcement o f A . 0

(3 ) I f every t o t a l l y ordered reinforcement o f A i s a well-orderina,and.jf there e x i s t s such a t o t a l order ing, then A i s a we l l p a r t i a l o rder ing (p roo f using ZF alone, POUZET 1979, unpublished). 0 L e t C be a we l l -o rder ing which re in fo rces A . Then A i s well-founded, f o r i f not then there would e x i s t a subset D o f the base w i th A/D having no min i - mal element, so C/D w i thout any minimal element. Suppose now t h a t there e x i s t s an i n f i n i t e f r e e (mod A) subset H . Le t I denote the i n i t i a l i n t e r v a l o f those elements x f o r which there ex i s t s an element y > x (mod A) w i t h y d H . S i m i - l a r l y denote by F t he f i n a l i n t e r v a l o f those x f o r which there e x i s t s an e le - ment Y < x w i t h y 6 H . F i n a l l y l e t L be the se t o f those elements which are incomparable (mod A) w i t h a l l elements o f H . The fou r se ts H, I, F, L are d i s j o i n t and form a p a r t i t i o n o f the base, Then the t o t a l o rder ing C/I t C/L .t

(C/H)- + C/F i s a reinforcement of A (where (C/H)- i s t he converse chain o f C/H ) . However, as H i s i n f i n i t e , the we l l -o rder ing C/H i s isomorphic t o an o rd ina l >/ r+, , and hence i t s converse i s no t a wel l -order ing. 0

--

3.6. (1) For every inf in i te well partial ordering A , we have Card H t A = Card A (uses axiom of choice for f i n i t e se t s ; ZF suffices i f A well-orderable base). 0 To each ordinal i H t A associate the f i n i t e s e t Ai of elements with height i , then the f i n i t e se t of total orderings based on each i one of these total orderings called C i (choice for f i n i t e s e t s ) . Denote by C the well-ordering which i s the sum of the C i according t o increasing values of i . To each component w of the ordinal H t A , bijectively associate the component w of C obtained by the substitution of Ci fo r each i . Thus the order type of C d i f fers from the ordinal H t A by a t most a f i n i t e ordinal: the i r cardinals are t h u s equal. 0

(2) If A i s a f in i te ly f ree par t ia l ordering, then ei ther Cof A i s f i n i t e and Cofh A = 1 Cofh A = Cof A (uses axiom of choice; ZF suffices i f A i s countable or has well-orderable base). 0 Recall t h a t , for any par t ia l ordering A , we have Cofh A 6 Cof A (see ch.2 5 7.3.(1) , axiom of choice). Take a cofinal s e t F of least in f in i te cardinal, so that Card F = Cof A , with A/F a well-founded ordering, hence a well partial ordering of minimum height. Hence Ht(A/F) = Cofh A : see ch.2 5 7.4. For each i < H t ( A / F ) , there e x i s t only a f i n i t e number of elements with height i (mod A/F) . Yet F i s assumed t o be inf in i te . Hence Card H t ( A / F ) = Card F =

(3) For every well par t ia l ordering A , we have Cof H t A < Cofh A (uses axiom of choice; ZF suffices i f A i s countable). 0 Take a cofinal res t r ic t ion B of A with least cardinal and leas t height. To each number) with height i (mod B ) . The se t of those maximum heights i s cofinal in the ordinal H t A . Hence Cof H t A < H t B = Cofh A . 0

I t i s necessary t o assume t h a t A i s a well partial ordering: recall that we defined in ch.2 5 7 . 7 . ( 1 ) , a well-founded par t ia l ordering with and H t A = ul , so Cof H t A = w . For a well par t ia l ordering, s t r i c t inequality may occur: see ch.7 5 3.13 below where Cofh A = W1 and H t A = Ldl. W thus Cof H t A = Cd .

--- i s countable or has

A i . Finally associate t o

-

= Cof A , SO Cof Ad Ht(A/F) = Cofh A . 0

i < H t B associate the maximum height (mod A) of elements ( in f i n i t e

Cofh A = 1

§ 4 - I N I T I A L INTERVALS OF A WELL PARTIAL ORDERING; RADO'S WELL PARTIAL ORDERING; HIGMAN'S THEOREM ON THE WELL PARTIAL ORDERING OF WORDS

4.1. THEOREM O N INITIAL INTERVALS OF A WELL PARTIAL ORDERING For a partial ordering A t o be a well partial ordering, i t i s necessary

Chapter 4 107

and s u f f i c i e n t t h a t the pa r t i a l o rder ins of inclusion of i n i t i a l i n t e rva l s of A be well-founded (HIGMAN 1952; the necess i ty uses dependent choice; ZF su f f i ces if A i s countable) . 0 Let A be a well pa r t i a l o rder ing , and suppose the statement of the proposi- t ion i s f a l s e . By ch.2 5 2.4 (dependent choice) , t he re e x i s t s an w -sequence of i n i t i a l i n t e rva l s Ai ( i i n t ege r ) which i s s t r i c t l y decreasing w i t h respect t o inclusion. For edch i , choose an element u i in A i - A i + l . Each u i i s < or I (mod A ) t o t he u w i t h indices -6 i . Hence our k, -sequence i s bad, con- t r ad ic t ing 3.2.(2). Conversely, suppose t h a t A i s not a well pa r t i a l ordering. Then e i t h e r A is not well-founded o r A i s not f i n i t e l y f r e e . In the f i r s t case, there e x i s t s a subset D of the base I A I such t h a t A/S has no minimal element. To each e l e - ment x of D , assoc ia t e the i n i t i a l in te rva l Dx of eiements Gx (mod A ) . The se t of t he ordering of the i n i t i a l i n t e rva l s i s not well-founded. I f A i s not f i n i t e l y f r e e , then there e x i s t s an i n f i n i t e s e t H o f incomparable elements (mod A ) . By the def in i r ion of f i n i t e n e s s (ch .1 5 l . i ) , there ex i s r s a s e t of subsets X of H , no one of which i s minimal w i t h respect t o inclusion. Complete each X in the i n i t i a l in te rva l X+ generated by X . Then X+n H = X

fo r every X , hence X # Y implies X+ p Y + , and even X c Y implies X t c Y+ f o r every X, Y . The X+ form a set of i n i t i a l i n t e rva l s of A , no one of which i s minimal w i t h respec t t o inc lus ion . Hence the pa r t i a l ordering of t he i n i t i a l i n t e rva l s i s no t well-founded (proof of suf f ic iency using only ZF and communicated by POUZET) . 0

Dx has no element minimal with respec t t o inclusion; so the pa r t i a l

4.2. R A D O ' S WELL PARTIAL ORDERING If A i s a well p a r t i a l ordering, then the pa r t i a l ordering of inclusion f o r the i n i t i a l i n t e rva l s of A i s not necessar i ly a well pa r t i a l ordering. In f a c t i t i s not necessar i ly f i n i t e l y f r ee . The following example i s ca l led Rado's well p a r t i a l o rde r ing (RADO 1954). I t will be used i n pa r t i cu la r i n ch.8 5 5.2 f o r the theory of b e t t e r pa r t i a l orderings. Consider t he couples of natural numbers x , y and set (x,y) 6 ( x ' . y ' ) i f f x = x ' and y d y ' in the usual ordering, o r i f X ' xty and y i s a r b i t r a r y . The comparison r e l a t ion thus defined i s a well pa r t i a l ordering. Yet there a re i n f i n i t e l y many i n i t i a l i n t e rva l s which a re mutually incomparable w i t h respect t o inclusion.

The reader can ver i fy r e f l e x i v i t y , antisymmetry and t r a n s i t i v i t y . Given a couple (x ,y ) , t he re a r e only f i n i t e l y many couples l e s s than i t , s ince e i t h e r t h e i r f i r s t term i s x and t h e i r second term i s .\< y , or the sum of t h e i r terms i s x . T h u s the pa r t i a l ordering i s well-founded.

Given (x,y) , any coup le which i s incomparable w i t h i t , has f i r s t t e rm s t r i c t l y l e s s than x+y . Hence t h e r e e x i s t f i n i t e l y many p o s s i b l e f i r s t terms. Moreover, two incomparable couples n e c e s s a r i l y have d i s t i n c t f i r s t terms.

Hence t h e p a r t i a l o r d e r i n g i s f i n i t e l y f r e e , and thus a w e l l p a r t i a l o r d e r i n g .

Consider now, f o r each i n t e g e r i , t h e i n i t i a l i n t e r v a l Ai formed o f t h e

couples ( i ,y ) w i t h y an a r b i t r a r y n a t u r a l i n t e g e r , and t h e couples

(x.y) w i t h x+y,(i : f o r an a r b i t r a r y i n t e g e r j + i , t h e i n i t i a i i n t e r v a l s

Ai and A a r e incomparabie w i t h r e s p e c t t o i n c l u s i o n . 0

A second example o f a w e l l p a r t i a l o r d e r i n g wi th t h e same incomparabie i n i t i a l i n - t e r v a l s : t a k e aga in t h e couples o f i n t e g e r s w i t h e i t h e r x = x i and y & y ' ,

o r x : > Max(x,y) dnd y ' a r b i t r a r y .

A t h i r d example, wnich w i l l be g e n e r a l i z e d i n ch.8 exe rc . 2: t e g e r s w i t h e i t h e r x = x ' and y,< y ' , o r X I > Max(x,y) and y < y ' .

j

couples o f i n -

4.3. L e t A be a p a r t i a l o r d e r i n g and u a minimal bad w-sequence .in A . Then t h e i n i t i a l i n t e r v a l o f t hose elements x of I A I such t h a t t h e r e e x i s t s an i n t e -

g p i fin x < ui (mod A ) i s a w e l l p a r t i a l o r d e r i n g (uses dependent choice; ZF s u f f i c e s i f A i s c o u n t a b l e j . Compare w i t h 2.5 above.

0 i4e prove f i r s t o f a i l t h a t , g i ven a te rm u ( p i n t e g e r ) sequence, t h e i n i t i a l i n t e r v a l o f t hose elements < u (mod A j i s a w e l l p a r t i a l

o r d e r i n g . By 2.5 t h i s i n t e r v a l i s wel l - founded (dependent c h o i c e ) . Suppose t h a t t h e r e e x i s t i n f i n i t e l y many m u t u a l l y incomparable elements, hence an (J -sequence of such elements a,,, al,. .., a.,. .. a . / a i l except a f i n i t e riumber o f terms of u (see 2.3) . Thus t h e w -sequence o f t h e a . i s embeddable i n u , and hence by 2.6 i s e x t r a c t e d f rom u , so t h a t t h e r e e x i s t terms u (q > p) which a r e 6 u . Yet a l l t h e terms o f u a r e m u t u a l l y incomparable by 2.3: c o n t r a d i c t i o n . Suppose now t h a t t h e p r o p o s i t i o n i s f a l s e . There e x i s t s a bad w -sequence

such t h a t , f o r each i n t e g e r i , t h e r e e x i s t s a j w i t h v i < uj (mod A ) : see 3.2.(2) u s i n g dependent cho ice . A t most a f i n i t e number o f terms o f v o r < ul, e t c . Hence t h e r e e x i s t s an (,d -sequence w e x t r a c t e d f rom v

e i t h e r s t r i c t i y l e s s than

f rom u . T h i s c o n t r a d i c t s t h e m i n i m a l i t y o f u . 0

o f t h e min imal bad P

P

a l l < u . Then each a . < up , so each J P J

J

J

q P

v

a r e < uo,

which i s

u (mod A) o r s t r i c t l y l e s s than a sequence e x t r a c t e d

4.4. THEOREM ON "ELL PARTIAL ORDERING OF WORDS

I f A

words i n A s u f f i c e s i f A i s coun tab le ) .

i s a w e l l p a r t i a l o r d e r i n g , t hen t h e p a r t i a l o r d e r i n q o f embeddab i l i t v f o r i s a w e l l p a r t i a l o r d e r i n g (HiGMAN 1952; uses dependent choice; ZF

Chapter 4 109

By 3.2.(2) (dependent cho ice ) , i t s u f f i c e s t o show t h a t t h e r e does n o t e x i s t any

bad &J -sequence o f words w i t h r e s p e c t t o embeddab i l i t y ( d e f i n e d i n 5 2) . I f such a bad sequence e x i s t s , t hen t h e r e e x i s t s a l s o a s t r o n g l y min imal bad sequence

(see 2.10). We s h a l l show t h a t t h i s i s imposs ib le . Denote by

ith term o f U . By hypo thes i s , Uo i s such t h a t no bad w-sequence o f words be- g ins by a s h o r t e r word, i . e . a word ob ta ined f rom

element. No bad sequence beg inn ing by Uo con t inues w i t h a word s h o r t e r t han U1 , e t c . N o t i c e t h a t no Ui i s empty, s i n c e t h e U . ( j > i) a r e < o r I Ui w i t h

respec t t o embeddab i l i t y : t h e f i r s t t e r m ui o f Ui always e x i s t s . The sequence

of t h e ui t akes i t s va lues i n t h e w e l l p a r t i a l o r d e r i n g A and so by 3.2. (3)

has an e x t r a c t e d 0 - s e q u e n c e which i s e i t h e r c o n s t a n t o r s t r i c t l y i n c r e a s i n g .

Denote by n ( i j ( i i n t e g e r ) t h e i n d i c e s co r respond ing t o t h i s cons tan t o r i n c r e a - s i n g sequence o f f i r s t terms, and l e t V term u

This sequence begins as U , b u t a t t h e p o s i t i o n h ( 0 ) , che h o r d U h (0 )

p laced by t h e s h o r t e r word Vh(o) . So by hypo thes i s , t h i s l a t t e r sequence i s

good. Now, two words i n p o s i t i o n C h(0 ) cannot admi t an embedding o f one i n t h e

o t h e r . S i m i l a r l y f o r two words i n p o s i t i o n 2 h(0 ) , which a r e o f t h e fo rm V

uh(i). < uh(j) (mod A) , such an and V

embedding would y i e l d Uh(i)'< Uh(j) . F i n a l l y , t h e r e remains t h e p o s s i b i l i t y t h a t a Ui ( i -= h ( 0 ) ) i s embeddable i n a Vh(k) ( k i n t e g e r ) and hence embedda- b l e i n U h ( k ) . However t h i s c o n t r a d i c t s t h e assumption t h a t U i s bad.

U

t h e Ui ( i i n t e g e r )

Uo by removing a t l e a s t one

J

w i t h i t s f i r s t

h ( 0 ) ' 'h( 1 ) h ( i )

be t h e word u h ( i )

* removed. Consider t h e sequence Uo, U1,..., Uh(o)-l, V h ( i ) i s re -

h ( i ) w i t h i< j : g i v e n t h e f a c t t h a t h ( j )

4.5. L e t A be a we l l - f ounded p a r t i a l o r d e r i n g and o( i t s h e i g h t . I f f o r eve ry o r d i n a l i < < , t h e r e a r e o n l y f i n i t e l y many-elements o f h e i g h t i , then A i s n o t n e c e s s a r i l y f i n i t e l y f r e e .

Take t h e t o t a l o r d e r i n g o f t h e i n t e g e r s , and f o r each i n t e g e r i , add an i' s e t t o be > i and incomparable w i t h i n t e g e r s > i , t h e i' m u t u a l l y incompa rab le .

LEMMA OF THE CHAIN MEETING EVERY HEIGHT L e t A be a we l l - f ounded p a r t i a l o r d e r i n g hav ing a t most f i n i t e l y many elements

o f each h e i g h t , and l e t c4 = H t A . Then t h e r e e x i s t s a w e l l o rde red r e s t r i c t i o n C of A which i s isomorphic w i t h o( . More p r e c i s e l y , t h e base I C I con ta ins f o r each o r d i n a l i < 4 a unique element o f h e i g h t i (mod A) . Uses u l t r a f i l t e r axiom; ZF s u f f i c e s i f A i s countable; communicated by POUZET 1979.

For o( = a , t h i s i s j u s t a fo rm o f KONIG's lemma: see below ch.5 5 2.5.

Compare t h e p resen t p r o p o s i t i o n w i t h 3.1, which uses t h e axiom o f choice; he re cho ice would be u s e f u l i f we had added t h e equipotence o f I A l and H t A .

0 For eve ry f i n i t e s e t h which assoc ia tes t o each i o f F an element h i o f h e i g h t i (mod A ) . The

h i ( i E F) o f A : i t s u f f i c e s t o s t a r t w i t h t h e l a r g e s t i i n F and t o r e c a l l t h a t f o r

each j < i t h e r e e x i s t s an element h j < h i (mod A) ; see ch.2 5 3.2. Consider t h e o r d i n a l o( and t h e base I A [ as d i s j o i n t s e t s . To each F asso-

c i a t e F+ = F p l u s those elements i n I A l whose h e i g h t be longs t o F . The F+

thus fo rm a d i r e c t system w i t h r e s p e c t t o o r d e r i n g by i n c l u s i o n . Given an t o each h w i t h domain F a s s o c i a t e t h e b i r e l a t i o n w i t h base Ft formed by t h e

unary r e l a t i o n t a k i n g t h e v a l u e + on F and - on F+-F , t o g e t h e r w i t h t h e b i n a -

ry r e l a t i o n t a k i n g t h e va lue + f o r o rde red p a i r s ( i , h i ) where i E F . Assoc ia te t o each F t h e non-empty s e t UF o f t hese b i r e l a t i o n s f o r a l l h w i t h domain F The

t h e u l t r a f i l t e r axiom). Hence t h e r e e x i s t s a b i r e l a t i o n on t h e un ion which d e f i n e s an isomorphism f rom c( o n t o a t o t a l l y o rde red r e s t r i c t i o n o f A . 0

A c l o s e l y connected p r o p o s i t i o n i s g i ven i n e x e r c i s e 4 below.

F o f o r d i n a l s < 4( , t h e r e e x i s t s a t l e a s t one f u n c t i o n

a r e m u t u a l l y comparable, so fo rm a f i n i t e t o t a l l y o rde red r e s t r i c t i o n

F ,

UF s a t i s f y t h e hypotheses i n t h e coherence lemma ch.2 5 1.3 ( e q u i v a l e n t t o

I A I u a( ,

5 - NET, OR DIRECTED P A R T I A L ORDERING: IDEAL, DECOMPOSITION OF A F I N I T E L Y FREE P A R T I A L ORDERING INTO IDEALS

A p a r t i a l o r d e r i n g i s s a i d t o be a net, o r d i r e c t e d p a r t i a l o r d e r i n q , i f g i v e n

any two elements i n t h e base, t h e r e i s a t h i r d element g r e a t e r t han bo th . Fo r example, a c h a i n ( t o t a l o r d e r i n g ) , o r a p a r t i a l o r d e r i n g w i t h a maximum.

Given a p a r t i a l o r d e r i n g A i s any i n i t i a l i n t e r v a l which i s d i r e c t e d . Fo r example, e v e r y i n i t i a l i n t e r v a l o f a c h a i n i s an i d e a l ; o r again, f o r an a r b i t r a r y p a r t i a l o r d e r i n g A , t h e i n i t i a l i n t e r v a l o b t a i n e d by t a k i n g

an element a and a l l e lements l e s s than a (mod A ) . I n t h e case o f t h e p a r t i a l o r d e r i n g o f i n c l u s i o n among subsets o f a s e t E , an i d e a l i s a s e t o f subsets o f E which i s c l o s e d under t a k i n g subsets and t a k i n g t h e un ion o f two subsets o f E : hence an i d e a l i s t h e complement o f a f i l t e r .

Given a s e t o f i d e a l s U which i s t o t a l l y o rde red by i n c l u s i o n , o r s imp ly i f i n c l u s i o n i s a n e t , t hen t h e un ion o f t h e U i s an i d e a l .

Consequently t h e MAXIMAL IDEAL AXIOM:

eve ry i d e a l i n a p a r t i a l o r d e r i n g A i s i n c l u d e d i n a maximal i d e a l o f A ( w i t h r e s p e c t t o i n c l u s i o n ) ,

i s e q u i v a l e n t t o t h e axiom o f cho ice .

0 It f o l l o w s f rom t h e maximal c h a i n axiom (ch.2 5 2.7) : s t a r t i n g w i t h an i d e a l U o f A , t a k e a maximal c h a i n i n t h e p a r t i a l o r d e r i n g o f i n c l u s i o n o f i d e a l s

A , an ideal i n

Chapter 4 1 1 1

of A and then take t h e un ion o f t h e i d e a l s i n t h i s cha in , which i s a maximal

i d e a l .. Conversely, t h e maximal i d e a l axiom i m p l i e s t h e w e l l - o r d e r i n g axiom: take a maximal i d e a l i n t h e i n t e r v a l - o r d e r i n g , d e f i n e d i n ch.2 5 2.6. Cl

5.1. I n a p a r t i a l o r d e r i n g , an i n i t i a l i n t e r v a l I i s an i d e a l i f f f o r any two

I n i t i a l i n t e r v a l s X, Y , if X u Y = I then X = I Y = I . 0 Suppose t h a t I i s an i d e a l which i s t h e un ion o f two i n i t i a l i n t e r v a l s X, Y

and t h a t X # I and Y # I . Thus t h e r e e x i s t s an element x E I - X and an e l e -

ment y ez I - Y . Consequently, t h e r e e x i s t s an element z such t h a t z E I and z 3 x and z 5,y . Hence z 4 X and z + Y : c o n t r a d i c t i o n . Conversely, suppose t h a t I i s an i n i t i a l i n t e r v a l which i s n o t an i d e a l . So

t h e r e e x i s t two elements u, v o f I w i t h o u t any common upper bound i n I . Def ine U t o be t h e i n i t i a l i n t e r v a l o f those elements x f o r which t h e r e e x i s t s

a common upper bound o f x and u i n I : hence u E U and v + U . De f ine V

t o be t h e i n i t i a l i n t e r v a l o f t hose y f o r which t h e r e e x i s t s a z ay i n I , such t h a t t h e r e i s no common upper bound o f z and u i n I : hence u I+ V and

v g V . F i n a l l y we have U u V = I . 0

5.2. Every w e l l p a r t i a l K d e r i n q i s a - f i n i t e un ion o f i d e a l s (uses dependent

choice; ZF s u f f i c e s f o r a coun tab le o r d e r i n g ) .

0 L e t A be a w e l l p a r t i a l o r d e r i n g ; r e c a l l t h a t t h e p a r t i a l o r d e r i n g o f i n i t i a l

i n t e r v a l s o f A i s we l l - f ounded (see 4.1, dependent c h o i c e ) . Suppose t h a t A i s n o t a f i n i t e un ion o f i d e a l s . Among t h e i n i t i a l i n t e r v a l s o f A which a r e n o t

f i n i t e un ions o f i d e a l s , t h e r e e x i s t s a min imal such i n i t i a l i n t e r v a l M , w i t h

respec t t o i n c l u s i o n . As M i s n o t an i d e a l , i t i s non-empty. By t h e preceding p r o p o s i t i o n 5.1, t h e r e e x i s t two i n i t i a l i n t e r v a l s U, V o f M , which a r e d i s - t i n c t f r o m M , and U u V = M . By m i n i m a l i t y o f M , t h e i n t e r v a l s U, V a r e

each a f i n i t e un ion o f i d e a l s ; so M as w e l l : c o n t r a d i c t i o n . 0

5.3. (1) F o r a p a r t i a l o r d e r i n g t o be f i n i t e l y f r e e , i t i s necessary and s u f f i -

c i e n t t h a t eve ry i n i t i a l i n t e r v a l be a f i n i t e un ion o f i d e a l s (BONNET 1975; t h e n e c e s s i t y uses axiom o f c h o i c e ) .

0 Suppose t h a t t h e r e e x i s t i n f i n i t e l y many elements u which a r e m u t u a l l y incom- p a r a b l e . Assoc ia te t o each u t h e i d e a l o f t hose elements < u : t h e un ion o f these i d e a l s i s an i n i t i a l i n t e r v a l which i s n o t decomposable i n t o a f i n i t e un ion of i d e a l s.

Conversely, l e t A be a f i n i t e l y f r e e p a r t i a l o r d e r i n g , and I an i n i t i a l i n t e r - v a l o f A . By ch.2 § 5.1, c o r o l l a r y (ax iom o f cho ice ) , t a k e a w e l l o r d e r i n g J which i s a c o f i n a l r e s t r i c t i o n o f I . By t h e p reced ing 5.2, J i s a f i n i t e un ion


of ideals. Complete each ideal X in J into an ideal of I by adding those elements in III which are bounded above by an element of 1x1 . T h u s I i s a f i n i t e union of ideals. 0

(2) Every f i n i t e l y f ree par t ia l ordering i s a f i n i t e union of ideals-.

( 3 ) Every inf in i te f in i te ly free partial ordering has an in f in i te ideal; consequences of (1) . The converse i s fa lse: take a non-finitely f ree par t ia l ordering which has a maximum. For further informations, see MILNER, 1982.

5 .4 . For every net A , there ex is t s a cofinal res t r ic t ion f A which i s a well-founded net (uses axiom of choice; ZF suffices i f A i s countable). 0 By ch.2 5 5.1, corollary (axiom of choice), there exis ts a cofinal A/F well-founded. For any two elements x , y o f F , by hypothesis there exis ts an element z satisfying z ax and z a y (mod A ) . Since F i s cofinal, there exis ts an element t i n F with t 3 z and so t x and t > / y : hence A/F i s directed; in other words, a net. 0

F with

5.5. For every denumerable net A , there ex is t s 2 to ta l ly p-de-r-d-restriction of- A , which i s cofinal and isomorphic t o LJ 1 . 0 Take an a-sequence of the elements a i ( i integer) bo = a. , bl = the element with leas t index which i s greater t h a n (mod A ) b o t h bo and a l , then b2 greater than b l and a2 , e tc . 0

On the other hand, the direct product w X (*1 defined in $ 7 below, has no to ta l ly ordered cofinal res t r ic t ion . 0 T h a t would require a total ordering of order-type w 1 formed o f ordered pairs ( i , j ) with i , j increasing and i running through cu and j running through a1 , which i s impossible. 0

of the base; then take

§ 6 - TREE

A partial ordering i s called a tree i f , for each element of a l l predecessors of x i s to ta l ly ordered. For example, every total ordering i s a t ree . Every free partial ordering, reduced t o the identity, i s a t ree . Another example: beginning with a se t a and a s e t A o f subsets of a , where any two elements of A are e i ther dis joint or one i s included in the other, and no element of A i s empty. Then reverse inclusion constitutes a t ree based on A . Conversely, l e t A be a t ree with base E . To each element x in E , associate the se t Ax of those elements >/ x (mod A ) : the Ax ordered by reverse inclusion form a t ree , isomorphic with A .

x in the base, the se t

Chapter 4 113

6.1. I f A i s a f i n i t e t r e e , t hen eve ry base o f a maximal c h a i n o f A and e v e r 1

maximal f r e e se t , have one and o n l y one e lement i n common. (KUREPA 1952) .

L e t F be t h e base o f a maximal t o t a l l y o rde red r e s t r i c t i o n o f A , and G a maximal f r e e se t ; l e t u be t h e maximum o f t h e c h a i n A/F . Then e i t h e r u i s i d e n t i c a l o r incomparable (mod A) t o each element o f G . I n t h a t case u belongs t o G , s i n c e G i s a maximal f r e e s e t . O r t h e r e e x i s t s an element v o f G , which i s d i s t i n c t f rom u and comparable w i t h u (mod A) . Then v i s a s t r i c t predecessor o f u ; f o r o t h e r w i s e v c o u l d be added t o F and so A/F would

n o t be a maximal cha in . Thus v i s comparable w i th eve ry predecessor o f u , and hence v be longs t o F s i n c e A/F i s maximal. Thus v i s common t o F and G . Th is r e s u l t does n o t h o l d f o r an i n f i n i t e t r e e .

Take t h e t o t a l o r d e r i n g o f t h e i n t e g e r s and t o each i n t e g e r i , a s s o c i a t e an element i '> i b u t i ' incomparable w i t h i n t e g e r s >i . F i n a l l y t h e i' a r e

s e t t o be m u t u a l l y incomparable. Then t h e s e t o f i n t e g e r s d e f i n e s a maximal chain,

and t h e s e t o f "primed" i n t e g e r s i s a maximal f r e e s e t (or d e f i n e s a maximal a n t i -

cha in ) ; example due t o KUREPA.

The r e s u l t no l o n g e r ho lds f o r an a r b i t r a r y f i n i t e p a r t i a l o r d e r i n g .

0 Take f o u r e lements and l e t a < b, a -= c, b ' c c, w i t h b I c, a I b ' , b I b ' . Then t h e c h a i n a < c i s maximal and t h e a n t i c h a i n b I b ' is maximal. 0

6.2. L e t A be a t r e e and E i t s base. There e x i s t s a c h a i n C based on E , such t h a t f o r each element x of E , t h e i n t e r v a l 3 x (mod A ) becomes an in- t e r v a l (mod C ) w i t h minimum x (uses u l t r a f i l t e r axiom: ZF s u f f i c e s i f A i s coun tab le ) .

Suppose f i r s t t h a t E i s f i n i t e . L e t ul, ..., uh be t h e minimal elements (mod A) and i n an a r b i t r a r y manner o r d e r these elements, o b t a i n i n g f o r example u1 < ... < uh . Then f o r each i = 1,. . .,h r e p l a c e ui by a sequence beg inn ing w i t h ui

and t o t a l l y o r d e r i n an a r b i t r a r y manner t h e immediate successors o f ui (mod A) . Cont inue i n t h i s f a s h i o n u n t i l a l l t h e elements o f E a r e con ta ined i n t h i s cha in : t hus we o b t a i n t h e c h a i n C . NOW cons ide r t h e case where E i s i n f i n i t e . To each f i n i t e subset F o f E , as

s o c i a t e t h e non-empty s e t UF o f cha ins w i t h base F which s a t i s f y t h e p ropos i - t i o n f o r t h e r e s t r i c t e d t r e e A/F . Then t h e UF s a t i s f y t h e hypotheses o f t h e

coherence lemma ch.2 5 1.3 ( e q u i v a l e n t t o u l t r a f i l t e r ax iom). Then we o b t a i n a cha in C w i t h base E , a l l o f whose f i n i t e r e s t r i c t i o n s s a t i s f y t h e p r o p o s i t i o n .

Hence A and C s a t i s f y t h e p r o p o s i t i o n . 0

Consequently, given a tree A , there exists a chain w i t h t hesame base , and a set- U of in t e rva l s (modulo the chain) which - a re mutually inc lus ive o r d i s j o i n t , such - t h a t A i s isomorphic t o reverse inclusion on U .

___.__. - ~

6.3. Given a s e t E and a s e t U of subsets of E , which a r e mutually inc lus ive o r d i s j o i n t , there e x i s t s a chain with base E , f o r which the e l e m e n t s f U are i n t e rva l s ( u l t r a f i l t e r axiom; ZF su f f i ces i f E i s countable). 0 Add t o U the s ing le tons of elements of E , and apply the preceding proposi- t ion t o the s e t thus obtained, which i s ordered by reverse inc lus ion . Replace the chain by i t s r e s t r i c t i o n t o the s e t of s ing le tons ; each singleton being iden t i f i ed with i t s unique element. 0

- .

6.4. REDUCED TREE

Let A be a t r e e . We introduce an equivalence r e l a t ion ca l led reduction (mod A ) . Two elements of t he base a r e sa id t o be equivalent i f they a r e comparable, f o r example i f x < y (mod A ) and a l s o every element a x i s comparable w i t h y (mod A ) . Equivalently, every t o t a l l y ordered r e s t r i c t i o n of A , which i s maximal with respect t o inclusion and which contains x , a l so contains y . The r e l a t ion thus defined i s r e f l ex ive , symmetric and t r a n s i t i v e . An equivalence c l a s s is a t o t a l l y ordered r e s t r i c t i o n of A . We say t h a t an equivalence c l a s s precedes another equivalence c l a s s , i f t he e l e - ments of t he f i r s t precede (mod A ) the elements of t he second. The pa r t i a l orde- r ing on the equivalence c l a s ses thus defined, i s a t r e e , and ca l l ed the reduced t r e e o r reduct of A . Given a t r e e A , assoc ia t e t o each element x of t he base the s e t A x of maxi-

x , y

- mal cha and y (mod A ) a t r e e

ns ( w i t h respec t t o inc lus ion) which contain x in t h e i r base. For x incomparable (mod A ) , t h e s e t s Ax and A a r e d i s j o i n t . For x b y , we have Ax 2 Ay . Hence these s e t s , ordered by reverse inc lus ion , form somorphic t o the reduct of A .

Y

6.5. INITIALLY MAXIMAL CHAIN Let A be a pa r t i a l ordering. We say t h a t a chain in A i s i n i t i a l l y maximal i f i t i s an i n i t i a l in te rva l of a maximal chain ( w i t h respec t t o inc lus ion ) . In o ther words, i f x belongs t o the base of an i n i t i a l l y maximal chain U , then every predecessor of x (mod A ) e i t h e r belongs t o the base U o r i s incomparab le (mod A ) t o some element i n t h i s base.

The s e t of i n i t i a l l y maximal cha ins , ordered by inc lus ion , forms a t r e e .

F ina l ly reca l l t h a t the amalgamation lemma ch.2 5 2 . 2 and 2.3, which holds f o r pa r t i a l orderings and f o r t o t a l o rder ings , no longer holds f o r t r e e s : a counter-

example has been given i n ch.2 § 2.3.

Chapter 4 115

§ 7 - DIRECT PRODUCT OF PARTIAL OR TOTAL ORDERINGS; D I M E N S I O N

Given a s e t o f p a r t i a l o r d e r i n g s i s

d e f i n e d t o be t h e p a r t i a l o r d e r i n g w i t h base t h e C a r t e s i a n p roduc t o f t h e bases, hence w i t h base t h e s e t o f f u n c t i o n s f w i t h domain I t a k i n g a va lue

f ( i ) c: Ai f o r each i ; t h e comparison r e l a t i o n be ing d e f i n e d by f Q g iff

f(i),( g ( i ) (mod Ai) an t i symmet r i c and t r a n s i t i v e .

I n p a r t i c u l a r , f o r two p a r t i a l o r d e r i n g s

A 5 B . Up t o isomorphism, t h e o p e r a t i o n x Another p a r t i c u l a r case: i f a l l t h e p a r t i a l o r d e r i n g s a r e i d e n t i c a l t o a f i x e d p a r t i a l o r d e r i n g A , where i runs th rough an o r d i n a l , t hen we o b t a i n t h e " l e s s than" comparison r e l a t i o n between sequences i n

l e n g t h (see 5 2 ) .

Ai ( i E I ) , t h e d i r e c t p roduc t o f t h e Ai

f o r eve ry i E I . T h i s comparison r e l a t i o n i s r e f l e x i v e ,

A, B t h e d i r e c t p roduc t i s denoted by i s commutative and a s s o c i a t i v e .

Ai

A o f t h e same o r d i n a l

7.1. (1) L e t A, B be two we l l - f ounded p a r t i a l o r d e r i n g s ; t hen t h e d i r e c t p ro - - duct A % B i s we1 1 -founded.

0 S t a r t w i t h a non-empty subset D o f t h e C a r t e s i a n p r o d u c t o f t h e two bases.

Then t a k e an element (x,y) o f D , where x 6 1 A l w i t h min imal h e i g h t (mod A),

then y E lBl w i t h min imal h e i g h t (mod B) : hence (x,y) i s minimal i n D . 0

( 2 ) L e t A, B be two w e l l p a r t i a l o r d e r i n g s ; t hen t h e d i r e c t p roduc t A x B 1s. a w e l l p a r t i a l o r d e r i n g (uses denumerable subset axiom: ch.1 5 2.6; ZF s u f f i c e s i f A and B a r e c o u n t a b l e ) . 0 By t h e p reced ing ( l ) , t h e d i r e c t p roduc t i s we l l - f ounded . Suppose t h a t t h e r e e x i s t i n f i n i t e l y many incomparable elements, and hence an w - s e q u e n c e of incom-

pa rab le e lements. Then we have an e x t r a c t e d cd -sequence o f elements

where t h e f i r s t terms x a r e i n c r e a s i n g (mod A) , and f rom t h i s sequence, ano-

t h e r e x t r a c t e d a - s e q u e n c e where t h e second terms y a r e i n c r e a s i n g (mod B) : t h i s c o n t r a d i c t s t h e i n c o m p a r a b i l i t y . 0

(x,y)

7.2. (1) L e t A, B be two wel l - founded p a r t i a l o r d e r i n g s , x an element o f t h e

base I A I and y an element of l B ( . Then Ht(x ,y) (mod Ax B) i s equal t o t h e commutative sum H t x (mod A) Q H t y (mod B) : see ch .1 5 9 . ( 2 ) I f A, B a r e non-empty we l l - f ounded p a r t i a l o r d e r i n g s , then:

Sup ( H t A, H t B) & Ht (A x B) < H t A @ H t B . 0 I f x i s minimal (mod A) and y min imal (mod B) t hen t h e coup le (x,y) i s min imal (mod A x B ) : so t h e above e q u a l i t y ho lds f o r H t x = H t y = 0 . L e t

8 be an o r d i n a l # 0 ; assume t h a t t h e e q u a l i t y ho lds f o r each x ' and each y '

w i t h H t x ' 0 H t y ' < y . Assume moreover t h a t t h e e q u a l i t y ho lds t o o i f

Ht(x',y') (mod A x B) < . Let I be the initial interval of those (x',y') with height 4 f , and J the complement of I . So (x,y) belongs to J iff Ht x 8 Ht y 8 . By definition of height, a couple (x,y) is minimal in J iff Ht(x,y) (mod A x B) = li . So it suffices to prove that (x,y) is minimal in J iff Ht x @ Ht y = . For (x,y) in J , either Ht x @ Ht y > $ . Then there exists for instance an ordinal j < Ht y with Ht x EI j = $ ; and so an element y'< y (mod B) with Ht y' = j (see ch.2 5 3.2). Hence (x,y') < (x,y) and (x,y') belongs to J , so that (x,y) is not minimal in J . Or Ht x Q Ht y = 8 . Then for any (x',y') < (x,y) (mod A K B) we have either x ' < x (mod A ) with y'+ y (mod B) , or conversely x'< x with y'< y . By the properties of commutative ordinal addition, this always gives Ht x'@ Ht y' < y , so that (x',y') belongs to I , and (x,y) is minimal in J . 0

0 (2) The first inequality is obvious. For the second inequality, notice that the height of a well-founded partial ordering A equals Sup(i+l) where i denotes the height of any element in A . Similarly for B we obtain Sup(j+l) ; and finally for A s B we obtain Sup(i @ j + l ) , taking in account the preceding (1). Then the second inequality immediately results from the following, which is a consequence of the definition of commutative sum , easily provable by the reader: Sup(i 0 j + 1) < (Sup i+l ) @ (Sup jtl ) . 0

7 . 3 . CONJUNCTION OF A SET OF PARTIAL ORDERINGS. Given a partial ordering A and a set of partial orderings Bi which are all reinforcements of A with common base IAI , we say that the partial ordering A is the conjunction of the Bi if, for any x, y in I A I , we have x \< y (mod A ) iff x,<y modulo each Bi . In the interesting particular case where the Bi are total orderings, if x < y (mod A) then x < y (mod Bi) for each i , and if x ly (mod A ) then there exists an i with x < y (mod Bi) and a j with x > y (mod B.) . DIMENSION The dimension of a partial ordering A is the least cardinal of a set of chains, each with base IAI and whose conjunction is A . Modulo the axiom of choice, every partial ordering has a dimension; in ZF at least every finite partial ordering has a dimension. The notion of dimension goes back to DUSHNIK, MILLER 1941. We shall denote by Dim A the dimension of a partial ordering A . - Let A be a partial ordering, B a restriction of A ; if dimensions exist for A and B , then Dim B 4 Dim A . Every chain has dimension 1 . Given a set E of cardinality 2 , the free

J

Chapter 4 117

o r d e r i n g , o r i d e n t i t y r e l a t i o n , has d imension 2 : t a k e a c h a i n based on E and

t h e converse cha in ( s o we use t h e o r d e r i n g axiom; ZF s u f f i c e s i f E i s coun tab le ) .

7.4. (1) L e t A be a p a r t i a l o r d e r i n g w i t h d imension h ( p o s i t i v e i n t e g e r ) . Add

a new element u which w i l l be t h e minimum o f a p a r t i a l o r d e r i n g , ex tens ion o f A t o t h e new base

l y i f we add a maximum.

0 Take t h e cha ins Ci whose c o n j u n c t i o n i s A , then add t o each Ci t h e minimum element u . 0

(2 ) L e t h be an i n t e g e r 3 2 , and l e t A, B be two p a r t i a l o r d e r i n g s w i t h d i s -

j o i n t bases, each one w i t h d imension & h . Then t h e p a r t i a l o r d e r i n g based on t h e

un ion o f t h e bases, common e x t e n s i o n o f A a d B and i n which eve ry element of I A l 0 Assume h = 2 ( t h e p r o o f immediate ly extends t o any g r e a t e r i n t e g e r ) . Consider

the two cha ins C, C ' whose c o n j u n c t i o n i s A , and t h e two cha ins D, D ' whose c o n j u n c t i o n i s B . Then, on t h e un ion o f t h e two bases, t a k e t h e chains C t D

and D ' + C ' . 0

\ A l u { u } . Then we o b t a i n an o r d e r i n g w i t h dimension h ; s i m i l a r -

i s i n c o m p a r a e w i t h eve ry element o f l B l , has d imension \< h .

7.5. Every f i n i t e t r e e , e i t h e r i s a cha in , o r has d imension 2 . 0 We c o n s t r u c t t h e t r e e f rom i t s maximal elements, by a f i n i t e sequence o f t h e

two f o l l o w i n g opera t i ons . (1) un ion o f two t r e e s w i t h d i s j o i n t bases, each e l e - ment o f one base be ing incomparable w i t h each element o f t h e o t h e r ; (2 ) add i -

t i o n o f a minimum; f i n a l l y use t h e p reced ing 7.4. 0

7.6. L e t a be a c a r d i n a l ; eve ry d i r e c t p roduc t o f cha ins whose s e t has c a r d i n a l i -

- ty cement axiom s u f f i c e s i f a

completed i n 7.9 below.

0 Denote t h e cha ins by Ai ( i € I w i t h c a r d i n a l i t y a) , and l e t A be t h e d i r e c t product o f t h e Ai . For each index i , c o n s i d e r t h e d i r e c t p roduc t Di o f t h e

A j ( j # i) . By t h e re in fo rcemen t axiom, t h e r e e x i s t s a t l e a s t one t o t a l l y o rde red

re in fo rcemen t Ci o f Di . To each i a s s o c i a t e a un ique Ci (ax iom o f cho ice ) , then a s s o c i a t e t h e c h a i n Bi w i t h base I A I , such t h a t f Q g (mod Bi) i f f f ( i ) < g ( i ) (mod Ai) o r f ( i ) = g ( i ) w i t h (sequence o f t h e f ( j ) f o r j # i) 4 (sequence o f t h e g ( j ) f o r j # i) (mod Ci) . Then t h e d i r e c t p roduc t A j u n c t i o n o f t h e cha ins

a , i s a p a r t i a l o r d e r i n g w i t h d imension 4 a (uses ax iom o f choice; r e i n f o r -

i s f i n i t e : see ch.2 Cj 4 .2 ) . Th i s s ta tement w i l l be

i s t h e con- Bi . 0

7.7. L e t A be a p a r t i a l o r d e r i n g and p a p o s i t i v e i n t e g e r . I f eve ry f i n i t e

r e s t r i c t i o n o f A has d imension & p , then A has d imension 4 p (uses

u l t r a f i l t e r axiom; ZF suffices i f A i s countable).

L l To each f i n i t e subset F of the base I A l , associate the s e t UF of multirelations every x , y in F , we have X Q y (mod A ) i f f x G y (mod C1) and ... and x g y (mod C ) . By hypothesis U F i s non-empty for every F . Moreover i f F'S F then every multirelation which i s an element of UF , when restr ic ted t o F ' , yields an element of U F , . By the coherence lemma ch.2 5 1.3 (equivalent t o the u l t r a f i l t e r axiom), there exis ts a multirelation with base I A l whose restr ic t ion t o each F belongs t o UF . Hence th i s multirelation i s a sequence of p chains, each of whose base i s I A 1 , and whose conjunction i s the partial ordering A . 0

I n par t icular , i t follows from 7 . 5 that every t ree ( f i n i t e or inf in i te ) i s e i ther a chain or has dimension 2 .

( C , ,..., C p ) , a sequence of p chains with cornon base F , such t h a t , for

P

7.8. Let E be a se t ; to each element a of E , associate the singleton a ' of a and the complement a" = E-a' se ts of E which contains a l l the a ' and a " as elements, the partial ordering of inclusion has dimension equal to Card E (DUSHNIK, M I L L E R 1941).

0 Let US denote by < of inclusion. Then there exis ts a t most one element a of E for which a " < a ' . Indeed, i f we have two d is t inc t elements a , b with a"< a ' and b " < b ' , hence we obtain a " < a ' s b " < b ' s a " thus a = b . For each element a of E , the two sets a ' and a" are incomparable with respect t o inclusion. Hence among the chains whose conjunction i s the partial ordering of inclusion, one a t l eas t sa t i s - f ies the inequality a " < a ' , with x ' < x" for a l l elements x # a . Associate th i s chain to a : then the s e t of those chains corresponding t o a l l elements of E gives the desired ordering of inclusion. To see t h i s , i t remains t o consider two subsets X, Y of E which are incomparable with respect t o inclusion. So there ex is t an element x of E with x c X and x I# Y , and an element y w i t h y € Y and y f X . Then the chain associated with x gives Y 5 x " < x ' t, X so Y < X ; similarly the chain associated with y gives X < Y . 0

of th i s singleton. Then for any given se t of sub-

any chain which i s a reinforcement of the partial ordering

The preceding statement extends as follows (POUZET, 1969'). Let A be a partial ordering with base E . To each element a of E , associate the in i t ia l interval a ' of elements x s a (mod A ) , and the in i t ia l interval a" of elements x < or I a (mod A ) . Then for any given se t of subsets o f E which contains a l l the a ' fi a " as elements, the partial ordering of inclusion has dimension equal t o the leas t cardinality of those se t s of chains which are res t r ic- tions of A and whose bases cover the whole s e t E (uses axiom of choice).

0 For each a , b in E , i f a " < a ' and b " < b ' (where c denotes a chain

Chapter 4 119

which re inforces the inc lus ion ) , then the preceding argument proves t h a t b a re comparable: a d b o r b,< a (mod A ) . Hence t o each chain which reinforces the inc lus ion , t he re corresponds a t o t a l l y ordered r e s t r i c t i o n I of A , such tha t the elements u of 111 s a t i s f y u " < u' , w i t h however x ' < x" f o r those elements x in E - I11 . Conversely, given an a r b i t r a r y t o t a l l y ordered r e s t r i c - t ion I of A , the inequa l i t i e s u"( u' f o r each u i n 1 1 1 , a re mutually compatible, and a r e compatible with inc lus ion . We can always take each I t o be maximal with respec t t o inclusion (axiom of choice) , and then these conditions have t o be completed by x ' ( x" f o r every x in E - 111 . We end a s in the preceding proof. 0

7.9. Given a d i r e c t product of a - many chains (where i s f i n i t e o r i n f i n i t e ) , each chain being reduced t o the elements 0 and 1, we obtain a pa r t i a l ordering isomorphic t o inclusion f o r subsets of a , by replacing each subset b of a by

i t s c h a r a c t e r i s t i c func t ion , taking the value 1 f o r each chain belonging t o b , and 0 f o r each chain belonging t o a-b . Hence, by the preceding, the d i r e c t product just considered has dimension a . Consequently, the d i r e c t product of a - many chains, each havinq a t l e a s t two elements, has dimension a . 0 I t has dimension 3 a by the preceding argument and 7 . 3 , and dimension 4 a by 7.6. 0

For several developments, e spec ia l ly concerning pa r t i a l orderings w i t h dimension 2 , see exerc ise 5 below.

Let us f i n a l l y c i t e a r e s u l t due t o HIRAGUCHI 1951 and 1955: for every pa r t i a l ordering w i t h f i n i t e ca rd ina l i t y pa 4 , we have dimension a t most equal t o p /2 .

a and

a

§ 8 - BOUND

Let A be a pa r t i a l ordering and B an i n i t i a l in te rva l of A . An element u in the d i f fe rence s e t I A l - I B I i s ca l l ed a -(more prec ise ly a minimal s t r i c t upper bound) o f B (mod A ) i f every element s t r i c t l y l e s s than u (mod A ) i s an element of l B l .

8.1. The bounds of an i n i t i a l in te rva l a r e pairwise incomparable. Consequently i f A i s f i n i t e l y f r e e , then there a r e only f i n i t e l y many bounds f o r each i n i t i a l in te rva l of A .

8.2. Let B be an i n i t i a l in te rva l of a pa r t i a l ordering A . If x 6 IBI , then x 3 u (mod A ) f o r every bound u of B .

Moreover if A i s wel l - founded, then t h e c o n d i t i o n x e I B I i s e q u i v a l e n t Lo-the

c o n d i t i o n x 2 u f o r eve ry bound u f B . 0 Assume t h a t A i s wel l - founded; i f x does n o t be long t o l B l , t h e r e e x i s t s a t l e a s t one element which i s min imal among those elements i n t h e d i f f e -

rence s e t I A I - I B I : t h i s u i s a bound. 0

I f A 0 Take A t o be t h e t o t a l o r d e r i n g L3 + W - (where CJ- i s t h e r e t r o - o r d i n a l

converse o f w ) , and B t o be t h e i n i t i a l i n t e r v a l u : then t h e r e does n o t e x i s t any bound. 0

u s x

i s n o t wel l - founded, then t h e p rev ious p r o p o s i t i o n i s f a l s e .

8.3. L e t A be a p a r t i a l o r d e r i n g and B an i d e a l i n A . For eve ry f i n i t e sequence .h) o f elements w i t h vi < ui (mod A ) f o r each i , t h e r e e x i s t s an element t

o f t h e base I B I , s a t i s f y i n g t h e c o n d i t i o n s t 3 vi and t2 ui f o r each i . 0 Consequence o f t h e d e f i n i t i o n s o f i d e a l and bound, t a k i n g i n t o account

t h a t vi < u . i m p l i e s vi E I B I f o r each i . 0

The p r o p o s i t i o n i s no l o n g e r t r u e when B i s an a r b i t r a r y i n i t i a l i n t e r v a l o f A. 0 Take A w i t h t h r e e elements a,b,c where a < c and b c c and a I b (mod A ) .

L e t B be t h e r e s t r i c t i o n t o (a,b) . Then c i s a bound and i t s u f f i c e s t o

t a k e u1 = u2 = c w i t h v1 = a and v2 = b . N o t i c e a l s o t h a t t h e converse o f t h e p r o p o s i t i o n i s f a l s e .

0 Take A w i t h f i v e elements a,b,c,d,e where d < a < e and d < b < e and d < c < e and a,b,c m u t u a l l y incomparable. L e t B be t h e r e s t r i c t i o n t o

(a,b,d}, so t h a t i s n o t an i d e a l . However t h e c o n c l u s i o n o f t h e p r o p o s i t i o n ho lds . Indeed t h e o n l y bound i s c and t h e o n l y e lement < c i s d : i t s u f f i -

ces t o t a k e t = a o r t = b , o r even t = d . 0

ui ( i = 1, ..., h ) o f bounds o f B , and f o r eve ry sequence vi ( i = 1, ...

1

B

8.4. L e t A be a we l l - f ounded p a r t i a l o r d e r i n g such t h a t eve ry element has o n l y f i n i t e l y many predecessors, and l e t B be an i d e a l i n A . Then f o r eve ry f i n i t e sequence o f bounds ui ( i = 1, ..., h ) B , t h e r e e x i s t s an element t of I B I

such t h a t t h e ui a r e a l s o some bounds of t h e i n t e r v a l 6 t . 0 It s u f f i c e s t o a s s o c i a t e t o each and t o enumerate them, r e p e a t i n g ui as many t imes as t h e r e a r e elements v l e s s than

One shou ld n o t t h i n k t h a t t h e i d e a l i s n e c e s s a r i l y reduced t o an element and

i t s predecessors; f o r i n t h i s case t h e base o f B would be f i n i t e . 0 To have an i d e a l w i t h i n f i n i t e base, t a k e A t o be t h e s e t o f words formed o f

t h e two l e t t e r s a, b, p a r t i a l l y o rde red by embeddab i l i t y (see 5 2, t h e elements

ui t h e f i n i t e s e t o f t hose elements v < ui

ui . Then a p p l y t h e p reced ing p r o p o s i t i o n . 0

B

Chapter 4 121

a , b assumed t o be incomparable). Then the s e t of words formed with the single l e t t e r a , repeated a f i n i t e number of times, constitutes an in f in i te ideal. The only bound of th i s ideal i s the word reduced t o the singleton of b . For t one can take any f i n i t e sequence of a . 0 The previous proposition no longer holds for an arbi t rary well-founded partial ordering. 0 Take A t o be the total ordering o +1 and’ B t o be the i n i t i a l interval w of A . The only bound i s the maximum element of A . 0

The converse of our proposition i s fa l se , as shown by the counterexample in the preceding 8.3.

8.5. Let A be a well partial ordering, and u l , ..., u h a f i n i t e sequence of elements. Suppose that there ex is t s an W -sequence of d i s t inc t elements t . (j integer) such t h a t , for each j , the elements ul, ..., u h are some bounds of the

J

in i t ia l interval \< t ul, ..., u h among i t s bounds (uses dependent choice; ZF suffices i f A countable).

. Then there ex is t s an inf in i te ideal i n A , which has j

0 Since A i s a well partial ordering, there exis ts a s t r i c t l y increasing w -sequence (mod A ) extracted from the sequence of the (see 3.2. (3) , dependent choice). Denote th i s sequence again by the terms t . , and l e t B be the set of those elements x for which there ex is t s a t l eas t one j with t . > x . Then f i r s t l y , B i s an in f in i te ideal since i t s base contains the t . Secondly, for each element u = u. (i = 1 or . . . or h ) , we have tha t u i s a bound of each interval of the form 6 t . Hence ugt for each j and so u 4 I B 1 . Moreover, for every element x ( u , we have x c to , hence x E: I B I . I t follows that u

i s a bound of B . 0

t j J

J

j 1

j j

8.6. Let A be a well partial ordering such that every element has a t most f i n i - te ly many predecessors. Let u l , ..., u h t h a t there ex is t s an &I -sequence of d i s t inc t elements t . ( j integer) such tha t , for each j , the bounds of the i n i t i a l i n t e r v a l s t are exactly ul,..., u h plus elements having each a t l eas t j predecessors (mod A ) . Then there exis ts an in f in i te ideal in A which has exactly the bounds ul, .... u h (same conditions as in preceding 8.5) .

0 As in the preceding proof, take a s t r i c t l y increasing &-sequence extracted from the t , and denoted again by the t . Let B be the ideal generated by these t . . By the preceding proposition, each u l , ..., u h i s a bound of B . Sup- pose t h a t there exis ts another bound v of B . Then v I+ 1 B I , so v & t for each j . Moreover, every element w e v belongs to I B I and hence sa t i s f ies

be a f i n i t e sequence of elements. Assume

J

j

j j

J

j

w4 tj f o r a l l j s u f f i c i e n t l y la rge . By hypothesis, there a re on ly f i n i t e l y many elements smal ler than v , so there e x i s t s an i n tege r j a f t e r which a l l the w are ,< t , hence a f t e r which v i s a bound of the i n t e r v a l 4 t . . This con- j J t r a d i c t s the hypothesis t h a t there are 3 j We sha l l see i n ch.12 0 2.2 and 2.3, t h a t the two preceding propos i t ions are s t i l l

t r ue f o r embeddabil ity between f i n i t e r e l a t i o n s o f a given a r i t y . Obviously embed- d a b i l i t y i s a well-founded p a r t i a l o rder ing f o r such re la t i ons , and s a t i s f i e s the cond i t ion t h a t each element has on ly a f i n i t e number o f predecessors; bu t i t i s no t a we l l p a r t i a l order ing. The proofs a re a l i t t l e l ess simple than here. More- over, the propos i t ion 8.5 becomes f a l s e i f A i s an a r b i t r a r y well-founded p a r t i a l Order i ig ; taRe A t o be the i d e n t i t y relatl 'on on an infi 'nl ' te base.

many predecessors o f v . 0

§ 9 - MAXIMAL REINFORCED CHAIN THEOREM FOR A WELL PARTIAL ORDERING: DE JONGH. PARIKH: EXTRACTION PROPERTY

9.1. Le t A be a we l l p a r t i a l order ing, B an i n i t i a l i n t e r v a l o f A , thus a we l l p a r t i a l order ing. Denote by A-B t he we l l p a r t i a l o rder ing r e s t r i c t i o n o f A t o the d i f f e rence se t I A I - I B I . Recal l t h a t 3 (A) denotes the p a r t i a l o rder ing o f i n i t i a l i n t e r v a l s o f A , which i s well-founded by 4.1 (dependent choice). L e t us denote by Ht"(A) the he igh t o f the maximum A o f g(A) , i . e . the he igh t o f 3 ( A ) minus one; analogous no ta t ions w i t h B and A-B . Then we have the fo l l ow ing i nequa l i t i es : Ht"(B) + Hto(A-B) 4 Ht"(A) & Ht"(B) @ Ht"(A-B) , where t i s the usual o rd ina l sum and @ i s the na tura l , commutative sum: see ch.1 5 9. 0 Consider the isomorphism which, t o each i n i t i a l i n t e r v a l X o f A-B , associa- t es the i n i t i a l i n t e r v a l BuX o f A . The p a r t i a l o rder ing 3 (A) i s an exten- s ion of 3 (B) fo l lowed by 3(A-B) : t h i s y i e l d s the f i r s t i nequa l i t y . On another hand, t o each i n i t i a l i n t e r v a l X o f A , associate the couple whose

f i r s t term i s t he r e s t r i c t i o n X / ( \ X ( n l B ( ) and whose second term i s the res- t r i c t i o n X/ ( lX \ A IA-BI) . Th is i s an isomorphism which transforms 3 ( A ) i n t o a r e s t r i c t i o n of the d i r e c t product inequa l i t y , by using 7.2. 0

3 ( B ) x 'J(A-B) : t h i s y i e l d s the second

9.2. L e t us s ta te as fo l lows the MAXIMAL REINFORCED CHAIN THEOREM. Given a wel l p a r t i a l o rder ing A and the associated well-founded p a r t i a l o rder ing

J(A) , there e x i s t s a wel l -ordered r e s t r i c t i o n o f 3 ( A ) which is isomorphic t o the he igh t of 3(A) , thus which has the maximum poss ib le height.

Equivalent ly, there e x i s t s a we l l -o rder ing which i s a reinforcement o f A , isomorphic t o the he igh t o f 3 ( A ) minus one, thus isomorphic t o the maximum

Chapter 4 123

p o s s i b l e o r d i n a l (DE JONGH, PARIKH 1977; uses axiom o f c h o i c e ) . 0 F i r s t o f a l l , t h e equ iva lence o f b o t h p reced ing s tatements i s an immediate consequence o f 1.3 (BONNET, POUZET 1969); n o t e t h a t f o r any o r d i n a l o r d e r i n g o f i n i t i a l i n t e r v a l s i s isomorphic w i t h o(t1 . We s h a l l prove t h e f i r s t s ta tement , conce rn ing t h e maximal c h a i n (up t o isomor-

phism) among r e s t r i c t i o n s o f Note t h a t t h e r e s u l t i s obv ious f o r A empty, t hus H t o ( A ) = 0 . I t i s obv ious

a l s o i f A i s based on a s i n g l e t o n , t hus Ht" (A) = 1 . Assume f i r s t t h a t H t " (A ) i s a decomposable o r d i n a l o( = /3+ . Moreover we

assume t h a t i s t h e l a s t component o f oc i n t h e Cantor normal form. Then t h e prev ious decomposi t ion /s + 'd sum as w i t h t h e usual o r d i n a l sum. Denote by B an i n i t i a l i n t e r v a l o f A w i t h h e i g h t 13 modulo 3 ( A ) . Then t h e

two i n e q u a l i t i e s i n t h e p rev ious 9.1 y i e l d hypothes is , t h e r e e x i s t s a c h a i n o f t y p e /3 which i s a r e s t r i c t i o n o f S i m i l a r l y w i t h J (A -B) and 8 . Hence t h e r e e x i s t s a c h a i n o f t ype 8 whose elements a r e i n i t i a l i n t e r v a l s o f A i n c l u d i n g B . F i n a l l y t h e sum o f these

two cha ins has t h e isomorphism type o( . Secondly we assume t h a t H t " (A )

form w ktl , where k h e i g h t CJ modulo g ( A ) . Then by 9 . 1 we have t h e f o l l o w i n g i n e q u a l i t y :

, t h e w e l l -

J ( A ) . We procede by i n d u c t i o n on t h e h e i g h t Ht" (A) .

ho lds as w e l l w i t h t h e n a t u r a l , o r commutative

Hto(A-B) = 8 . By o u r i n d u c t i o n

3 (B) .

i s an indecomposable o r d i n a l o f t h e p a r t i c u l a r

B o f A w i t h i s an o r d i n a l . Take an i n i t i a l i n t e r v a l

k) k+l 4 w 8 Ht"(A-B) , thus Ht"(A-B) = L3 k+l . By i t e r a t i o n , we o b t a i n an c ~ - s e q u e n c e o f i n i t i a l i n t e r v a l s

A which i s s t r i c t l y i n c r e a s i n g under i n c l u s i o n , w i th Hto(Bitl-Bi) = W f o r

each i (obvious n o t a t i o n Bitl-Bi f o r t h e r e s t r i c t i o n o f A t o t h e d i f f e -

rence s e t o f t h e bases). By o u r i n d u c t i o n hypo thes i s , we a s s o c i a t e t o each i a cha in o f o r d e r t ype 0 o f i n i t i a l i n t e r v a l s , each one s i t u a t e d between Bi and Bitl . Hence t h e

w -sum o f t hese cha ins y i e l d s a c h a i n o f i n i t i a l i n t e r v a l s w i t h t y p e w k+l . T h i r d l y we assume t h a t Ht" (A) i s an a r b i t r a r y indecomposable o r d i n a l o( . Denote by u = Cof d and t a k e a u-sequence o f i n c r e a s i n g o r d i n a l s o( (i < u) w i t h Sup o(i = K . F o r each i < u , t a k e an i n i t i a l i n t e r v a l Ai o f A w i t h h e i g h t C t i modulo r(A) (axiom o f c h o i c e ) . Hence H t o ( A i ) = o( f o r each i < u . L e t bi be an element i n t h e d i f f e r e n c e s e t I A ( - lA i l . Denote

by hi t h e h e i g h t modulo (A) o f t h e i n i t i a l i n t e r v a l Bi o f elements

s t r i c t l y l e s s than o r incomparable w i t h bi . Then o( 4 fi < 4 t hus

Sup /si = o( . From t h e u-sequence o f i n d i c e s i K u , we e x t r a c t another u-sequence f o r which a l l e lements bi and a l l o r d i n a l s bi a r e d i s t i n c t .

Bi ( i i n t e g e r ) o f

k

Since A i s a well par t ia l ordering, by 3.4 we may require that the u-sequence of the bi intervals Bi i s s t r i c t l y increasing with respect t o inclusion. By a new extrac- t ion, we may require t h a t fs i+l >, pi 0 p i (natural , or commutative sum)

for each index i / pi for each i , using 9 . 1 above. By the induction hypothesis, we can take in each difference B i + l - B i a chain of i n i t i a l intervals of A , situated between B i and B i t l and whose length i s a t l eas t hi , The sum of these chains yields a def ini t ive chain whose length i s Oi = 4 by ch.2 5 5.5. CI

be s t r i c t l y increasing (mod A ) . Thus the sequence of corresponding

9.3. THE EXTRACTION PROPERTY Given an aleph , we shall say that a par t ia l ordering A has the extraction property for w,, i f for every &,-sequence u of d is t inc t elements in the base A , there exis ts a s t r i c t l y increasing (mod A ) sequence extracted from u and with the same length as By 3 .2 . (3) , an in f in i te par t ia l ordering A i s a well par t ia l ordering i f f A has the extraction property for 0 . By 3.4, any well partial ordering has the extraction property for any inf in i te aleph.

Let A be a well par t ia l ordering, J ( A ) the par t ia l ordering of i n i t i a l intervals of A (with respect t o inclusion). Then 'j ( A ) has the extraction property for every regular aleph s t r i c t l y greater than w ( Z A G U I A 1983). 0 For any i < U* (regular aleph), l e t B i be an in i t ia l interval of A . Denote by Mi the f i n i t e se t of minimal elements (mod A ) in the complement s e t I A l - l B i l . For each integer p , denote by C the class of those Mi whose cardinal i s p . Since a, i s regular, then there exis ts an integer p such that C has cardinality . Reindexing the sequence of Mi , we consider i t as an G)&-sequence in the direct Droduct Ax ... % A (p times) (choice for f i n i t e s e t s ) . This direct product being a well par t ia l ordering by 7.1.(2) (denumerable subset axiom), the proposition follows by 3.4. 0

I n the cited thesis , i t i s proved that any dis t r ibut ive la t t ice which has the extraction property for every regular aleph s t r i c t l y greater than w , admits a chain isomorphic t o i t s height: th i s i s another proof for theorem 9 . 2 .

u (notion communicated by Z A G U I A 1983).

P

P

§ 10 - COFINALITY OF A FINITELY FREE PARTIAL ORDERING

10.1. INCREASING SEQUENCE LEMMA Let A be a par t ia l ordering, and l e t k = Cof A . Then a necessary and suff ic ient condition that A be generated by a s t r i c t l y increasing sequence ( i . e . tha t every element have an upper bound in the sequence) i s t h a t every subset of the base with

Chapter 4 125

c a r d i n a l 4 k have an upper bound i n A . Moreover k i s n e c e s s a r i l y r e g u l a r

(uses axiom o f choice; communicated by POUZET) . 0 If A i s generated by a s t r i c t l y i n c r e a s i n g o r d i n a l - i n d e x e d sequence C , then

k = Cof C t hus k i s r e g u l a r . Moreover e v e r y subset o f t h e base, w i t h c a r d i n a l s t r i c t l y l e s s than k , has an upper bound i n C . Conversely suppose t h a t eve ry subset w i t h c a r d i n a l < k has an upper bound. Take a c o f i n a l subset H o f A w i t h Card H = k . We l l -o rde r H ; c o n s i d e r k as an o r d i n a l and denote by ai

(i C k ) t h e elements o f H . Then d e f i n e as f o l l o w s t h e i n c r e a s i n g k-sequence o f elements b i n H . L e t bo = a. . L e t 16 i < k ; by hypo thes i s t h e r e e x i s t s

an upper bound o f a l l a and a l l a l r e a d y d e f i n e d b . ( j < i). Then d e f i n e bi j J

as be ing such an upper bound, be long ing t o

.-

H . 0

10.2. REGULAR INCREASING SEQUENCE THEOREM

L e t A be a p a r t i a l o r d e r i n g and k = Cof A . I f eve ry p roper i n i t i a l i n t e r v a l o f A has c o f i n a l i t y < k , then ( 1 ) eve ry subset o f t h e base, w i th c a r d i n a l i t y < k , has an upper bound; ( 2 ) k i s a r e g u l a r a leph and A i s generated by an i n c r e a -

sing k-sequence . (uses axiom o f choice; communicated by POUZET). 0 By t h e p reced ing lemma, ( 2 ) f o l l o w s f rom (1). Moreover we can always reduce A

t o a c o f i n a l r e s t r i c t i o n , s t i l l denoted by A , hav ing c a r d i n a l i t y k . To each element x i n t h e base, a s s o c i a t e Ax = i n i t i a l i n t e r v a l o f elements < o r I x (mod A) ; then l e t u < k , denote by

Hu t h e s e t o f e lements x such t h a t k x G u . F i r s t we see t h a t , g i v e n u c k , every subset F o f Hu w i t h Card F < k , has an upper bound. Indeed f o r a l l x i n F , t h e u n i o n o f co r respond ing Ax has a t most c o f i n a l i t y uxCard F < k . So t h a t t h i s u n i o n does n o t cove r t h e e n t i r e base: hence t h e i n t e r s e c t i o n o f co r - responding f i n a l i n t e r v a l s 3 x , i s non-empty.

Now l e t us prove t h e r e g u l a r i t y o f k . For each u < k , s i n c e Card Hu< k , cons ide r Hu as t h e u n i o n o f an a r b i t r a r y i n c r e a s i n g (Co f k)-sequence o f subsets,

each o f which hav ing c a r d i n a l i t y < k : so t h a t each has an upper bound. We o b t a i n

a s e t Ku o f these upper bounds, w i t h Card KU 6 Cof k . Now when u v a r i e s by t a k i n g (Cof k ) many va lues, we o b t a i n a c o f i n a l s e t modulo A , hav ing c a r d i -

n a l i t y l e s s than o r equal t o (Co f k ) = Cof k : t h i s proves t h a t k = Cof k . We a r e f i n i s h e d p r o v i d e d we n o t e t h a t each s e t F w i t h Card F < k , i s necessa-

r i l y i n c l u d e d i n some = k many d i s j o i n t non-empty subsets : c o n t r a d i c t i o n . 0

k x = Cof Ax < k . F o r each c a r d i n a l

2

Hu (u < k ) ; f o r o the rw ise F would be t h e un ion o f (Cof k )

10.3. COFINALITY THEOREM FOR FINITELY FREE PARTIAL ORDERINGS

Every f i n i t e l y f r e e p a r t i a l o r d e r i n g , e i t h e r has f i n i t e c o f i n a l i t y , o r i n f i n i t e r e g u l a r c o f i n a l i t y (POUZET 1979; uses axiom o f c h o i c e ) .

0 Wi th p rev ious n o t a t i o n s , suppose t h a t k = Cof A i s an i n f i n i t e s i n g u l a r aleph. We can always suppose t h a t Card A = k and t h a t A i s we l l - f ounded (ch.2 5 5.1,

c o r o l l a r y ) . Then by t h e p reced ing 10.2, t h e r e e x i s t s a p roper i n i t i a l i n t e r v a l w i t h c o f i n a l i t y

t i a l i n t e r v a l s (under i n c l u s i o n ) , each o f which hav ing c o f i n a l i t y k . Take an element i n each d i f f e r e n c e s e t : we o b t a i n a bad w - s e q u e n c e (mod A) . S ince A i s wel l - founded, we have i n f i n i t e l y many m u t u a l l y incomparable elements. 0

k . Thus t h e r e e x i s t s a s t r i c t l y dec reas ing a - s e q u e n c e o f i n i -

E X E R C I S E 1 - MAC NEILLE COMPLETION

L e t A be a p a r t i a l o r d e r i n g w i t h base E . To each subset X o f E , a s s o c i a t e

t h e s e t X ' o f upper bounds (each element o f X ' i s >/ eve ry element o f X ) , then t h e s e t X+ o f l ower bounds o f X ' . N o t i c e t h a t X+ i s an i n i t i a l i n t e r -

v a l o f A and t h a t t h e a s s o c i a t i o n o f X+ t o X i s a c l o s u r e o p e r a t i o n ; i . e . we have t h e i n c l u s i o n X+z X , idempotence: i n c r e a s i n g : i f X E Y then X+C Y + . 1 - Give examples t o show t h a t ( X u Y)+ can p r o p e r l y i n c l u d e X+ u Y+ , and t h a t

( X n Y ) + can be a p roper subset o f X + A Y+ . I n any case, eve ry i n t e r s e c t i o n o f

c losed s e t s i s c losed ( c l o s e d s e t = i d e n t i c a l t o i t s c l o s u r e ) .

2 - L e t x be an element o f E . The c l o s u r e o f t h e s i n g l e t o n o f x i s t h e i n i - t i a l i n t e r v a l 4 x (mod A ) . I d e n t i f y each x w i t h t h i s i n i t i a l i n t e r v a l : t h e

s e t o f c losu res , p a r t i a l l y o rde red by i n c l u s i o n , becomes a p a r t i a l o r d e r i n g ex ten - d i n g A , c a l l e d t h e MAC-NEILLE comp le t i on o f A (see MAC-NEILLE 1937).

Fo r each s e t X , t h e c l o s u r e X+ i n c l u d e s t h e i n i t i a l i n t e r v a l generated by X . I n genera l , i t i s d i s t i n c t f rom i t . Indeed, t a k e a,b,c w i t h a < c and b < c and a I b and cons ide r t h e i n i t i a l i n t e r v a l 4a.b) . Another example: t a k e t h e o r d i n a l W +1 and i t s i n i t i a l i n t e r v a l G, . F i n a l l y t a k e t h e d i r e c t p roduc t

t i a l i n t e r v a l s o f (,J . 3 - L e t Ci be cha ins w i t h base E which r e i n f o r c e A . Suppose t h a t A i s t h e

c o n j u n c t i o n o f t h e Ci : see 7.3. L e t T be c losed (mod A) . For each index i , c a l l t h e component Ti w i t h i ndex i t h e c l o s u r e o f T (mod Ci) : then Ti 3 T . The components o f a l l T r e l a t i v e t o a g i v e n index i a r e i n i t i a l i n t e r v a l s o f

t h e cha in Ci , hence t h e i r s e t i s a c h a i n ( w i t h r e s p e c t t o i n c l u s i o n ) assoc ia ted t o Ci . Moreover, eve ry c l o s e d s e t T i s t h e i n t e r s e c t i o n o f i t s components Ti. Indeed, i f t i s an element o f each Ti , then t &(mod Ci) eve ry upper bound

of T (mod Ci) f o r each i . Hence t $ ( m o d Ci) eve ry upper bound o f T (mod A) f o r each Consequently t h e comp le t i on o f A has t h e same d imension as A .

X++ = X+ ; and t h e o p e r a t i o n i s

u w and n o t i c e t h a t t h e c l o s u r e s a r e those r e c t a n g l e s d e f i n e d by two i n i -

i . Hence t ,< (mod A) eve ry u. b. o f T (mod A) .

EXERCISE 2 - BOUNDEDLY FINITELY FREE PARTIAL ORDERING L e t A be a p a r t i a l o r d e r i n g w i t h base E , whme f r e e subsets a re assumed t o

be f i n i t e and a l l o f c a r d i n a l i t i e s l e s s than a maximum f i n i t e i n t e g e r . L e t D be

one of t h e f r e e subsets w i t h maximum c a r d i n a l . Then t o each e lement i f D ,

Chapter 4 127

t h e r e corresponds a t o t a l l y o rde red r e s t r i c t i o n Ai of A whose base i n t e r s e c t s

D w i t h t h e s i n g l e t o n o f i , such ______ t h a t t h e un ion o f t h e bases l A i l 12 E (see DILWORTH 1950; t h e p r o o f q i v e n he re i s due t o PERLES 1963). We assume t h a t t h e

bases o f these chains, o r t o t a l l y o rde red r e s t r i c t i o n s , a r e d i s j o i n t , which i s

obv ious l y p o s s i b l e .

1 - F i r s t assume t h a t E i s f i n i t e , and argue by i n d u c t i o n on Card E . F i x A and D , and examine t h e f i r s t case where D i s d i f f e r e n t f rom t h e s e t o f m i n i -

mal elements (mod A) and f rom t h e s e t o f maximal e lements. Then cons ide r t h e res -

t r i c t i o n A+ o f A t o those x f o r which t h e r e e x i s t s i n D an element i 6 x ;

analogous r e s t r i c t i o n A- w i t h i x . By t h e i n d u c t i o n hypo thes i s , t h e p ropos i - t i o n i s t r u e f o r A+ and A- , which a r e p r o p e r r e s t r i c t i o n s o f A . Then t o each i as t h e i r common ex tens ion . I n t h i s f i r s t case, n o t e t h a t t h e s e t o f min imal elements modu- l o A i s always a subset o f t h e s e t o f minimums f o r a l l cha ins

Examine t h e second case where D

moreover t h e r e e x i s t s ano the r f r e e s e t D ' w i t h same c a r d i n a l i t y , which i s n e i - t h e r t h e s e t o f min imal , n o r t h e s e t o f maximal e lements. Then by o u r l a s t re -

mark i n t h e f i r s t case, we can a s s o c i a t e w i t h D t h e same s e t o f chains Ai as w i t h D ' . Analogous argument w i t h maximal e lements.

Now examine t h e t h i r d case where D i s , f o r i ns tance , t h e s e t o f min imal elements,

b u t t h e o n l y o t h e r p o s s i b l e f r e e s e t w i t h maximum c a r d i n a l i t y i s t h e s e t o f maximal elements. Take i i n D , then a maximal c h a i n (under i n c l u s i o n ) whose m i n i -

mum i s i . Then remove f rom E t h e elements o f t h i s maximal chain. Then use t h e i n d u c t i o n hypo thes i s on t h i s p r o p e r r e s t r i c t i o n o f A and t h e subset D - t i } . 2 - Suppose t h a t E i s i n f i n i t e and assume t h e u l t r a f i l t e r axiom. L e t D be a

f r e e s e t w i t h maximum ( f i n i t e ) c a r d i n a l i t y . To each f i n i t e subset F o f E i n - c l u d i n g D , a s s o c i a t e as p r e v i o u s l y a l l p o s s i b l e p a r t i t i o n s o f F y i e l d i n g f o r each i i n D a t o t a l l y o rde red r e s t r i c t i o n o f A/F c o n t a i n i n g i as an e l e -

ment. Consider each p a r t i t i o n as an equ iva lence r e l a t i o n , and c a l l of t hese equ iva lence r e l a t i o n s v e r i f y i n g o u r p r o p o s i t i o n : i t s u f f i c e s t o app ly t h e coherence lemma ch.2 5 1.3.

N o t i c e t h a t t h e p r o p o s i t i o n does n o t ex tend t o t h e case o f an a r b i t r a r y f i n i t e l y

f ree p a r t i a l o r d e r i n g . Fo r example, t a k e A t o be t h e d i r e c t p roduc t o f w1 w i t h i t s e l f , which i s a w e l l p a r t i a l o r d e r i n g by 7 .1 . (2 ) . Yet eve ry decomposi- t i o n i n t o a sum o f cha ins c o n s i s t s o f W1-many cha ins (PERLES).

3 - L e t us take up aga in t h e case o f a p a r t i a l o r d e r i n g A w i t h a f i n i t e base E ; cons ide r aga in t o t a l l y o rde red r e s t r i c t i o n s Ai o f A , b u t no l o n g e r assu-

me t h a t t h e i r bases a r e m u t u a l l y d i s j o i n t . L e t x, y be elements o f E : we say t h a t y covers x i f y i s an immediate successor (mod A) o f x . We say t h a t

i n D , t h e r e i s assoc ia ted a c h a i n A t and a c h a i n AT : d e f i n e Ai

Ai . i s t h e s e t o f min imal elements (mod A ) , and

UF t h e s e t

A i s t r a n s i t i v e l y generated by t h e Ai i f f o r x and y > x (mod A) , t h e r e

e x i s t s a f i n i t e sequence go ing f rom x t o y , each te rm o f which covers t h e

p reced ing term, any two consecu t i ve terms of t h e sequence b e l o n g i n g t o the base

o f a same Ai . For t h i s i t s u f f i c e s t h a t , f o r eve ry x and el 'ery y c o v e r i n g

x , t h e r e e x i s t s an Ai c o n t a i n i n g x and y as e lements. I n o r d e r t o a v o i d useless comp l i ca t i ons , we say t h a t an element o f

pa rab le (mod A) w i t h eve ry o t h e r e lement : l e t us remove a l l i s o l a t e d elements.

Consider t h e o rde red p a i r s (x,y) where y covers x . We say t h a t (x,y) i s

l e s s t h a n o r equal t o ( x ' , y ' ) where y ' covers x ' , i f e i t h e r y,< x ' (mod A)

o r ( x ' , y ' ) = (x,y) . N o t i c e t h a t t h i s r e l a t i o n forms a p a r t i a l o r d e r i n g denoted

by A" . The c o m p a r a b i l i t y (mod A " ) o f two o rde red p a i r s i s e q u i v a l e n t t o t h e

c o m p a r a b i l i t y (mod A) L e t Do be a f r e e s e t o f o rde red p a i r s , which i s maximal w i t h r e s p e c t t o i n c l u -

s ion, f o r t h e p a r t i a l o r d e r i n g A" . Then t o each e lement i of Do we can b i - j e c t i v e l y a s s o c i a t e a t o t a l l y o rde red r e s t r i c t i o n Ai ef A , such t h a t A is. t r a n s i t i v e l y qenerated by t h e l e r c a r d i n a l i t y generates A (see BOGART 1970). Th i s l a t t e r p o i n t i s obv ious .

Thus we o n l y have t o prove t h e e x i s t e n c e o f t h e cha ins

a p p l y t h e s tatement i n paragraph 1 t o t h e p a r t i a l o r d e r i n g o f o rde red p a i r s w i t h one element c o v e r i n g t h e o t h e r , s t a r t i n g w i t h t h e f r e e s e t Do . For each o f t h e

cha ins obta ined, i t s u f f i c e s t o t a k e t h e e lements i n t h e o rde red p a i r s , and t o complete t r a n s i t i v e l y .

E i s i s o l a t e d i f i t i s incom-

o f a l l t h e f o u r elements which c o n s t i t u t e these p a i r s .

Ai . Moreover no s e t o f cha ins w i t h s t r i c t l y smal-

Ai . For t h a t purpose,

EXERCISE 3 - CONJUGATE PARTIAL ORDERINGS

L e t E be a s e t and A, B two p a r t i a l o r d e r i n g s w i t h base E . We say t h a t A

and B a r e conjugates i f , f o r each p a i r o f e lements x, y o f E , i f x and y a r e comparable (mod A) , t h e n t h e y a r e incomparable (mod B) , and converse ly .

Example: A i s a t o t a l o r d e r i n g , o r c h a i n on E , and 6 i s t h e f r e e p a r t i a l

o r d e r i n g ( i d e n t i t y r e l a t i o n ) on E . Another example: l e t C, D be two chains;

t h e d i r e c t p roduc t C x D d e f i n e d i n 5 7 has as a con juga te t h e p a r t i a l o r d e - r i n g d i r e c t p roduc t C x D- where D i s t h e converse o f D (see ch.2 5 1 .7 ) . No t i ce t h a t i f A i s a con juga te o f B , then A i s a l s o a con juga te o f t h e converse p a r t i a l o r d e r i n g B- . 1 - L e t A, B be conjugates; show t h a t t h e d i s j u n c t i o n r e l a t i o n Au B d e f i n e d

by x g y iff x ( y (mod A o r mod B) , i s a cha in . N o t i c e t h a t A i s t h e con- j u n c t i o n of A u B and A u B- . Deduce t h a t A has a con juga te i f f i t s dimen-

s i o n i s 1 o r 2 (see 7.3). O r e q u i v a l e n t l y i f f A has a t o t a l l y o rde red r e i n f o r - Cement C w i t h t h e f o l l o w i n g c o n d i t i o n : i f x < y and x I z and y I z (mod A) , then z < x or z 7 y (mod C ) .

Chapter 4 129

2 - S t a r t w i t h a c h a i n C and a s e t U o f i n t e r v a l s o f C , p a r t i a l l y ordered by i n c l u s i o n . Show t h a t one o b t a i n a con juga te o f U by o r d e r i n g i n t e r v a l s which

a re incomparable w i t h r e s p e c t t o i n c l u s i o n , by means o f t h e p o s i t i o n i n o f t h e i r l e f t c u t ( i . e . t h e i n i t i a l i n t e r v a l o f C s i t u a t e d on t h e l e f t ) . One can as w e l l o r d e r these incomparable i n t e r v a l s by t h e p o s i t i o n o f t h e i r r i g h t c u t .

Deduce t h a t a p a r t i a l o r d e r i n g has dimension 1 o r 2 i f i t i s isomorphic t o t h e

p a r t i a l o r d e r i n g o f i n c l u s i o n between i n t e r v a l s o f a g i v e n cha in . I n p a r t i c u l a r

u s i n g 6.2 ( u l t r a f i l t e r axiom), eve ry t r e e has d imension 1 o r 2, which was a l ready mentioned i n 7 .7 .

3 - Conversely, l e t A and B be con juga tes . Consider t h e cha in A- u B w i t h

base E , f o l l o w e d by A u B w i t h base E ' (an isomorphic copy o f E ) . Associate t o each element x o f t h e p a r t i a l o r d e r i n g A , hence x be long ing t o E , t h e i n t e r v a l ( x , x ' ) where x ' i s t h e r e p l i c a o f x i n E ' (communicated by POUZET

i n 1968, unpub l i shed) .

C

EXERCISE 4 - WELL-ORDERED RESTRICTION OF MAXIMAL LENGTH (LEMMA 4.5 REVISITED)

L e t A be a we l l - f ounded p a r t i a l o r d e r i n g . Then e i t h e r t h e r e e x i s t i n f i n i t e l y

many elements, a l l o f d i f f e r e n t h e i g h t s (mod A) which a r e m u t u a l l y incomparable,

o r t h e r e e x i s t s a w e l l - o r d e r e d r e s t r i c t i o n o f A w i t h t h e same h e i g h t ( u s e s u l t r a -

f i l t e r axiom; POUZET 1979). 1 - Suppose t h e f i r s t c o n c l u s i o n does n o t ho ld , and argue by i n d u c t i o n on t h e h e i g h t o f A . Assume f i r s t t h a t H t A i s decomposable, hence o f t h e form

A t o elements whose h e i g h t (mod A ) x i n

H t C = . By t h e i n d u c t i o n hypo thes i s , t h e r e e x i s t s a we l l -o rde red r e s t r i c t i o n C" o f C w i t h h e i g h t . L e t u be t h e minimum o f C " . The r e s t r i c t i o n o f A t o e lements s t r i c t l y l e s s than u i s a we l l - f ounded p a r t i a l o r d e r i n g B w i t h

h e i g h t >/ /3 (see ch.2 5 3.5) . By t h e i n d u c t i o n hypo thes i s , t h e r e e x i s t s a w e l l - ordered r e s t r i c t i o n B" o f B w i t h h e i g h t ('3. Then B" + C" i s a r e s t r i c t i o n o f A w i th h e i g h t fi + 8 . 2 - Assume now t h a t H t A = c( , an indecmposab le o r d i n a l ; and suppose t h a t t h e f i r s t c o n c l u s i o n does n o t h o l d . By 4.5 ( u l t r a f i l t e r axiom), i t s u f f i c e s t o prove

t h a t t h e r e e x i s t s a r e s t r i c t i o n o f A w i t h t h e same h e i g h t d, , which i s f i n i t e l y f r e e and hence a w e l l p a r t i a l o r d e r i n g . L e t k be t h e c o f i n a l i t y o f o( ; decorn-

pose d i n t o a sum o< ( ic k ) w i t h t h e o( i nc reas ing , a l l s t r i c t l y l e s s than o( , and Sup o( = o( (ch.2 5 5.5) .

For each i c. k , cons ide r t h e r e s t r i c t i o n Ai o f A t o elements whose h e i g h t

i s b o t h g r e a t e r t han o r equal t o 2 OC

/3 + 8 w i t h /3 non-zero and I < /3 + J' . L e t C denote t h e r e s t r i c t i o n o f i s g r e a t e r t han o r equal t o A . For each

ICI , we have H t x (mod A ) = 0 + H t x (mod C) ( p r o o f by i n d u c t i o n ) . Hence

j (j< i) and s t r i c t l y l e s s than o(

( j 6 i) . This Ai has height H . By the induction hypothesis, and since there do not exist infinitely many mutually incomparable (mod Ai) elements with different heights, there exists a well-ordered restriction Ci of Ai with the same height M i . Let D denote the restriction of A to the union of the bases lCil . Firstly Ht D is strictly greater than each o( hence greater than or equal to o( = Sup M i ; so that Ht D = o( . Secondly, there do not exist infinitely many mutually incomparable elements in D , since they would all have different heights (mod A), which contradicts our hypothesis. Hence D is finitely free, so that we can apply 4.5.

3 - As opposed to what happens in 4.5, we cannot require that the well-ordering having the same height as Consider the following counterexample due to POUZET. Take the points (XJ) of coordinates natural numbers with y~ x . Let (x,y) < (x',y') iff either x = x' and y < y' , or x d x ' and y' 3 y+2 . Then the height of each point (x,y) is y , hence Ht A = w . However, a totally ordered restriction of A isomorphic with w can have at most a finite sequence of points with heights 0,1,2, ... ; for example the points (u,O), ( u , l ) , ... , (u,u) (where u is an integer). After which one must pass to a point with ordinate greater than or equal to u t 2 , hence with height greater than or equal to u+2 .

A , have one and only one element of each height (mod A).

EXERCISE 5 - REINFORCEMENT OF A WELL PARTIAL ORDERING (REVISITED) 1 - Given a well partial ordering A , there exists a well-ordered reinforcement of A . This is an immediate consequence of the reinforcement axiom (ch.2 5 4.2) by using proposition 3.5.(1) above. We propose to obtain this result in a more economic manner, using only the axiom of choice for finite sets (ch.1 5 2.10).

For this, to each ordinal u strictly less than Ht A , associate the finite set Fu of those elements with height u in A ; then the finite set of chains with base u one of these chains. Finally it suffices to take the sum along the u . 2 - Notice that with the considered axiom, a partial ordering A is a well partial ordering iff: (i) every totally ordered reinforcement of A (ii) there exists a totally ordered reinforcement of More precisely, the preceding proposition is equivalent to the following weakening of the axiom of choice for finite sets: "for every well-orderable set of finite mutually disjoint sets, there exists a choice set"(communicated by POUZET in 1979).

Fu . Then the axiom of choice for finite sets associates to each

is a well-ordering, and A (use 3.5.(3): ZF suffices).

131

CHAPTER 5

EMBEDDABILITY BETWEEN PARTIAL OR TOTAL ORDERING

§ 1 - EMBEDDABILITY, IMMEDIATE EXTENSION, FAITHFUL EXTENSION

EMBEDDABILITY, EQUIMORPHISM L e t R, S be two r e l a t i o n s o f t h e same a r i t y . We say t h a t R i s embeddable i n S o r i s s m a l l e r t han S under embeddab i l i t y , o r t h a t S admi ts an embedding o f R o r i s g r e a t e r t han R , i f f t h e r e e x i s t s a r e s t r i c t i o n o f S

isomorphic w i t h R ; we w r i t e R + S o r S R . We say t h a t R i s s t r i c t l y embeddable i n S o r s t r i c t l y s m a l l e r t han R , o r

t h a t S admits a s t r i c t embedding o f R o r i s s t r i c t l y g r e a t e r t han R , denoted

by R < S o r S > R , i f f R $ S b u t S $ R .

We say t h a t R i s equimorphic w i t h S , denoted R 5 S , i f f R 6' S and S 6 R . The comparison r e l a t i o n 4 i s r e f l e x i v e and t r a n s i t i v e , hence d e f i n e s a quasi - o r d e r i n g on each s e t o f r e l a t i o n s . Moreover equimorphism i s symmetric and hence d e f i n e s an equ iva lence r e l a t i o n . Embeddab i l i t y i s n o t an t i symmet r i c , even up t o isomorphism: see t h e f o l l o w i n g examples.

L e t Q be t h e c h a i n o f t h e r a t i o n a l s , and Q+l t h e ex tens ion o b t a i n e d by adding a l a s t element: t hen Q I Q+l . L e t LJ be t h e c h a i n o f t h e n a t u r a l numbers, and a- t h e converse cha in ( see ch.2 5 1.7). Then u-.G) E 1 + ( W - . w ) ( t h e o r d i n a l p roduc t i s d e f i n e d i n ch.2 5 3.7). I n t h e c h a i n o f n a t u r a l numbers, r e p l a c e each even number by Z ( t h e cha in of p o s i t i v e and n e g a t i v e i n t e g e r s ) and each odd number by a f i n i t e chain. We o b t a i n continuum many m u t u a l l y non- isomorphic chains, a l l o f which a r e equimorphic .

1.1. L e t R, S be two equimorphic r e l a t i o n s . Then t h e r e e x i s t s a p a r t i t i o n o f t h e base I R I i n t o two d i s j o i n t subsets D, D ' , and a p a r t i t i o n o f 1s t i n t o two d i s j o i n t subsets E, E ' w i t h R/D isomorphic t o S/E and R / D ' isomor-

p h i c t o S / E ' . Repeat t h e p r o o f o f BERNSTEIN-SCHRODER's theorem (ch.1 5 1.4), where f and g become isomorphisms from one r e l a t i o n o n t o a r e s t r i c t i o n o f

t h e o t h e r . The converse i s f a l s e , even f o r cha ins . Indeed, t h e c h a i n w o f t h e n a t u r a l numbers and t h e c h a i n w+1 t i o n s . S i m i l a r l y f o r t h e incomparable cha ins U+ c.4- and Z = W- + .

g i v e r i s e t o p a r t i t i o n s s a t i s f y i n g t h e above condi -

1.2. IMMEDIATE EXTENSION Given a r e l a t i o n R , we say t h a t S i s an immediate extension o f R i f S i s an extension, and furthermore i f there does no t e x i s t any s t r i c t l y intermediate r e l a t i o n T such t h a t R < T < S w i t h respect t o embeddabil ity. We say a l so t h a t S , o r any r e l a t i o n equimorphic w i t h S , i s an immediate successor o f R w i t h respect t o embeddabil ity.

For each r e l a t i o n R , there e x i s t s an immediate extension o f R . Moreover .if R has a r i t y >/ 1 , then there e x i s t a t l e a s t two immediate extensions (HAGENDORF 1977, p ropos i t ion VI.5.6).

0 Suppose f i r s t t h a t R i s a 0-ary r e l a t i o n , say R = ( E , + ) : then i t su f f i ces t o replace the base E by a s e t w i t h immediately g rea ter c a r d i n a l i t y : see ch.2 5 3.10. Suppose t h a t R i s a unary r e l a t i o n . L e t a be the c a r d i n a l i t y o f the s e t o f e le - ments g i v i n g the value (+) t o R , and b the analogous c a r d i n a l i t y f o r ( - ) . Then i t suf f i ces t o replace e i t h e r a o r b by an immediately g rea ter card ina l , again using ch.2 5 3.10. Suppose now t h a t R has a r i t y n >/ 2 . Add t o the base E o f R a se t D d is - j o i n t w i t h E , and def ine R' t o have base E u D w i t h R+/E = R and R+/D always (t), and f i n a l l y w i t h R+ t ak ing the value (+) f o r those n-tuples contai- n ing a t l e a s t one term i n D . F i n a l l y choose f o r d = Card D the l e a s t aleph f o r which Rt i s no t embeddable i n R , hence R+> R . Le t us prove t h a t R' i s an immediate extension o f R ; the r e l a t i o n R- s i m i l a r l y def ined by exchanging (t) and ( - ) , being another immediate extension, obviously incomparable w i t h R' w i t h respect t o embeddabi 1 i ty. Suppose f i r s t t h a t d i s an i n f i n i t e aleph, and t h a t there e x i s t s a s t r i c t l y i n t e r - mediate r e l a t i o n T w i t h R < T < R' . Consider T as a r e s t r i c t i o n o f Rt . Then

the i n t e r s e c t i o n D n IT1 has c a r d i n a l i t y d . Indeed i f i t had c a r d i n a l i t y c d , then T would be embeddable i n R (more prec ise ly , i n a r e s t r i c t i o n o f Rt t o E increased w i t h (c: d) many elements o f D ). Now p a r t i t i o n DnITI i n t o two d i s j o i n t subsets, each w i t h c a r d i n a l i t y d , say D ' and D" . Then T i s isomorph ic w i t h i t s r e s t r i c t i o n t o I T 1 - D" , so t h a t R i s embeddable i n t h i s r e s t r i c - t ion , and f i n a l l y R+ i s embeddable i n T : cont rad ic t ion . Now i t remains t o consider the case where d = Card D i s f i n i t e . We f i r s t see t h a t d = 1 . Indeed assume t h a t d f i n i t e and >, 2 . Then by hypothesis, the extension of R obtained by adding on ly one element u t o the base I R 1 , w i t h value (t) f o r a l l n- tuples conta in ing u , i s embeddable i n R , thus equimorphic w i t h R . I t e r a t i n g t h i s , the s i m i l a r extension obtained by adding 2 elements, i s s t i l l equimorphic w i t h R , and so on u n t i l we add d elements: con t rad ic t ion .

Now examine the case where d = 1 . Cal l u the supplementary element, whose s ing le ton cons t i t u tes D . Consider again the intermediate r e l a t i o n T as a

Chapter 5 133

r e s t r i c t i o n o f R+ ; obviously u belongs t o the base I T . We say t h a t an e le - ment x i n the base i s a (+)-element (mod R ) i f f every n - tup le which contains x gives value (+) t o R . Analogous d e f i n i t i o n f o r a p a r t i c u l a r , u i s a (+)-element (mod T) . Every (+)-element (mod R) belongs t o the base I T 1 and i s a (+)-element (mod T). Indeed otherwise, i f x i s a (+)-element (mod R ) and does no t belong t o I T I , then by rep lac ing u by x we could embed T i n R : cont rad ic t ion . E i the r there e x i s t Dedek ind- in f in i te ly many (+)-elements (mod R ) . Then R+ i s isomorphic w i t h R : cont rad ic t ion . O r t he s e t o f (+)-elements (mod R ) has Dedek ind- f in i te c a r d i n a l i t y , say h . Then there e x i s t a t l e a s t h + l many (+)-elements (mod T) . Consider a r e s t r i c t i o n R ' o f T which i s isomorphic w i t h R . Since we have exac t l y h many (+)-elements (mod R ' ) and a t l e a s t h t l many (+)-elements (mod T) , there e x i s t s a t l e a s t one (+)-element (mod T) , say v , which does no t belong t o the base I R ' I . Then the r e s t r i c t i o n o f T t o the base I R'I p lus the element v i s isomorphic w i t h R' : cont rad ic t ion . 0

(+)-element (mod T) ; i n

1.3. FAITHFUL EXTENSION

(1) Le t R , S be two n-ary r e l a t i o n s ( n >r 1) . Assume t h a t S does not admit an embedding o f R . Then there e x i s t s a s t r i c t l y g rea ter extension T

of S which does no t _admit an embedding o f R . We c a l l i t a f a i t h f u l extensio? o f S modulo R . Moreover we can choose T t o be an immediate extension o f S : see HAGENDORF 1977.

The statement i s obviously f a l s e f o r a r i t y zero.

0 Suppose f i r s t t h a t R aqd S are unary. L e t at be 'the c a r d i n a l i t y o f the se t o f elements g i v i n g the value (+) t o R , and a- the analogous c a r d i n a l i t y f o r (-); s i m i l a r l y l e t b+ and b- be the analogous c a r d i n a l i t i e s fo r S . Since R $ S , e i t h e r b+< a+ o r b-< a- . Suppose the f i r s t case holds, the argument being analogous f o r the second case. It s u f f i c e s t o take an extension of S i n which bt i s preserved and b- i s replaced by an immediately l a r g e r card ina l . Suppose t h a t R and S have a r i t y n 3 2 . Add t o the base E o f S a s e t 0' which i s d i s j o i n t from E , and de f ine the extension T+ o f S w i t h base EvD',

+ t a k i n g t h e va lue (+) f o r those n - t u p l e s c o n t a i n i n g a t l e a s t one te rm o f D . A lso choose D+ w i t h c a r d i n a l (a leph ) s u f f i c i e n t l y l a r g e t o have T + > S . Do t h e same w i t h t h e va lue ( - ) , thus o b t a i n i n g 0- and T -> S . We c l a i m t h a t R 4:' o r R $T- , which y i e l d s o u r conc lus ion . Indeed suopose t h e c o n t r a r y , and c o n s i d e r

R as a r e s t r i c t i o n o f T+ , The base o f R

hence t h e r e e x i s t s an element u+ such t h a t R takes t h e va lue (+) f o r each n - t u p l e c o n t a i n i n g a t l e a s t one t e r m equal t o

ment f o r t h e va lue ( - ) : c o n t r a d i c t i o n . 0

( 2 ) L e t R be a r e l a t i o n w i t h a r i t y 5 2 and S1& R and S2 & R . Then there e x i s t s a common e x t e n s i o n o f S1 and S2 which does n o t a d m i t an embedding o f R. The c o n t r a p o s i t i v e i s : i f R,< X f o r eve ry X which s a t i s f i e s b o t h X >,S1 and X 3 S2 , then S1,. . . ,Sh.

0 Take t h e case o f a b i n a r y r e l a t i o n , and suppose t h a t S1 and S2 have d i s j o i n t bases El and E2 . L e t St be t h e common e x t e n s i o n w i t h base El E2 t a k i n g

t h e va lue (+) f o r those o rde red p a i r s w i t h one te rm i n El and t h e o t h e r i n E2 . Analogously d e f i n e S- f o r t h e va lue ( - ) . It s u f f i c e s t o see t h a t e i t h e r S+ o r S- does n o t admi t an embedding o f R . Suopose t h e c o n t r a r y ; t hen t h e r e e x i s t s a

p a r t i t i o n o f t h e base o f R i n t o two non-empty d i s j o i n t subsets such t h a t , f o r

eve ry element u i n one subset and v i n t h e o t h e r , we have R(u,v) = R(v,u) =

= + ; same conc lus ion w i t h t h e va lue ( - ) . Note t h a t , g i v e n two p a r t i t i o n s o f t h e base, each w i t h two non-empty d i s j o i n t s e t s , t h e r e e x i s t two elements t h e base which a r e separated b o t h by t h e f i r s t p a r t i t i o n and by t h e second. Thus

t h e r e e x i s t two elements u, v g i v i n g s imu l taneous ly R(u,v) = + and - :

c o n t r a d i c t i o n . 0 The p r o p o s i t i o n i s o b v i o u s l y f a l s e f o r unary r e l a t i o n s .

i s n o t a subset o f E , s i n c e R $ S,

u+ . There e x i s t s an analogous e l e -

R 6 S1 o r R 6 S2 . Extend t h i s t o any f i n i t e sequence

u, v i n

( 3 ) L e t S have a r i t y >/ 2 and non-empty base. Suppose t h a t S & R1 , S $ R2 and S % R 3 . Then t h e r e e x i s t s a p roper ex tens ion S+ f S which respec ts t h e non-

e m b e d d a b i l i t i e s S+* R1 , S+$ R 2 and S+$ R3 . For t h e a r i t y 1 o r f o r empty base, t h e p r o p o s i t i o n i s o b v i o u s l y f a l s e , even w i t h o n l y R1 and R2 . 0 Consider t h e 4 f o l l o w i n g ex tens ions of element a and s e t t i n g S,(a,x) = S (x,a) = + f o r eve ry x i n t h e base IS1 . L e t S2 be s i m i l a r l y ob ta ined w i t h ( - ) ' i n s t e a d o f (+) . L e t S3 be o b t a i n e d w i t h S (a,x) = + and S (x,a) = - f o r eve ry x i n IS ( , and moreover S3(a,a) = + . F i n a l l y S4 w i t h t h e same c o n d i t i o n s , except t h a t S4(a,a) = - . Suppose o u r conc lus ion i s f a l s e . Then t h e r e e x i s t two

an embedding o f a same R . Hence i n t h e base o f t h i s R , t h e r e e x i s t s an element

al , say t o f i x t h e ideas, p l a y i n g t h e r o l e o f a i n S1 and an a3 p l a y i n g t h e r o l e of a i n S3 . Suppose f i r s t l y t h a t t h e base I R I has c a r d i n a l i t y 3 2 .

S . L e t S1 be o b t a i n e d by add ing a new

1

3 3

Si ( i = 1,2,3,4) which admi t

Chapter 5 135

Then al and a3 a r e d i s t i n c t , s i n c e R(x,al) = R(al,x) = + f o r eve ry x # al

and R(x,a3) = - f o r eve ry x # a3 . Moreover R(al,a3) takes s imu l taneous ly t h e va lue (+) and t h e va lue ( - ) : c o n t r a d i c t i o n . Analogous argument f o r S1 and S2 , S1 and S4 , S2 and S3 , S2 and S4 , S3 and S4 . Suppose now t h a t t h e base I R I takes t h e va lue ( - ) . S ince S > R and by hypo thes i s S non-empty, n e c e s s a r i l y S

i s r e f l e x i v e . As p r e v i o u s l y d e f i n e ex tens ions S1, S2, S3 which now a r e a l l t h r e e r e f l e x i v e . E i t h e r R1 = R 2 = R3 = R and then o u r conc lus ion ho lds . O r R1 and

p o s s i b l y R2 a r e d i s t i n c t f rom R , thus have c a r d i n a l i t i e s 2 . Again suppose

ou r c o n c l u s i o n i s f a l s e : t hen R1 f o r i n s t a n c e i s embeddable i n a t l e a s t two Si

( i = 1,2,3) , and t h e argument t e r m i n a t e s as p r e v i o u s l y . 0

has c a r d i n a l i t y 1, and t o f i x t h e ideas, t h a t R

1.4. Consider an 13 -sequence o f r e l a t i o n s Ri ( i i n t e g e r ) o f common a r i t y 3 2

and w i t h m u t u a l l y d i s j o i n t bases. L e t A be a r e l a t i o n o f t h e same a r i t y . have Ria A f o r eve ry i , then t h e r e e x i s t s a comnon ex tens ion o f a l l t h e Ri

which does n o t admi t an embedding o f A . 0 L e t R+ denote t h e common e x t e n s i o n o f t h e Ri on t h e un ion o f t h e bases, which takes t h e v a l u e (+) f o r a l l t hose n - t u p l e s ( n = a r i t y ) c o n t a i n i n g a t l e a s t two

terms taken f rom two d i s t i n c t bases. Analogously d e f i n e t h e ex tens ion R- . Suppose t h a t A i s embeddable b o t h i n R+ and R- . Since A i s n o t embeddable i n any

Ri , t h e r e n e c e s s a r i i y e x i s t two elements t rans fo rmed i n t o two elements i n two d i s t i n c t

and second embedding. tience f o r an n - t u p l e c o n t a i n i n g b o t h x and y , we have f o r A t h e va lue (+) and t h e v a l u e ( - ) : c o n t r a d i c t i o n . 0

.-

x, y i n t h e base I A I , which a r e l R i l , s imu l taneous ly f o r t h e f i r s t

5 2 - EMBEDDABILITY BETWEEN PARTIAL ORDERINGS; WELL PARTIAL ORDERING OF FINITE TREES (KRUSKAL); CANTOR'S THEOREM FOR PARTIAL ORDERINGS

(DILWORTH, GLEASON); TOURNAMENT

2.1. There e x i s t i n f i n i t e l y many f i n i t e p a r t i a l o r d e r i n g s which a r e m u t u a l l y incom- -parable w i t h r e s p e c t t o embeddab i l i t y .

0 L e t A1 be t h e p a r t i a l o r d e r i n g on 5 elements a,b,a',b',v w i t h a ( b < v , a ' < b ' < v and i n c o m p a r a b i l i t y e lsewhere. Now more g e n e r a l l y f o r each i n t e g e r

i , l e t Ai be t h e p a r t i a l o r d e r i n g based on 2 i+3 elements a,b,a',b',u l , . . . , U ~ ~ ~ , V ~ , ~ , V ~ , ~ , . . . , V ~ ~ ~ , ~ w i t h a < b and a ' < b;< vi-l,i and compara-

b i l i t i e s u1 < v ~ , ~ and u1 < v ~ , ~ and then u2 < v1,2 and u i and so

u n t i l ui-l < V i - 2 , i - 1 and u i - 1 < vi-l,i ; and i n c o m p a r a b i l i t y elsewhere. The p a r t i a l o r d e r i n g s thus d e f i n e d a r e m u t u a l l y incomparable w i t h r e s p e c t t o em- b e d d a b i l i t y (example comnunicated i n 1969 by JULLIEN). 0

I36 THEORY OF RELATIONS

2.2. (1 ) There e x i s t s a s t r i c t l y decreas ing (under embeddab i l i t y ) w - s e q u e n c e o f denumerable p a r t i a l o r d e r i n g s . 0 To each s e t I o f i n t e g e r s , a s s o c i a t e t h e p a r t i a l o r d e r i n g AI o b t a i n e d as an

e x t e n s i o n o f t h e f i n i t e p a r t i a l o r d e r i n g s Ai ( i e I) , taken t o be m u t u a l l y incomparable. Then take an i n f i n i t e s t r i c t l y dec reas ing sequence o f s e t s I . 0 (2 ) There e x i s t continuum many denumerable p a r t i a l o r d e r i n g s which a r e m u t u a l l y

incomparable under embeddab i l i t y .

0 Take t h e p reced ing AI and no te t h a t , f o r two s e t s I , J o f i n t e g e r s n e i t h e r

o f which i n c l u d e s t h e o t h e r , t hen AI and AJ a r e incomparable. 0

Problem posed by HAGENDORF. Ex i s tence o f a s t r i c t l y dec reas ing G1-sequence o f denumerable p a r t i a l o rde r ings . Same problem f o r r e l a t i o n s , posed by POUZET.

2.3. THEOREM OF THE WELL PARTIAL ORDERING OF FINITE TREES Embeddabi l i ty between f i n i t e t r e e s i s a w e l l p a r t i a l o r d e r i n g (KRUSKAL 1960).

0 We can always assume t h a t each o f t h e cons ide red f i n i t e t r e e s has a minimum

element: i t s u f f i c e s t o add a minimum, and even a new minimum i f t h e r e a l r e a d y e x i s t s a minimum, t o each f i n i t e t r e e ; t hen embeddab i l i t y o r non-embeddabi l i ty i s preserved.

Suppose t h a t embeddab i l i t y between f i n i t e t r e e s w i t h a minimum, i s n o t a w e l l pa r - t i a l o r d e r i n g . Then t h e r e e x i s t s an w -sequence o f such t r e e s , which i s bad w i t h

r e s p e c t t o embeddab i l i t y : see ch.4 5 3 .2 . (2 ) coun tab le case. Hence t h e r e e x i s t s a

s t r o n g l y minimal bad w-sequence: ch.4 5 2.iO. I n t h i s sequence, t h e terms a r e

m u t u a l l y incomparable w i t h r e s p e c t t o embeddab i l i t y : ch.4 5 2.3 and 2.8. L e t U denote t h i s sequence, and a minimum, which a r e s t r i c t l y embeddable i n a

see ch.4 5 4.3 and 2.8.

L e t denote t h i s coun tab le s e t (up t o isomorphism) o f f i n i t e t r e e s w i t h a m i n i - mum. Embeddabi l i ty between words, o r f i n i t e sequences o f elements o f H , i s a w e l l p a r t i a l o r d e r i n g : ch.4 5 4.4 (HIGMAN). To each t r e e a s s o c i a t e one o f t h e

f i n i t e sequences ob ta ined by t o t a l l y o r d e r i n g i n an a r b i t r a r y manner t h e immediate successors o f t h e minimum, then by r e p l a c i n g each o f them by t h e sub - t ree o f t hose

g r e a t e r o r equal elements. Then t h e p reced ing sequence U becomes a sequence o f words i n H , hence i s good. Moreover, i f t h e word thus s u b s t i t u t e d f o r t h e t r e e

Ui i s embeddable i n t h e word s u b s t i t u t e d f o r U . (i,j i n t e g e r s ) , then t h e t r e e Ui i t s e l f i s embeddable i n U . . Hence t h e sequence U i s good: c o n t r a d i c t i o n . 0

Ui ( i i n t e g e r ) each term. The s e t o f f i n i t e t r e e s w i t h

Ui , i s a w e l l p a r t i a l o r d e r i n g :

H

Ui

J

3

2.4. A b i n a r y r e l a t i o n A i s a p a r t i a l o r d e r i n g i f f A does n o t admi t an embed- d i n g o f t h e f o l l o w i n g f i n i t e r e l a t i o n s :

t h e b i n a r y r e l a t i o n w i t h c a r d i n a l i t y 1 and va lue ( - ) ( t h i s ensu res r e f l e x i v i t y ) ;

t h e r e l a t i o n always (+) w i t h c a r d i n a l i t y 2 ( a n t i s y m n e t r y ) ; t h e r e f l e x i v e b i n a r y

c y c l e w i t h c a r d i n a l i t y 3 , and f i n a l l y t h e c o n s e c u t i v i t y r e l a t i o n on 3 elements

Chapter 5 137

associated w i t h t h e c h a i n o f c a r d i n a l i t y 3 : see ch.2 5 8.6 ( these two non-embed-

dab i l i t i e s ensure t r a n s i t i v i t y ) .

2.5. Every denumerable p a r t i a l o r d e r i n g admi ts an embedding o f t h e o r d i n a l w or i t s converse (J- o r t h e denumerable f r e e p a r t i a l o r d e r i n g . I n p a r t i c u l a r , eve ry denumerable f i n i t e l y f r e e p a r t i a l o r d e r i n g , hence eve ry denu-

merable chain, admi ts an embedding o f w o r L,I- . Th is i s KONIG's lemma: compare

WiXh ch.4 5 4.5 i n t h e case o f a we l l - f ounded p a r t i a l o r d e r i n g .

0 Enumerate t h e elements ai ( i i n t e g e r )

in tegers i, j where we assume i < j , i n t o 3 c o l o r s , acco rd ing t o whether

ai < a o r 7 a ch.3 5 1.1, t h e r e e x i s t s a denumerable s e t o f i n t e g e r i n d i c e s i n which a l l p a i r s

have the same c o l o r . Accord ing t o whether i t i s t h e f i r s t , second o r t h i r d c o l o r , the g iven p a r t i a l o r d e r i n g admi ts an embedding o f cc) o r w - o r a denumerable r e l a t i o n o f i d e n t i t y . 0

Modulo t h e denumerable subset axiom (ch.1 5 2.6), eve ry i n f i n i t e p a r t i a l o r d e r i n g

admits an embedding o f w o r w - o r t h e denumerable f r e e p a r t i a l o r d e r i n g .

A wel l - founded p a r t i a l o r d e r i n g does n o t admi t an embedding o f w - . Conversely,

a denumerable p a r t i a l o r d e r i n g ( o r more g e n e r a l l y a p a r t i a l o r d e r i n g w i t h w e l l - orderable base) which does n o t admi t an embedding o f w - , i s wel l - founded. Wi th

the axiom o f dependent choice, e v e r y p a r t i a l o r d e r i n g which does n o t admi t an embedding o f w - i s wel l - founded: see ch.2 5 2.4.

o f t h e base, and p a r t i t i o n t h e p a i r s o f

o r l a . (modulo t h e g i ven p a r t i a l o r d e r i n g ) . By RAMSEY's theorem j j J

2.6. CANTOR'S THEOREM FOR PARTIAL ORDERINGS

Given a p a r t i a l o r d e r i n g A , t h e p a r t i a l o r d e r i n g o f i n c l u s i o n among i n i t i a l i n t e r - vals o f A , admits a s t r i c t embedding o f A (DILWORTH, GLEASON 1964).

I n p a r t i c u l a r , i f A i s a chain, t h e n t h e c h a i n o f c u t s o f A admi ts a s t r i c t embedding o f A . 0 Let B denote t h e p a r t i a l o r d e r i n g o f i n i t i a l i n t e r v a l s o f A . To see t h a t

A g B , i t s u f f i c e s t o a s s o c i a t e t o each element x o f I A I t h e i n i t i a l i n t e r v a l

o f elements 6 x (mod A ) . NOW suppose t h a t BG A and l e t f be an isomorphism o f B i n t o A . Some i n i - t i a l i n t e r v a l s X o f A s a t i s f y t h e r e l a t i o n f ( X ) E X : f o r example t h e e n t i r e

i n t e r v a l X = \ A [ . L e t U be t h e i n t e r s e c t i o n o f a l l t hese i n t e r v a l s . We s h a l l

prove t h a t f ( U ) 4 U . Indeed i f f ( U ) E U , t h e n t h e i n t e r v a l V o f those e l e - ments x < f ( U ) (mod A) s a t i s f i e s V c U ( s t r i c t i n c l u s i o n ) . Hence by t h e isomor-

Phhm f ( V ) < f ( U ) (mod A ) , hence f ( V ) E V and so Vz U : c o n t r a d i c t i o n .

Thus f (U ) + U and s i n c e U i s t h e i n t e r s e c t i o n o f those X such t h a t f ( X ) i5 X, there e x i s t s one o f t hese X , say Xo , which does n o t c o n t a i n t h e element f ( U ) .

Thus we have Xoz U and # U , hence by isomorphism f (U) < f(Xo) (mod A) Moreover f (U) + Xo and so f (Xo) $ Xo : cont rad ic t ion .

2.7. TOURNAMENT A b inary r e l a t i o n A i s c a l l e d a tournament i f f i t i s re f l ex i ve , ant isymmetr ic and comparable: f o r a l l x, y e i t h e r A(x,y) = + (and thus A(y,x) = - by a n t i -

symnetry) o r A(y,x) = t (and A(x,y) = - ) . I n the f i r s t case we say t h a t y

fo l lows x (mod A) ; i n the second case y precedes x (mod A) . For each element u o f the base E o f A , the elements i n E - u are d i v ided i n t o two complementary sets, formed respec t i ve l y o f those elements which f o l l o w u and those which precede u . Every r e s t r i c t i o n o f a tournament i s a tournament. A tournament i s a chain i f f i t i s t r a n s i t i v e .

The b inary cyc le on 3 elements (see ch.2 5 8.6), obviously mod i f ied t o be r e f l e x i - ve, i s a tournament. Yet a b ina ry r e f l e x i v e cyc le w i t h c a r d i n a l i t y 3 4 i s not

antisymmetric, thus i s no t a tournament.

A tournament i s a chain i f f i t does no t admit an embedding o f the b inary r e f l e x i v e cyc le w i t h c a r d i n a l i t y 3.

A b inary r e l a t i o n i s a tournament i f f i t admits no embedding o f the b inary r e l a t i o n w i t h c a r d i n a l i t y 1 and value (-) (ensures r e f l e x i v i t y ) ; no embedding o f t he r e l a - t i o n always (+) w i t h c a r d i n a l i t y 2 (ensures antisymmetry); f i n a l l y no embedding o f the i d e n t i t y r e l a t i o n w i th c a r d i n a l i t y 2 (ensures comparab i l i t y ) .

§ 3 - DENSE CHAIN: A-DENSE CHAIN, FOR AN INFINITE CARDINAL A 3.1. DENSE CHAIN A chain i s sa id t o be dense i f f i t s base i s i n f i n i t e , and i f between any two d i s - t i n c t elements x < y there e x i s t s an element z : x c z < y . Hence between any two elements x and y 7 x there e x i s t i n f i n i t e l y many e le - ments.

To each z ( x < z c y ) associate the i n t e r v a l (x,z) : there i s no minimal

i n t e r v a l w i th respect t o i nc lus ion (use ch.1 tj 1.1). 0 (x,z)

Every denumerable dense chain w i thout any minimum o r maximum, i s isomorphic to_ the chain Q o f the ratj.1..

Every countable chain i s embeddable i n Q . Every dense chain admits an embedding o f Q (uses dependent choice; ZF su f f i ces i f the base i s denumerable, o r on l y we l l -o rderab le ) .

- -- _ _

Chapter 5 139

U Consider t h e s e t o f f i n i t e s t r i c t l y i n c r e a s i n g sequences o f elements o f t h e chain,

say sequences wi th a r b i t r a r y l e n g t h h , assoc ia tes any sequence w i t h l e n g t h 2h + 1 o f

t h e form vo < u,, c v1 c u 1 < . . . < vh < uh< v ~ + ~ . Then a p p l y dependent cho ice t o

t h e r e l a t i o n R . The ob ta ined w-sequence y i e l d s a r e s t r i c t i o n isomorphic w i t h Q. 0

u o c u1 < ... < uh ( h i n t e g e r ) , and t h e r e l a t i o n R which t o each o f these

3.2. L e t A be a denumerable cha in . I f A admi ts an embedding o f eve ry coun tab le

o r d i n a l , t hen A admi ts an embedding o f t h e c h a i n Q (uses coun tab le axiom o f cho ice ) .

0 To each coun tab le o r d i n a l @ , a s s o c i a t e 4 +1+a which by hypo thes i s can be embedded i n A . Hence a s s o c i a t e an element a ( & ) such t h a t 4 i s embeddable

i n t h e l ower i n t e r v a l and i n t h e i n t e r v a l above a ( * ) . The c h a i n A i s denumera- b le , y e t t h e r e a r e

element a such t h a t eve ry coun tab le o r d i n a l i s embeddable bo th below and above a

( coun tab le axiom o f choice: we use t h e f a c t t h a t eve ry denumerable un ion o f denume- r a b l e o r d i n a l s i s a denumerable o r d i n a l : see ch .1 5 2 . 5 ) . By i t e r a t i o n , we o b t a i l i

a r e s t r i c t i o n o f A which i s isomorphic w i t h Q . 0

wl-many coun tab le o r d i n a l s 6 . So t h e r e e x i s t s a t l e a s t one

3.3. L e t A be a non-empty c h a i n w i t h an i n i t i a l i n t e r v a l and a f i n a l i n t e r v a l ,

bo th d i s j o i n t and i n each of which A i s embeddable. Then Q i s embeddable i n A

(uses dependent choice; ZF s u f f i c e s i f A i s denumerable o r w i t h w e l l - o r d e r a b l e base) .

0 By i t e r a t i o n , t h e hypo thes i s a l l o w s one t o d i v i d e A i n t o t h r e e d i s j o i n t i n t e r -

va l s , i n each o f which A i s embeddable. Hence t h e r e e x i s t s an element a such

t h a t A i s embeddable b o t h b e f o r e and a f t e r a . By i t e r a t i o n u s i n g dependent

choice, we o b t a i n a r e s t r i c t i o n o f A which i s isomorphic w i t h Q . 0

3 . 4 . a-DENSE CHAIN

Given an i n f i n i t e c a r d i n a l a , we say t h a t a c h a i n i s a-- i f i t s base i s i n f i - n i t e and between any two d i s t i n c t elements x and y , t h e i n t e r v a l has c a r d i n a l i t y

equal t o a . Analogous d e f i n i t i o n f o r a cha in

Sense o f 3.1, i s ( w )-dense (uses denumerable subset ax iom).

( 3 a)--. A dense chain, i n t h e

Given an i n f i n i t e c a r d i n a l a , t h e r e e x i s t s a c h a i n w i t h c a r d i n a l a which

- i s a-dense (uses axiom o f c h o i c e ) .

c3 S t a r t w i t h t h e o r d i n a l p roduc t Q.a where Q i s t h e c h a i n o f t h e r a t i o n a l s , and

a i s an o r d i n a l , more p r e c i s e l y an a leph ( r e c a l l t h a t , w i t h choice axiom, eve ry c a r d i n a l i s an a leph: ch .1 0 6 .1 ) . The c h a i n Q.a has c a r d i n a l i t y a and i s ob-

v i o u s l y dense w i t h o u t any minimum or maximum. A lso t h e s e t o f f i n i t e sequences o f

elements i n Q.a has c a r d i n a l i t y a . We o r d e r t h i s s e t l e x i c o g r a p h i c a l l y . Now i t

s u f f i c e s t o see t h a t i t i s an a-dense o r d e r i n g . Indeed, cons ide r two f i n i t e se-

quences u and v s t r i c t l y a f t e r u w i t h r e s p e c t t o l e x i c o g r a p h i c a l comparison.

Denote by ui and vi ( i i n t e g e r ) t h e i - t h terms. E i t h e r t h e r e e x i s t s a l e a s t i ndex i w i t h ui < vi (mod Q.a ) ; then we take wi which i s s t r i c t l y between

ui and vi , and we have a-many f i n i t e sequences beg inn ing w i t h U ~ , . . . , U ~ - ~ , W ~ :

they a l l a r e s i t u a t e d between u and v . O r u i s an i n i t i a l i n t e r v a l o f v , and we have t h e same c o n c l u s i o n by t a k i n g sequences beg inn ing w i t h an element wh < vh ( h = l e n g t h o f u ) . 0

u f o l l o w e d by

3.5. L e t w, be an i n f i n i t e r e g u l a r a leph . Every c h a i n w i t h c a r d i n a l i t y admi ts an embedding e i t h e r o f t h e o r d i n a l a,, o r i t s converse CJ, ,

0 L e t A se i s embeddable. L e t x, y be elements o f I A l ; we p u t x e q u i v a l e n t w i t h y

w, ,

W , -dense cha in .

be a cha in w i t h c a r d i n a l i t y w 4 , i n which n e i t h e r ad n o r i t s conver-

i f t h e i n t e r v a l between them has c a r d i n a l i t y < LJ& . The equ iva lence c lasses a r e i n t e r v a l s o f A . Every equ iva lence c l a s s has c a r d i n a l i t y < L d M . Indeed l e t T be an equ iva lence c l a s s . Take an o r d i n a l indexed sequence o f e lements o f T which i s

s t r i c t l y i n c r e a s i n g (mod A) and c o f i n a l i n T . Th is sequence has l e n g t h s t r i c t l y

l e s s than ~3~ , s i n c e w, i s n o t embeddable i n A . The subset o f T above t h e

f i r s t t e r m o f t h e sequence, i s a un ion o f < w,-many i n t e r v a l s , each w i t h c a r d i -

n a l i t y < ui . By r e g u l a r i t y o f w , , t h i s un ion has c a r d i n a l i t y s t r i c t l y l e s s than ad. Same argument f o r t h e subset o f T

quence, by c o n s t r u c t i n g a dec reas ing o r d i n a l - i n d e x e d sequence. S ince t h e equ iva lence c lasses a l l have c a r d i n a l i t y s t r i c t l y l e s s than wd and ad i s r e g u l a r , t h e r e e x i s t C d d -many equ iva lence c lasses . Take a r e p r e s e n t a t i -

ye element f rom each c l a s s . Again by r e g u l a r i t y , between two r e p r e s e n t a t i v e s t h e r e

n e c e s s a r i l y e x i s t LJ, -many equ iva lence c lasses , hence as many r e p r e s e n t a t i v e s .

F i n a l l y , t h e r e s t r i c t i o n o f A t o t h e s e t o f r e p r e s e n t a t i v e s i s an W, -dense

cha in . 0

Th is p r o p o s i t i o n i s no l o n g e r t r u e i f ad i s a s i n g u l a r a leph .

0 Take t h e sum cc,- + LJ + ... + i~ .- + ... where i i s an a r b i t r a r y i n t e g e r

and (Ji designates t h e r e t r o - o r d i n a l converse o f o i . The sum i s thus a c h a i n w i t h c a r d i n a l i t y W . F i r s t l y , t h e o n l y we l l -o rde red r e s t r i c t i o n s of

t h i s sum a r e t h e f i n i t e cha ins and u . No dense c h a i n i s embeddable i n t h i s sum. Secondly, a r e t r o - o r d i n a l i s embeddable i n t h i s sum o n l y i f i t i s embeddable i n one

o f t h e w i , so w, i s n o t embeddable. 0

b e f o r e t h e f i r s t t e rm o f t h e se-

1

3.6. L e t A be a p a r t i a l o r d e r i n g whose base has c a r d i n a l i t y an i n f i n i t e r e q u l a r

a leph (dw . Suppose t h a t , f o r each element u o f t h e base I A l , t h e s e t o f

elements < u I u has c a r d i n a l i t y s t r i c t l y l e s s than c d K . Then t h e r e e x i s t s

Chapter 5 141

a t o t a l l y ordered r e s t r i c t i o n of A which i s isomorphic with the ordinal L J ~ . a Well-order the base I A I i n order-type and l e t C denote the chain thus obtained. Pa r t i t i on the pa i r s of elements x, y of the base i n t o two colors . I f x < y (mod C ) , we say t h a t the pa i r i s (+ ) i f f x < y (mod A ) and ( - ) i f f x > y or xly (mod A ) . For a given element a i n the base, there a re s t r i c t l y l e s s than wA-many elements x < a (mod C ) o r x < a o r l a (mod A ) . Now apply ch.3 5 3.2: there e x i s t s a subset equipotent t o the base, i n which a l l pa i r s have color (+) . This y i e lds a r e s t r i c t i o n isomorphic t o the ordinal dd . 0 The proposit ion i s f a l s e f o r a s ingular aleph:

For each in t ege r i , take a s e t Ei with ca rd ina l i t y oi , the E i being mutually d i s j o i n t . Define a p a r t i a l ordering on the union, which has ca rd ina l i t y aW , by taking each E i a s a f r e e s e t , each element i n E i being l e s s than

each element of E f o r any two in tegers j > i . Then every t o t a l l y ordered r e s t r i c t ion has order-type a t most G) .

x f o r which the p a i r { a , x ) has co lor ( - ) : so t h a t e i t h e r

j

3.7. (1) Let W, be a regular l i m i t aleph. Then each ad -dense chain admits e i the r an embedding of wd or i t s converse, o r an embedding of a l l o rd ina ls s t r i c t l y l e s s than ad and t h e i r converses. ( 2 ) Let Ud be a regular aleph. Then each L3 d+l -dense chain admits e i t h e r a n embeddinq of equipotent w i t h and their converses,(ERDOS, RADO 1953; (1) and ( 2 ) use the generalized continuum hypothesis; f o r o( = 0 , ZF p l u s choice su f f i ces ; fo r d = 1 , ZF plus choice plus continuum hypothesis su f f i ces ) .

we can assume A t o have ca rd ina l i t y a&. Let C denote a well-ordering with base 1 A I two co lors : assuming t h a t x < y (mod C ) , the pa i r will have co lor (+) i f f x ( y (mod A ) and ( - ) i f f x > y (mod A ) . By ch.3 5 3 .5 . (1 ) (gen. continuum hypothesis), e i t h e r there e x i s t s a base: thus A admits an embedding of the retro-ordinal . Or f o r each cardinal b < W, , there e x i s t s a (+)-monochromatic subset w i t h c a rd ina l i t y b : thus A admits an embedding of every ordinal s t r i c t l y l e s s than proof ends by interchanging the co lors .

( 2 ) Let A be an Wbc+l -dense chain. Since ud i s regular by hypothesis, the l e a s t cardinal b sa t i s fy ing b ( G), ) > md u s i n g gen. continuum hypothesis. In order t o prove ( 2 ) , we now denote by A those

P a r t i t i o n the pa i r s of elements of I A I i n t o two colors , exac t ly as in (1) above.

o r i t s converse, o r an embedding of a l l o rd ina ls

(1) Let A be an -dense chain. By r e s t r i c t i n g t o one o f i t s i n t e rva l s ,

and order-type a0( . Par t i t i on the pa i r s { x,y} i n the base i n t o

(-)-monochromatic subset equipotent t o the

C C ) ~ . The

i s b = &IM : see ch.2 5 6.5(1)

-dense chains which embed ne i the r the ordinal W o(+l nor i t s converse. oc+1

By ch.3 § 3.5 . (2 ) , e i t h e r there e x i s t s a (-)-monochromatic subset equipotent w i th the base: thus A admits an embedding of @ N + l - . O r a (+)-monochromatic subse t w i t h c a r d i n a l i t y Cdd : thus A admits an embedding o f C d M . Hence i n every case, A admits an embedding o f the o rd ina l ae and i t s converse. Le t ao,al ,..., ai ,... ( i < wo( ) be a s t r i c t l y inc reas ing (mod A ) a,-sequence. Le t C be the l e a s t o rd ina l f o r which there e x i s t s an w , + ~ -dense chain which admits no embedding o f c , By the preceding we have c 3 wOc . Assume, i n order t o ob ta in a cont rad ic t ion , t h a t c has c a r d i n a l i t y ud . Then c i s the l i m i t o f a s t r i c t l y inc reas ing sequence o f o rd ina l s ci < c , indexed by i running through a t most (30( . Each ci i s embeddable i n every aH+l -dense chain. For each i , the o rd ina l , o r the aleph equal t o ci , i s embeddable i n the i n t e r v a l (ai,ai+l) . Hence the sum o f the ci i s embeddable i n A and so c i s embeddable i n A . This con t rad i c t i on shows t h a t A admits an embedding o f every o rd ina l equipotent w i t h W M . The same argument proves t h a t A admits as we l l an embedding of the corresponding converse o rd ina l s . 0

5 4 - IMMEDIATE EXTENSION OF A CHAIN: FAITHFUL EXTENSION OF A CHAIN

An o rd ina l U i s indecomposable i f f ever.y o rd ina l V < U s a t i s f i e s V.2 < U (indecomposable o rd ina l i s def ined i n ch.1 5 3 . 6 ) .

0 Let U be an indecomposable o rd ina l and \I < U . Then V i U = U so V . 2 < U . Conversely, suppose t h a t U i s decomposable: there e x i s t V < U and W < U w i t h V+W = U . Le t T = Max(V,W) . Then T < U and T.Z&V+W = U . 0

I n next chapter, indecomposabil ity w i l l be extended t o chains (ch.6 5 3 ) . The preceding propos i t ion must there be mod i f ied (ch.6 5 3.4).

4.1. Le t A , A ' be two chains and U t he l e a s t ordi-nal such t h a t A + U + A ' 4 A+A' . Then U i s indecomposable (HAGENDORF 1971) .

0 Let Y < U , so t h a t A+V+A' I A+A'(equimorphism L def ined i n 5 1 above). Take an isomorphism from A+V+A' i n t o A+A' . L e t V = W+W' w i t h W embeddable i n A and W ' embeddable i n A ' . I n order t o f i x our ideas, suppose t h a t W $ W ' . Then W'+A',( A ' so (W' .2)+A',< A ' and so (W' .4)+A' < A' , hence (V.2)+A1 4 A ' and so A+(V.2)+A',< A+A' and f i n a l l y V.2, W ' . Thus U i s indecomposable, by the prece- d ing propos i t ion . 0

4.2. IMMEDIATE EXTENSION OF A CHAIN

We say t h a t a chain B i s an immediate extension o f A (up t o isomorphism) or i s immediately q rea ter than A ( w i t h respect t o embeddabil ity) i f B > A and

Chapter 5 143

t h e r e does n o t e x i s t any c h a i n X s a t i s f y i n g A < X < B . L e t A be a c h a i n and U the l e a s t o r d i n a l such t h a t A+U $ A . Then A+U i L a L

immediate e x t e n s i o n o f A (HAGENDORF 1972).

N o t i c e t h a t t h e p r o p o s i t i o n no l o n g e r h o l d s i f we r e p l a c e U by a r e t r o - o r d i n a l . Fo r example, l e t A = U- , t h e converse o f w . Then U = W - ; y e t between A and A+U = IJ - . 2 we have 1+ W - , 2+ W - , e t c .

0 The o r d i n a l

decomposable. I n o r d e r t o o b t a i n a c o n t r a d i c t i o n , l e t us suppose t h a t t h e r e e x i s t s

a c h a i n B s a t i s f y i n g A < B < A+d . Consider an isomorphism o f B o n t o a r e s t r i c - t i o n o f A t U , which thus decomposes B i n t o an i n i t i a l i n t e r v a l B ' and a com-

p lementary f i n a l i n t e r v a l B" such t h a t B ' G A and B" < U . The c h a i n B" i s an o r d i n a l . I f 8" < U , i t f o l l o w s t h a t B = B ' + B " , C A+B" 6 A t hus c o n t r a d i c t i n g

t h e d e f i n i t i o n o f B . Hence B" = U and B = B'+U (by = we mean " isomorphic t o " ) .

Consider an isomorphism f rom A o n t o a r e s t r i c t i o n o f B = B ' + U , which decomposes A i n t o an i n i t i a l i n t e r v a l A ' and a complementary f i n a l i n t e r v a l A" such t h a t A ' 6 B ' and A",( U . The c h a i n A" i s an o r d i n a l . I f A" < U , s i n c e U i s i n - decomposable we have A"+U = U so A+U = A ' t A " + U = A ' + U hence A+U,< B ' + U = B , thus c o n t r a d i c t i n g t h e d e f i n i t i o n o f B . Hence A" = U and A = A ' + U . Then e i t h e r U i s an i n f i n i t e indecomposable o r d i n a l , i n which case 1 < U so A+1 5 A (equimorphy). The preceding argument, w i t h A rep laced by A+l , proves

t h e e x i s t e n c e o f a f i n a l i n t e r v a l o f A+1 which i s isomorphic w i t h U : c o n t r a - d i c t i o n .

O r U = 1 and so A = A ' + 1 and B = B ' + 1 , hence A ' + 1 ( B ' + l < A ' + Z , so t h a t A ' < B ' < A ' + 1 . I t e r a t e t h e p reced ing argument: we see t h a t A i s t h e sum o f a c e r t a i n i n i t i a l i n t e r v a l and o f t h e f i n a l i n t e r v a l (.d - . Hence A+1 i s isomor-

p h i c w i t h A , which c o n t r a d i c t s t h e hypo thes i s t h a t U = 1 . 0

U o f t h e p r o p o s i t i o n i s non-zero and by t h e p reced ing 4.1 i t i s i n -

4.3. Every i n f i n i t e c h a i n admi ts a t l e a s t two immediate ly g r e a t e r c h a i n s ( w i t h r e s - p e c t t o e m b e d d a b i l i t y ) , which a r e m u t u a l l y incomparable (HAGENDORF 1972) .

0 L e t A be an i n f i n i t e c h a i n and U t h e l e a s t o r d i n a l such t h a t A+U i s n o t

embeddable i n A ; s i m i l a r l y l e t U ' be t h e l e a s t r e t r o - o r d i n a l such t h a t U ' + A i s n o t embeddable i n A . By t h e p reced ing 4.2, t hese sums a r e immediate ly g r e a t e r t han A , It remains t o prove t h a t t h e y a r e m u t u a l l y incomparable w i t h r e s p e c t t o

embeddab i l i t y . We s h a l l argue ad absurdum: t o f i x t h e ideas, suppose U'+A,( A+U . Take an isomorphism f rom U ' + A o n t o a r e s t r i c t i o n o f A+U . Th is isomorphism can-

n o t embed U ' + A i n A , n o r i n A+V f o r any V < U . Thus t h e r e e x i s t s a f i n a l

i n t e r v a l o f A which i s c o f i n a l l y embeddable i n U . T h i s f i n a l i n t e r v a l i s i s o - morphic w i t h U : o t h e r w i s e t h e r e would e x i s t V < U such t h a t U ' + A $ A + V $ A , which c o n t r a d i c t s t h e d e f i n i t i o n o f U ' . F i n a l l y by 4.1, U i s indecomposable.

If U is an infinite indecomposable ordinal, then 1 < U so A+1 I A . The preceding argument, where A is replaced by A+l , proves the existence of a final interval of A+1 which is isomorphic with U : contradiction. Hence U = 1 and U'+A d A + 1 . Thus every isomorphism of U'+A into A+l has in its range the maximum element of A+1 ; for otherwise U'+A would be embeddable in A . Thus A has a maximum. By iteration, we see that A has a final interval which is isomorphic with the retro-ordinal w - . Thus A+1 is isomorphic with A , contradicting the inequality A+l = A+U > A . 0

4.4. An infinite chain can have 3, 4, ... immediately greater chains. For example the product A = b -. has the 3 immediately greater chains B = A+l , C = W + A ,

ordinal W . These are the only immediate extensions of A . 0 The reader easily see that there does not exist any chain strictly intermediate between A and B , A and C , A and 0 . Let us see that every chain strictly greater than A admits an embedding of B or C or 0 . Indeed, every chain which is an extension of A can be obtained as follows. Either by adding new elements at the end, so admitting an embedding of B . Or by adding a chain at the beginning, in which w is embeddable (embedding of C ) , or in which the converse of an ordinal is embeddable (from the ordinal Or by partitioning the new elements into a finite number of components W - of A , which is the same as adding them at the beginning. Or by modifying infinitely many components, by adding an ordinal CC, (embedding of C ) , or by replacing these components by a retro-ordinal ( s o getting an embedding of D from the converse

D = ( w 2 ) - + A , where ( W 2 ) - is the converse of the

w 2 we get an embedding of D ) .

of u 2 ) . 0

Another example: the chain W1 + Q , where the 5 following immediate extensions: the addition of &J1 end or between k, and Q , the addition of W1- at the beginning.

For each integer p 3 2 , the chain G, .(p-1) has exactly p immediate extensions (among chains).

There exist chains having infinitely many immediately greater chains. For example, every chain A which has no strict restriction isomorphic with A : see the method of SIERPINSKI 1950, used in 5.3 below.

Q is the chain of the rationals, has or its converse at the

4.5. FAITHFUL EXTENSION BETWEEN CHAINS Let A be an infinite chain, B a chain in which A is not embeddable. lhen there exists a chain, immediate extension of B , in which A is not embeddable (uses axiom of choice; HAGENDORF 1972; the case for scattered chains .e. chains

Chapter 5 145

without any embedding of Q , was a l r e a d y proved by JULLIEN 1969).

0 I f B i s a f i n i t e chain, t hen i t s u f f i c e s t o take B + l . Suppose t h a t B i s

i n f i n i t e . We d i s t i n g u i s h two cases: B < A and B I A ( w i t h r e s p e c t t o embeddabi-

l i t y ) .

I n the case B A , l e t B ' and B" be two m u t u a l l y incomparable chains, each imme- d i a t e l y g r e a t e r t han B (see 4 . 3 ) . Suppose t h e p r o p o s i t i o n i s f a l s e . Then A,< B' o r A,( B" . It i s imposs ib le t o have b o t h equimorphisms A E B' and A z B " , since B ' l B" . Hence A < B ' f o r i ns tance , and so A i s a s t r i c t l y i n t e r m e d i a t e between B and B ' : c o n t r a d i c t i o n .

It remains t o examine t h e case where B i s i n f i n i t e and B I A . We argue ad absur-

dum by supposing t h a t A i s embeddable i n eve ry cha in s t r i c t l y g r e a t e r t han B . I n an a r b i t r a r y manner decompose B i n t o C+C' , where C i s an i n i t i a l i n t e r v a l and C ' t h e complementary f i n a l i n t e r v a l . L e t U be t h e l e a s t o r d i n a l such t h a t C+U+C '> B ; s i m i l a r l y V t h e l e a s t r e t r o - o r d i n a l such t h a t C+V+C' > B . By 4.1,

the o r d i n a l U and t h e converse o f V a r e indecomposable o r d i n a l s . By hypo thes i s

A < C+U+C' . Consider an isomorphism o f decomposes A i n t o t h r e e i n t e r v a l s D , UA , D' w i t h A = D+UA+D' and 0 4 C , UA 6 U , 0 ' 4 C ' . I f U A < U t hen A < C+UA+C',< B , c o n t r a d i c t i n g t h e hypothe-

s i s t h a t B i s incomparable w i t h A . Thus UA = U and then A = D+U+D' . S i m i l a r -

l y we d e f i n e t h e decomposi t ion o f A i n t o i n t e r v a l s E , E ' and an i n t e r v a l i s o - morphic t o t h e r e t r o - o r d i n a l V such t h a t A = E+V+E' and E,< C , E ' Q C ' . Envisage a l l t h e r e l a t i v e p o s i t i o n s o f U and V , i n t e r v a l s o f A . I f t h e i n t e r - v a l U i s i n c l u d e d i n E , then D+U \< E gC so D+U+D',( C+C' = B : i n o t h e r

words

U i n E ' , then U+D',< E ' 4 C ' so A = D+U+D'& C+C' = B , c o n t r a d i c t i n g t h e hypo thes i s . Thus we must suppose t h a t t h e i n t e r v a l

posable o r d i n a l , i s i n c l u d e d i n t h e i n t e r v a l

Hence U = 1 , and s i m i l a r l y V = 1 : t h e i n t e r v a l s U = V = 1 a r e t h e same, thus D = E and 0 ' = E' . L e t x be an a r b i t r a r y element o f t h e base IS1 . L e t Bx be t h e i n i t i a l i n t e r v a l

o f those elements 6 X (mod 6 )

preceding, we have Bx+l+B; > B hence > / A . Moreover, by t h e axiom of choice,

a s s o c i a t e t o each x an element f x o f t h e base des igna te t h e i n i t i a l i n t e r v a l o f t hose elements < f x (mod A ) , then we have

A = Ax+l+A;( w i t h A x < Bx and A;,< B; . We s h a l l prove t h a t t h e f u n c t i o n f which t o each e:ement x o f t h e base I B I assoc ia tes f x , an element o f A , i s s t r i c t l y i n c r e a s i n g . From t h i s we w i l l deduce t h a t B,( A , which c o n t r a d i c t s t h e hypo thes i s o f i n c o m p a r a b i l i t y o f A and B . TO do t h i s , we argue ad absurdum by supposing t h a t f i s n o t s t r i c t l y i n c r e a s i n g : t hus t h e r e e x i s t two elements x, y o f I S ( w i t h x < y (mod B) , and so w i t h

k o n t o a r e s t r i c t i o n o f t h i s sum: t h i s

A \< B , c o n t r a d i c t i n g t h e hypo thes i s . I f t h e r e e x i s t s a f i n a l i n t e r v a l o f

U , isomorphic t o an indecom-

V , isomorphic t o a r e t r o - o r d i n a l .

and B; t h e f i n a l i n t e r v a l > x (mod B) . By t h e

I A I , such t h a t i f we l e t Ax

Bxtl& B . Yet f y < f x (mod A) and so A i d A ' 6 B;I t hus A = Ax + 1 + A;

< Bx t 1 + B' 4 B + B ' = B , c o n t r a d i c t i n g t h e hypo thes i s . 0 \ Y Y Y

Y Y

4.6. I f two cha ins A , B have t h e same s t r i c t l y g r e a t e r cha ins ( w i t h r e s p e c t t o

embeddab i l i t y ) , then A and B a r e equimorphic . I f A and B a r e i n f i n i t e , t hen i t even s u f f i c e s t h a t e v e r y c h a i n s t r i c t l y qrea- t e r t han B be ( o n l y ) g r e a t e r t han A , and converse ly .

0 Obvious f o r A f i n i t e o r B f i n i t e . Suppose t h a t A , B a r e i n f i n i t e and t h a t

A 4 B . Then by t h e p reced ing p r o p o s i t i o n , t h e r e e x i s t s a c h a i n s t r i c t l y g r e a t e r

t han B and n o t g r e a t e r t han A . 0

4.7. N o t i c e (by JULLIEN 1969) t h a t t h e f a i t h f u l ex tens ion no l o n g e r h o l d s i f we

r e p l a c e A by two cha ins A1 and A2 . 0 For example n e i t h e r However, eve ry c h a i n which i s s t r i c t l y g r e a t e r t han w admits an embedding o f

Al o r o f A2 . 0 The e x i s t e n c e o f a f a i t h f u l e x t e n s i o n between chains, no l o n g e r h o l d s i f we r e p l a c e

B by two cha ins B1 and B2 . 0 For example A = W + c3 - i s n e i t h e r embeddable i n B1 = U-. n o r i n i t s converse B2 = w . W - . Yet A i s embeddable i n eve ry cha in i n which b o t h B1

and B2 a r e embeddable: t h i s w i l l be proved i n ch.6 3.7. 0

We w i l l have an o p p o s i t e r e s u l t i f we a l l o w ex tens ions by a r b i t r a r y b i n a r y r e l a -

t i o n s , i n s t e a d o f r e s t r i c t i n g ou rse l ves t o cha ins : see exe rc . 2 below.

Problem posed by SABBAGH i n 1975. Ex i s tence o f two chains A, B which a r e incompa rab le w i t h r e s p e c t t o embeddab i l i t y , and have a supremum c h a i n

l y , f o r e v e r y c h a i n X we would have X >/C i f f X > / A and X ,B . It has been proved by HAGENDORF 1977 t h a t t h i s supremum c h a i n does n o t e x i s t f o r

A or B sca t te red ; t h e genera l case remains unsolved.

For t h e dual o f t h e above s tatement , i . e . t h e e x i s t e n c e o f t h e inf imum, we have

t h e easy example A = w+1 and B = Z w i t h t h e in f imum C = w .

A1 = w +1 n o r A2 = Z = W -+ W i s embeddable i n W .

C . More p r e c i s e -

5 5 - DECREASING SEQUENCES AND SETS OF INCOMPARABLE CHAINS OF REALS: DUSHN I K, MILLER, S I ERP I NSKI

5 .1 . L e t A, B be two chains, each o f which i s embeddable i n t h e r e a l s . Then

t h e r e a r e a t most cont inuum many r e s t r i c t i o n s o f B isomorphic w i t h A . 0 Consider t h e base l B l as a subset o f t h e s e t E o f r e a l s , and l e t F be

Chapter 5 147

a subset o f I B I such t h a t B/F i s isomorphic w i t h A ( i f t h e r e e x i s t s such) . For eve ry subset X o f 1 B l such t h a t B/X i s isomorphic w i t h A , t h e r e e x i s t s

a s t r i c t l y i n c r e a s i n g map fX f rom F o n t o X , hence f rom F i n t o E . Now t h e r e a r e continuum many s t r i c t l y i n c r e a s i n g maps f rom F i n t o E : see ch .2

5 8.4; and f i n a l l y X = f X o ( F ) i s determined by fX ( n o t a t i o n O i n ch .1 5 1 .2 ) . U

5.2. L e t A be a c h a i n o f cont inuum c a r d i n a l i t y , which i s embeddable i n t h e

cha in o f t h e r e a l s .

(1) There e x i s t s a s t r i c t l y s m a l l e r ( w i t h r e s p e c t t o embeddab i l i t y ) r e s t r i c t i o n o f A which has continuum c a r d i n a l i t y (DUSHNIK, MILLER 1940; uses t h e axiom o f c h o i c e ) .

( 2 ) Even s t ronger , t h e r e e x i s t s a r e s t r i c t i o n B of A w i t h continuum ca rd ina -

l i t y , such t h a t no i n t e r v a l o f A , o f continuum c a r d i n a l i t y , i s ernbeddable i n B (HAGENDORF 1977, unpub l i shed) .

( 3 ) F o r eve ry denumerable c h a i n U we have A $ B.U (where B s a t i s f i e s t h e p reced ing ( 2 ) ) .

0 ( 2 ) L e t X be any subset o f t h e base I A 1 such t h a t A / X i s isomorphic w i t h

an i n t e r v a l o f A o f continuum c a r d i n a l i t y . For each such i n t e r v a l , by t h e p re -

ced ing p r o p o s i t i o n , t h e r e a r e a t most continuum many corresponding s e t s over , i f we c o n s i d e r A as a r e s t r i c t i o n o f t h e c h a i n R o f r e a l s , each i n t e r - v a l o f A i s t h e r e s t r i c t i o n t o I A l o f an i n t e r v a l o f R , which i s i t s e l f de f i ned by i t s endpo in ts . Consequently t h e r e a r e continuum many such i n t e r v a l s .

F i n a l l y , t h e s e t o f a l l t h e X has a t most continuum c a r d i n a l i t y . Apply ch.2 5 8.1 (ax iom o f cho ice ) . There e x i s t s a s e t C i n c l u d e d i n I A I , where C and D = I A l -C b o t h a r e e q u i p o t e n t w i t h t h e continuum, as w e l l as

a l l t h e i n t e r s e c t i o n s C n X and D n X . Consequently no X i s i n c l u d e d i n C , t h u s t h e r e s t r i c t i o n B = A / C admi t s no embedding o f any i n t e r v a l o f A which has continuum c a r d i n a l i t y . 0

0 ( 3 ) Suppose A 6 B . U . Then t h e base I A I i s p a r t i t i o n e d i n t o coun tab ly many

i n t e r v a l s , each co r respond ing w i t h an element o f U . A t l e a s t one o f t hese i n - t e r v a l s i s e q u i p o t e n t w i t h t h e continuum: see ch .1 § 4.3, axiom o f cho ice . More-

ove r i t i s embeddable i n

N o t i c e t h a t (1) i m p l i e s t h e ex i s tence , s t a r t i n g f rom t h e cha in o f t h e r e a l s , o f a s t r i c t l y decreas ing w -sequence o f cha ins . We s h a l l see t h a t such a sequence does n o t e x i s t f o r s c a t t e r e d cha ins , i . e . those i n which t h e c h a i n Q o f t h e

r a t i o n a l s i s n o t embeddable: see ch.8 5 4.4.

-

X . More-

B , c o n t r a d i c t i n g t h e p reced ing ( 2 ) . 0

5.3. L e t E be t h e s e t o f r e a l s and R t h e cha in o f t h e r e a l s . There e x i s t two subsets C, D of E which a r e d i s j o i n t , equ ipo ten t w i t h t h e continuum, dense

- in R and such t h a t , f o r each subset X f D , every r e s t r i c t i o n of R isomorphic w i t h R / ( C u X ) , e i t h e r has base C u X , o r i t s base contains a t l e a s t one

element o f E - (C u D ) . Consequently f o r For X and Y subse ts of D which a r e incomparable with respec t t o inc lus ion ,

the preceding two restrictions are incomparable with respect to embeddability (SIERPINSKI 1950; see a l s o ROSENSTEIN 1982; uses axiom of choice) .

0 Take the s e t s a l l isomorphisms from R i n t o R , d i s t i n c t from the i d e n t i t y . For such an isomorphism f , i f a real x i s mapped t o fx # x , f o r example i f fx > x (mod R ) , then every real i n the in te rva l (x , fx ) i s mapped t o a s t r i c t l y g rea t e r r e a l , hence f y # y f o r continuum many r e a l s y . Notice t h a t C and 0 a r e d i s j o i n t and each equipotent with the continuum. Moreover by the same proposit ion we have

f" (C) # C 5 8.5 , the s e t s C and D a r e both dense (mod R ) . Take an a r b i t r a r y subset X of D and an isomorphism g of R / ( C u X ) i n to R , which i s d i s t i n c t from the i d e n t i t y . Since C , hence C u X , i s dense, there e x i s t s an isomorphism gt of R i n t o R , which extends g t o the domain E of a l l r ea l s : see ch.2 5 8.3. Hence morphisms ded in C u D . 0

This immediately implies t he ex is tence of a s t r i c t l y decreasing (with respec t t o embeddability) sequence, indexed by the continuum, of chains of r e a l s . Also the ex is tence of a s e t , equipotent w i t h the continuum, of mutually incomparab le chains of r e a l s (SIERPINSKI 1950).

Y C X cD , we have the s t r i c t embeddability R / ( C uY)< R/(CuX).

C , D in ch.2 5 8.2.(2) (axiom of choice) , where the f i designate

f o r each considered isomorphism f ; s imi l a r ly with D . T h u s by ch.2

g+ i s one of the previously considered i so- f . By ch.2 5 8.2, t he s e t of images g"(C) = ( g + ) " ( C ) i s not inc lu-

§ 6 - SUSLIN C H A I N A N D S U S L I N TREE

Given a chain A :

a subset D of the base f o r which, given any two elements x < y (mod A ) , t he re e x i s t s an element t of D w i t h x st d y (mod A ) The chain of r e a l s , and more genera l ly any chain chain of r e a l s , s a t i s f i e s the two following conditions: (1) there e x i s t s a countable s e t which i s dense in A ; (2) every s e t of mutually d i s j o i n t i n t e rva l s of A , none o f which a re s ing le tons , i s countable. The condition (2) follows from (1) . 0 I f D i s countable and dense i n A , then every non-singleton in te rva l contains a t l e a s t one element of D . Two d i s j o i n t i n t e rva l s cannot contain a same element

A , the reader i s acquainted w i t h the notion of a s e t - in

A which i s embeddable in the

Chapter 5 149

of D , so there are countably many intervals . 0

SUSLIN'S HYPOTHESIS (see SUSLIN 1920) The axiom called Susl in 's hypothesis, asser ts that the preceding condition (2) implies ( l ) , hence t h a t (1)& (2) are equivalent. This axiom i s neither provable nor refutable in ZF, even with the axiom of choice and even with the generalized continuum hypothesis. More precisely JECH and TENNENBAUM have proved the consistency of the existence o f a Suslin t ree ( i . e . the negation of the axiom) with ZF (modulo the consistency of Z F ) . Whereas SOLOVAY and TENNENBAUM have proved the relative consistency of the axiom: see JECH 1978. For a detailed discussion of Suslin chains and Suslin t rees , as well as for the advanced resul ts o f JENSEN, see for example D E V L I N , JOHNSBRiTEN 1974.

6.1. SUSLIN CHAIN I t i s more convenient t o work with the negation of Suslin's hypothesis, rather than the hypothesis i t s e l f . We say that a chain i s a Suslin chain i f i t sa t i s f ies ( 2 ) and n o t ( l ) , i . e . i f every se t of non-singleton mutually dis joint intervals i s countable, yet there ex is t s no countable s e t which i s dense in the chain.

A Suslin chain i s uncountable; moreover i t admits an embedding of t h e c h a i n Q rationals (uses axiom of choice). 0 The inexistence of any countable dense se t implies that the chain i t s e l f be uncountable: i t s cardinality i s a t l eas t W1 (axiom of choice). If Q i s n o t embeddable in i t , then ei ther the ordinal i t : see 3.5. Hence there ex is t uncountably many non-singleton mutually dis joint intervals. 0

of

W 1 or i t s converse i s embeddable in

6 .2 . Every Suslin chain has cardinality exactly w 1 the continuum hypothesis). 0 Let A be a Suslin chain; we already know t h a t A i s uncountably inf in i te , so has cardinality a t l eas t w 1 (axiom of choice). Suppose t h a t A has cardinality a t least w . Replace A by a res t r ic t ion o f cardinality O 2 and l e t B be a well-ordering of type o2 on the same base. Partition the pairs of elements x, y of the base into two colors: l e t t ing x < y (mod A ) , we say t h a t the pair has color (+) i f x < y (mod B ) , and color (-) i f x ) y (mod B ) . By the ERDOS partition lemma (ch.3 5 3.4) for d = 0 (hence using only the continuum hypothesis), there exis ts a subset of the base of cardinality G, , a l l of whose pairs have a same color. Hence there exis ts a s t r i c t l y increasing or a s t r i c t l y decreasing

(uses axiom of choice and

0 l-sequence, and hence wl-many non-singleton mutually dis joint intervals: contradiction. 0

6.3. SUSLIN TREE

We say t h a t a t ree i s a Sus l in t ree i f i t has c a r d i n a l i t y w1 and i f every chain

( o r t o t a l l y ordered r e s t r i c t i o n ) and every an t i cha in i s countable.

The existence o f a Sus l in chain o f c a r d i n a l i t y Sus l in t ree ( the add i t i ona l assumption o f c a r d i n a l i t y using the axiom o f choice: ZF s u f f i c e s ) .

0 Let A be a Sus l i n chain o f c a r d i n a l i t y w1 . To each countable o rd ina l i , associate an i n t e r v a l def ined by i t s two endpoints ui < vi (mod A) , where

a l l ui and vi are d i s t i n c t . To do t h i s , begin w i t h A. = (uo,vo) an a r b i t r a r y i n t e r v a l . Le t i be a non-zero countable o rd ina l , and suppose t h a t the A f o r

j < i contained i n the other. The se t o f endpoints uj, v j ( j < i ) hypothesis i t i s no t dense i n A , hence there e x i s t two elements u, v i n the

base o f A , between which there i s no endpoint u j o r v . ( j < i ) . L e t Ai =

(u,v) so ui = u and vi = v : t h i s i n t e r v a l must be e i t h e r d i s j o i n t o r included i n each A . ( j < i ) .

J The se t o f i n t e r v a l s Ai def ines a t r e e on the se t o f these Ai . Every ant ichain, i . e . every se t o f i n t e r - va ls Ai i n t e r v a l s Ai ordered by the o rd ina l ind ices w i t h A . c Ai

J i, j ( i < j ) . Such a chain i s countable; f o r i f i t had c a r d i n a l i t y w 1 , then

using the endpoints o f preceding i n t e r v a l s , we could ob ta in W l-many mutua l l y d i s j o i n t i n te rva l s . 0

W 1 imp l ies the existence o f a

w1 al lows us t o avoid

Ai

j have already been def ined so t h a t they are mutua l l y e i t h e r d i s j o i n t o r one

i s countable: by

J

thus obtained has c a r d i n a l i t y w 1 . Reverse i nc lus ion

which are mutua l l y d i s j o i n t , i s countable. F i n a l l y , a chain, o r s e t o f

which are mutual ly comparable w i t h respect t o i nc lus ion , i s we l l - f o r every p a i r o f countable o rd ina ls

6.4. The existence o f a Sus l i n t r e e imp l ies , and hence i s equ iva len t w i t h the

existence o f a Sus l i n chain (here we use the r e g u l a r i t y o f W 1 , thus f o r example the countable axiom o f choice).

0 Let A be a Sus l i n t ree ; we can assume t h a t A i s a well-founded p a r t i a l orde-

r i ng , i f necessary by rep lac ing A by a c o f i n a l well-founded p a r t i a l order ing. To see t h i s , apply ch.2 5 5.1 wh i le no t ing tha t , by hypothesis, the base o f A i s

wel l -orderable w i t h order type w 1 . More p rec i se l y l e t ui ( i < be an indexat ion o f the base; then remove each u f o r which there e x i s t s an i< j

j w i th ui > u j (mod A) . To see t h a t what remains i s s t i l l a Sus l i n t ree , note t h a t

f o r each index i , the removed elements u are a l l < ui (mod A) , and hence j

there are countably many such; thus there remain w l-many elements. We can f u r t h e r assume t h a t every non-empty f i n a l i n t e r v a l o f A has c a r d i n a l i t y

W . To see t h i s , l e t x be an element o f the base which has on ly countably many successors (mod A ) . Those x o f minimal he igh t are mutua l l y incomparable,

td 1)

Chapter 5 151

and so there are countably many such: i t suffices t o remove these x and their successors. Neither A nor any f inal interval of A i s f i n i t e l y f ree . For otherwise, by ch.4 5 3.1 (using the regularity of LJ there would ex is t a to ta l ly ordered restr ic- tion of A with cardinality w contradicting our hypotheses. We can assume that A has a minimum, whose singleton will be denoted by Eo . For each countable non-zero ordinal i , l e t Ei be the denumerable se t of elements of height i : we require that th i s s e t be inf in i te . For each element x of E i we require that there ex is t denumerably many elements which are immediate succes- sops of x (mod A ) . This s e t i s denoted must be E i t l . For each countable l imit ordinal i and each x of height < i , we require that there ex is t in f in i te ly many elements in Finally, for each countable l imit ordinal X of A containing elements of a l l heights < i , we require that there exis t either a unique element of none such. These requirements are easy t o sa t i s fy . For example for the minimum, take an arbitrary element having elements x with height i , note that for each x in E i , the final interval > x has cardinality w 1

free subset as E i + l , x and among the successors of x , retain only those which are identical t o or successors of an element of table limit ordinal i and each chain X containing elements of a l l heights 4 i , i f there ex is t elements above X , then decide t o retain one such plus

the wl-many successors of th i s element. For each element x of the base IA l with height i , to ta l ly order the denumerable set E i + l set of a l l maximal to ta l ly ordered res t r ic t ions , or maximal chains of A . This set i s total ly ordered by the preceding dense chains. Indeed, given two dis t inct maximal chains U and V : none of the two bases i s included in the other. More- over there exis ts a leas t element u among those elements of I U 1 which do not belong t o I V I , and a l eas t element v among those elements of I V I which do no t belong t o I U I . By the preceding, there exis ts a l a s t element x whose height will be denoted i , common t o bo th bases of U and V , and having u and v as immediate successors. Let U < V i f u < v (mod C i + l , x ) : th i s total ly orders the s e t of maximal chains. Let H be the chain thus obtained. We shall prove that H i s a Suslin chain. =i-r+ - 1 1 a rnt n irrhirh i c h n c o i n H rannot be countable. For i f i t were,

E i t l , X and the union of these sets

E i which are 7 x (mod A ) . i and each to ta l ly ordered restr ic t ion

E i which i s a successor of a l l elements of X , or

W l-many successors. Then having obtained Ei , the s e t of

and i s not f i n i t e l y f ree . Hence take a denumerable

E i + l , X . Finally, for each coun-

with the order type of a dense chain C i + l , x . Now consider the , X

the i n t e r v a l o f maximal chains passing through z does n o t con ta in any element o f D . Now suppose t h a t there e x i s t o f H . I n each i n t e r v a l take two elements, o r maximal chains U and V . As

before, take an element x whose he igh t i s denoted i and the elements u, v immediate successors o f x (mod A) ; and take w between u and v modulo the chain Ci+l,x . Then these w thus associated w i t h our d i s j o i n t i n t e r v a l s o f H , are mutua l l y incomparable (mod A): they must be countably many; con t rad i c t i on . 0

o l-many non-singleton mutua l l y d i s j o i n t i n t e r v a l s

§ 7 - ARONSZAJN TREE, SPECKER CHAIN

7.1. ARONSZAJN TREE

This i s a well-founded t ree o f c a r d i n a l i t y ~3~ whose chains and he igh t l eve l s are countable. It i s no t requ i red t h a t every an t icha in be countable. Hence every well-founded Sus l i n t r e e i s an Aronszajn t ree; bu t the converse poss ib l y depends on se t - theo re t i c axioms: see the problem a t the end o f 7.4. The fo l l ow ing cons t ruc t ion o f an Aronszajn t ree , using ZF p lus choice, goes back t o KUREPA 1935 p . 96, c i t i n g a l e t t e r from ARONSZAJN i n 1934. The elements o f the t r e e w i l l be o rd ina l sequences o f in tegers ai ( i < ) y

wi thout r e p e t i t i o n , where o( var ies over a l l countable o rd ina l s . We say t h a t such a sequence u precedes v o r t h a t v fo l lows u , i f u i s an i n i t i a l i n t e r v a l o f v . Moreover we requ i re the fo l l ow ing cond i t ions o f convergence and denumerabi l i ty . Convergence. For each sequence ai (i < o() , the sum o f the inverses i s f i n i t e . Furthermore for each sequence u w i t h length o( , each countable o rd ina l

and each p o s i t i v e r e a l number r , there must e x i s t , i n our set , a sequence w i t h length oi + /s , fo l l ow ing u , and such t h a t the sum of the inverses f o r o( i </3 be less than r . Denumerabil ity. For each non-zero countable o rd ina l o( , we take i n our s e t denumerably many o( -sequences (uses axiom o f choice). Using the preceding convergence cond i t ion , we see t h a t , g iven a l i m i t countable o rd ina l 4 we can r e t a i n only o( -sequences u such tha t , f o r each i (4 , the i n i t i a l i n t e r v a l o f u

w i t h l eng th i al ready belongs t o our set; more exac t ly , we r e t a i n denumerably many such o( -sequences. F i n a l l y f o r each sequence u i n our set , every i n i t i a l i n t e r v a l o f u w i l l belong t o our set . These cond i t ions a l l ow us, f o r each o( , t o cons t ruc t a " d e f i n i t i v e " denumerable s e t o f o( -sequences, which w i l l n o t be unduly increased by the u l t e r i o r construc- t i o n o f longer sequences.

Because the non- repet i t ion of values

l /a i

l/ai

ai , every t o t a l l y ordered r e s t r i c t i o n , o r

Chapter 5 153

chain, of the preceding Aronszajn t ree , i s countable. In the preceding t ree , for each countable ordinal o( and each o( -sequence u , there exis t true for a l l Aronszajn t rees; for instance we can add t o the preceding tree new ordinal sequences without successors, or with countably many successors.

u1-many sequences which are successors of u . However th i s i s not

7 .2 . A chain A subset which i s dense in A

i s embeddable in the chain of reals , i f f there exis ts a countable (uses countable axiom of chsice).

0 If A i s a res t r ic t ion o f the chain of rea ls , then for each integer n take a partition o f the reals into intervals of length l / n , and choose an element of I A l in each interval (countable axiom o f choice). Conversely, i f there i s a res t r ic t ion I o f A , which i s denumerable and dense in A , then i f necessary complete I in to J , the l a t t e r which i s isomorphic with the chain of rationals. Then in each cut o f J , we take the unique element o f I A I ( i f i t exists) defined by i t s position with respect t o the elements of I I I . Thus we embed A into the reals. 0

7.3. SPECKER CHAIN This i s an uncountable chain A such t h a t neither w nor i t s converse are embeddable in A ; moreover any chain which i s embeddable both in A and in the chain of reals, i s countable. A Specker chain has no countable dense subset. For i f so , then by the preceding, i t would be embeddable in the chain of reals as well as in i t s e l f , and consequently i t would be countable.

Every Specker chain has cardinality W

0 Suppose on the contrary t h a t there exis ts a Specker chain of cardinality 3 w 2 . By 3.5, this chain admits an embedding ei ther of w 2 , or i t s converse: contradiction with the definition o f Specker chains. Or our chain admits an embedding of an W *-dense chain. By 3.7 ( 2 ) (choice and continuum hypothesis), the l a t t e r admits

an embedding o f w 1 : contradiction. 0

(uses choice plus continuum hypothesis).

7.4. Construct as follows a Specker chain, s tar t ing with the Aronszajn t ree of 7 .1 , hence on a se t of ordinal sequences of integers, without repetition. Apply ch.4 5 6 .2 (using u l t r a f i l t e r axiom) which associates t o the given t ree A , a chain with the same base. For each element u (ordinal sequence of integers without repe- t i t ion) , the final interval 3 u (mod A ) mum u .

C

becomes an interval (mod C ) with mini-

We s h a l l show f i r s t o f a l l t h a t t he cha in C thus def ined does no t admit an

embedding o f u1 , nor o f i t s converse,

By ch.4 5 6.2, each corresponding t o an i n tege r a , and formed from some sequences which begin

w i t h a . Consequently, g iven a s t r i c t l y inc reas ing (mod C ) W1-sequence whose terms are denoted ui ( i countable o rd ina l ) , there e x i s t s an i n tege r al such tha t , from a c e r t a i n o rd ina l on, the sequence ui begins w i t h al . By i t e r a t i o n , f o r every countable o rd ina l k , there e x i s t s a k-sequence o f in tegers al, a*, ... a

a s t r i c t l y inc reas ing wl-sequence o f elements o f the t r e e A , thus an o l-sequence o f in tegers w i thout r e p e t i t i o n : con t rad i c t i on . 0

L e t us now show t h a t if H jA_an uncountable chai_n-_which has a denumerable res- t r i c t i o n D dense i n H ,then H i s no t embeddable i n C . 0 Since D i s denumerable, it i s formed o f ord'nal sequences o f in tegers w i thout r e p e t i t i o n , a l l o f l eng th less than a c e r t a i n countable o rd ina l . L e t o( countable o rd ina l s t r i c t i y g rea ter than a l l these lengths. Since H i s uncounta-

ble, there e x i s t s a t l e a s t one sequence u w i t h length o( having ul-many sequences i n I H I which extend u . By ch.4 4 6.2, there e x i s t s an i n t e r v a l (mod C ) formed o f a l l those elements, o r sequences extending u . Thus any two o f them are n o t separated (mod C) by any sequence belonging t o I Dl : t h i s contra- d i c t s the dens i ty o f D i n H . 0 Problem. I n 7.1 we constructed a well-founded Aronszajn t ree , based on c e r t a i n o rd ina l sequences o f in tegers (w i thout r e p e t i t i o n ) , us ing ZF p lus choice. Whereas a Sus l i n t r e e can on ly be constructed by us ing the negat ion o f Sus l i n ' s hypothe- s i s . Consequently, w i t h ZF p lus choice p lus the axiom c a l l e d Sus l i n ' s hypothesis, we are insured t h a t there e x i s t s an Aronszajn t r e e which i s no t a Sus l i n t ree . However i f we admit ZF p lus choice p lus the negat ion o f Sus l i n ' s hypothesis, then we do no t know whether the Aronszajn t r e e i n 7.1 i s o r no t a Sus l i n t ree . And more genera l l y w i t h these se t - theore t ic axioms, we do n o t know whether there e x i s t Aronszajn t rees which are no t Sus l i n t rees .

C can be decomposed i n a denumerable sum o f d i s j o i n t i n te rva l s ,

( j < k ) such tha t , from a c e r t a i n o rd ina l index on, the sequence ui j' * *

begins w i t h the above k-sequence. F i n a l l y , these al, a2, ... , aj, . . form

be a

Chapter 5 155

5 8 - FAITHFUL INFINITE EXTENSION: MALITZ' AND LOPEZ' COUNTEREXAMPLES

8.1. FAITHFUL INFINITE EXTENSION Let A1, ..., Ah

be a f i n i t e r e l a t i o n w i th I f there e x i s t ex tens ions o f are 3 A1 and ... and 3 Ah , then t h e r e e x i s t s a denumerable ex tens ion o f R which respects t h e same c o n d i t i o n s .

0 We can assume t h a t R i s d e f i n e d on t h e i n t e g e r s 1, ... ,p and t h a t , f o r each

in tege r i , t h e r e e x i s t s an e x t e n s i o n Ri o f R based on t h e i n t e g e r s 1 t o p+i and r e s p e c t i n g t h e c o n d i t i o n s . Fo r i n f i n i t e l y many i n t e g e r s i , t h e Ri

have a same r e s t r i c t i o n S1

many i n t e g e r s i f o r which t h e Ri have a same r e s t r i c t i o n S2 t o 1, ...,p+ 2 . I t e r a t i n g t h i s , we t h u s d e f i n e S f o r each i n t e g e r j . It now s u f f i c e s t o take t h e common e x t e n s i o n o f t h e S

be a f i n i t e s e t o f f i n i t e r e l a t i o n s w i t h common a r i t y , and R

R 2A1 R w i t h a r b i t r a r y l a r g e f i n i t e c a r d i n a l i t i e s , which

and ... and $ A h .

- - -

t o 1,. . . ,p+1 . Among these, t h e r e a r e i n f i n i t e l y

j , based on t h e s e t o f a l l i n t e g e r s . 0

j

8.2. L e t A1, ..., Ah and B1, ..., Bk be two f i n i t e s e t s o f f i n i t e r e l a t i o n s o f common a r i t y , and l e t R s a t i s f y R 3 A A 1 and ... and # A h as w e l l as R > / E l and ... and >/ Bk . Then t h e r e e x i s t s an i n t e g e r u such t h a t every R w i t h c a r d i n a l i t y a t l e a s t equal t o u s a t i s f y i n g t h e p reced ing c o n d i t i o n s has a r e s t r i c t i o n R ' respec- t i n g t h e same c o n d i t i o n s , and such t h a t R ' has a denumerable e x t e n s i o n s t i l l r e s p e c t i n g t h e c o n d i t i o n s .

0 L e t v be t h e sum o f t h e c a r d i n a l i t i e s of t h e r e l a t i o n s B1 through Bk . F o r each R s a t i s f y i n g t h e c o n d i t i o n s , t h e r e e x i s t s a r e s t r i c t i o n R ' o f R

w i t h c a r d i n a l i t y a t most equal t o v , which s a t i s f i e s t h e c o n d i t i o n s . Consider a l l these R ' , which a r e o n l y f i n i t e l y many, up t o isomorphism. F o r each, e i t h e r t h e r e e x i s t s a denumerable e x t e n s i o n s a t i s f y i n g t h e c o n d i t i o n s . O r t h e r e

e x i s t s an i n t e g e r u ( R ' ) which i s s t r i c t l y g r e a t e r t han t h e c a r d i n a l i t i e s o f a l l ex tens ions o f R ' r e s p e c t i n g t h e c o n d i t i o n s . Then i t s u f f i c e s t o s e t u t o be t h e maximum o f t hese u ( R ' ) . 0

___-.

8.3. MALITZ' COUNTEREXAMPLE Can we require that R ' = R ; in other words, does there ex is t an integer u such tha t , i f R has cardinality greater than or equal t o u and s a t i s f i e s the conditions, then there ex is t s a denumerable extension of R A negative answer i s due t o MALITZ 1967. 0 Take the base of integers from 0 t o n-1 . Let I n be the usual chain of these integers; l e t C n be the consecutivity re la t ion ( y = x + l ) ; l e t O n be the unary relation called the singleton of zero, i . e . the relation taking (+) for 0 and ( - ) elsewhere; and l e t U n be the relation singleton of n -1 . Finally l e t R n be the quadrirelation (In,Cn,On,Un) . From n = 7 on , a l l the R have the same restr ic t ions of cardinal i t ies 1, 2 and 3 , up t o isomorphism. Let A1, ..., Ah be those quadrirelations of the same ar i ty and cardinal i t ies 1, 2 , 3 which are n o t embeddable in R n ( n 3 7 ) . We see t h a t every extension of an A 1 , . . . , A h . An analogous b u t rather complicated counterexample i s obtained for binary relations by LOPEZ 1973. 0

satisfying them.

n

R, , and hence in

R n t o a new element added t o i t s base admits an embedding of one of the

8.4. LOPEZ ' COUNTEREXAMPLE Given the f i n i t e relations A1, ..., Ah and B 1 , . . . , B k , one can ask whether there ex is t two integers u , v such tha t , for every R with cardinality greater t h a n or equal t o u , there e x i s t v elements of the base, such that the rest r ic t ion of R t o i t s base with these v elements removed respects the embedding inequa- l i t i e s in the 6 ' s and has an extension of arbi t rary large cardinality respecting the non-embedding inequalities in the A's . Negative answer by LOPEZ 1973. 0 For the base, take the s e t of points, or ordered pairs of integers called the abscissa and ordinate, and which vary from 0 t o n-1 . Let R n be the multirelation on th i s base, which i s composed of the following 4 unary relations and 6 binary relations. The unary relation O n with abscissa 0 . The relation U n takes (+) for the points with abscissa n-1 . Similarly 01; and U,', are defined by interchanging the abscissas and ordinates. The s t r a t i f i e d partial ordering takes the value (+) for each ordered pair of points ( i , x ) , ( j , y ) whose abscissas sa t i s fy i -c j < n , with arbi t rary ordinates x , y ; moreover I n i s reflexive. The equivalence relation E n takes (+) for any two points with a same abscissa and arbi t rary ordinates. The equivalence classes of th i s relation are thus the classes of elements which are pairwise incomparable modulo In . The binary relation Cn , which by abuse of notation we shall cal l a consecutivity, takes the value (t) for each ordered pair of points

takes the value (+) for the points

I n

Chapter 5 157

( i ,x ) , ( i+ l ,y) ordering I,', , the equivalence relation E,', and the consecutivity C,', are obtained from the preceding by interchanging abscissas and ordinates. From n = 7 on , every R n has the same restr ic t ions B1, ..., B k with cardinali- ties 1, 2 , 3 ( u p t o isomorphism). Let A1, ..., Ah be the other multirelations of the same ar i ty and cardinal i t ies 1, 2 , 3 . We see t h a t every proper extension of R n ( n 3 7 ) admits an embedding of a t l eas t one of the A's . Indeed, add a new element t t o the base of R n . Consider the case where e i ther O n or U n or 0,', or U,', takes the value (+) for t , and reduce th i s t o the preceding 5 8.3. Now consider the case where a l l the preceding unary relations take the value ( - ) for t . Then e i ther there exis ts an equivalence class of t belongs: again reduce t o 5 8.3. Or t occurs between two consecutive equivalence classes of extension of R n t h u s obtained admits an embedding of one o f the A's . Now suppose the existence of u and v satisfying our hypothesis; take n > u and > v . Let Sn be a res t r ic t ion of R n in which the B's are embeddable, and which i s obtained by removing v points. Then in each equivalence class of En , there remains a t l eas t one element of I S n l new element t t o the base o f Sn , and attempt to require t h a t the extension of Sn t o i t s base with t added admit only embeddings of the B's and not of the A's . This leads us t o s i tua te t in the chain of the equivalence classes of En . By using Cn , one sees that t necessarily belongs t o one o f the equivalence classes: t cannot be situated between two consecutive classes. Thus we obtain an element in the base of Sn , which i s equivalent with t (mod E n ) , and another element equivalent with t (mod E,',) . From t h i s , we deduce that t is the unique element common t o both equivalence classes. Thus,we have again a restriction of R n obtained by removing v-1 points: th i s i s our extension of Sn . Iterating th i s , we obtain i t s e l f , and a t the following step we obtain a proper extension o f

whose abscissas are consecutive. Finally, the s t ra t i f ied partial

En t o which

En . I n t h i s case, use the consecutivity C n t o see t h a t the

; similarly for E,', . Add a

R n R n , in which necessarily one of the A's i s embeddable. 0

EXERCISE 1 - RESTRICTIONS OBTAINED BY REMOVING ONE OR TWO ELEMENTS 1 - Existence of a relation R with base E and two elements a , b o f E such t h a t R z R / ( E - j a } ) I R / ( E - {b}) yet R > R / ( E - { a , b ) ) . Let R be the binary relation based on the s e t formed with negative integers -1, - 2 , ... and ordered pairs of natural numbers value (+) in only following cases. R ( u - 1 , u ) = + for each negative u ; and R ( - l , ( O , O ) ) = t ; and R ( ( i , j ) , ( i + l , j ) ) = R ( ( i , j ) , ( i , j + l ) ) = + for a l l natural numbers i , j . Show t h a t R sa t i s f ies the stated conditions with a = ( 0 , l ) and b = ( 1 , O ) (communicated by POUZET in 1975). 2 - Similarly with

i , j = 0 , 1, 2 , .. , taking the

R > R / ( E - \ a j ) > R / ( E - {a,b}) yet R L R / ( E - { b ) ) .

L e t R be t h e b i n a r y r e l a t i o n based on t h e s e t o f p o s i t i v e and n e g a t i v e i n t e g e r s , which we denote by O,l,-1,2,-2,.. and n a t u r a l numbers, which a re assumed t o be

d i s t i n c t from t h e i n t e g e r s and denoted by 0 ' ,1' ,2' ,. . . The va lue (+) i s t aken on

o n l y i n f o l l o w i n g cases: R ( i , i + l ) = R ( i , i + 2 ) = + f o r a l l i n t e g e r s i ; and R ( i ,i') = + f o r eve ry i n t e g e r i 3 0 and f o r t h e n a t u r a l number i' w i t h equal va lue; f i n a l l y R ( i ' , ( i + l ) ' ) f o r e v e r y n a t u r a l number i ' and i t s immediate suc-

cessor ( i + l ) ' . Show t h a t R s a t i s f i e s t h e s t a t e d c o n d i t i o n s w i t h a = 0 and b = 0 '

3 - S i m i l a r l y w i t h

Fo r R t ake t h e c o n s e c u t i v i t y on p o s i t i v e and n e g a t i v e i n t e g e r s , w i t h a, b two consecu t i ve i n t e g e r s (communicated by HAGENDORF i n 1977).

(communicated by LOPEZ i n 1977).

R > R / ( E - { a } ) : R/(E - { b ) ) : R/(E - {a,b} ) .

E X E R C I S E 2 - L e t A, B, C be t h r e e cha ins such t h a t C < B and C i n i t i a l i n t e r - Val o f B s a t i s f y i n g A+B 5 A+C ; then A+B 1 A (see HAGENDORF 1977 lemma 1.10). D e f i n i n g D by B = C+D , we have e q u i v a l e n t l y t h a t if A+C+D 3 A+C ,then e i t h e r A+C I A o r C+D 5 C . L e t f he an isomorphism f rom A+B o n t o a r e s t r i c t i o n o f A+C , and l e t Do be t h e non-empty f i n a l i n t e r v a l o f B which i s t h e complement o f C . The image s e t

f " (Do) i s a t t h e l e f t of Do i n A+B . L e t D1 be t h e i n t e r v a l o f those elements which a r e a t l e f t o f y e t g r e a t e r t han o r equal t o some element of I t e r a t i n g t h i s , we o b t a i n D2 and more g e n e r a l l y Di f o r each i n t e g e r i . E i t h e r a l l o f t h e Di a r e i n c l u d e d i n B , i n which case t h e i d e n t i t y on those

elements o f B p reced ing t h e Di , when completed by f , g ives an isomorphism o f B i n t o C , c o n t r a d i c t i n g t h e hypo thes i s . O r a t l e a s t one o f t h e Di h i t s A , hence a l l t h e D . (j > i) a r e i n c l u d e d i n A . I n t h i s case, f i t e r a t e d ( i + l ) t imes i s an isomorphism o f A+B o n t o a r e s t r i c t i o n o f A , hence A+B 2 A . Note t h a t o u r s ta tement no l o n g e r h o l d s i f we remove t h e hypo thes i s t h a t

i n i t i a l i n t e r v a l o f B , even i f C < B . F o r i n s t a n c e take A = B = l+w- and

C = CJ- . O r t a k e A = B = Q+O1+Q and C = W1+Q , where Q i s t h e c h a i n o f t h e r a t i o n a l s . O r aga in A = converse o f W2 and B = Z and C = 0 .

.-

Do f " (Do) .

J

C i s an

E X E R C I S E 3 - IMMEDIATE EXTENSIONS (HAGENDORF 1983, unp.; compare w i t h 5 4.4) . Prove t h a t A = U- + W1 has 3 immediate ex tens ions : 1+A , A+1 and a l s o 0 . W - + which i s n o t o b t a i n e d by s i m p l y add ing an i n t e r v a l t o A .

Prove t h a t B = Q + has 5 immediate ex tens ions : B+1, wl+B , w + B , Q + "; + 0 ; and Q + W - . W 1 , n o t o b t a i n e d by add ing an i n t e r v a l .

159

CHAPTER 6

SCATTERED CHAI 11, NEIGHBORHOOD, INDECOMPOSAB I LITY

§ 1 - SCATTERED CHAIN

A chain i s said t o be scat tered i f the chain Q of rationals i s n o t embeddable in i t . For example, every well-ordering or converse well-ordering, the chain Z of positive and negative integers, the product G J - . ~ or i t s converse. I f a chain has no dense res t r ic t ion , then i t i s scattered. Conversely by ch.5 5 3.1 (using dependent choice), a scattered chain has no dense restr ic t ion.

1.1. Every sum of scattered chains, ordered alona a scattered chain, i s scattered. I n par t icular , the ordinal sum and the ordinal product of two scattered chains i s scattered. 0 Let Ai be the chains in consideration, and A the sum of the Ai along the chain I . If Q i s embeddable in A , then e i ther there ex is t two elements of Q in the same interval Ai , in which case Q i s embeddable in Ai , or there exis ts a t most one element of Q in each Ai , in which case Q i s embeddable in I . 0

1 . 2 . Let A be a chain; i f every i n i t i a l interval of A i s scattered, then A i s scattered. Same statement with "final interval".

1.3. (1) Given a s c a t t e r e d chain A , the chain A . 2 = A+A i s scattered and s t r i c t l y greater than A with respect t o embeddability (uses dependent choice; ZF suffices i f A i s denumerable or with well-orderable base). This follows from the preceding 1.1 and ch.5 5 3.3.

( 2 ) For every countable s e t of denumerable scattered chains, there exis ts a denumerable scattered chain in which a l l are embeddable. Particular case of 1.1.

1.4. Let A be a denumerable chain. If. A i s scattered, then there are only countably many i n i t i a l intervals (uses countable axiom of choice). If A i s not scattered, then there are exactly continuum many i n i t i a l intervals o f A . 0 For the second conclusion, i t suffices t o note tha t Q i s embeddable in A and there are exactly continuum many i n i t i a l intervals of Q . Conversely, suppose tha t A has uncountably many i n i t i a l intervals: we shall prove t h a t Q i s embeddable i n A . For t h i s , we shall prove that there exis ts an element u of the base 1 A I , such t h a t uncountably many i n i t i a l intervals do n o t

conta in u , and uncountably many i n i t i a l i n t e r v a l s do contain u . Suppose the contrary. We say t h a t an element o f the base i s "on the r i g h t " i f there are uncountably many i n i t i a l i n t e r v a l s which do no t con ta in it. Analogously define an element t o be "on the l e f t " . The elements on the l e f t form an i n i t i a l i n t e r v a l

B , and the elements on the r i g h t form a f i n a l i n t e r v a l C . Every element i s

e i t h e r on the l e f t o r r i o h t ; moreover by our assumption o f the non-existence o f u , no element i s both on the l e f t and r i g h t . Thus B and C are complements. A t l e a s t one o f the two, say B , has uncountably many i n i t i a l i n te rva l s . Yet B i s countable: enumerate the elements o f i t s base. Now B has no maximum element, f o r

otherwise i t would be preceded by uncountably many i n i t i a l i n t e r v a l s . Hence there

e x i s t s a s t r i c t l y inc reas ing a-sequence c o f i n a l i n B . P a r t i t i o n B according

t o t h i s sequence. We then have denumerably many i n t e r v a l s , each o f which has coun- t a b l y many i n i t i a l i n te rva l s , and y e t t h e i r union has uncountably many i n i t i a l i n t e r v a l s . This con t rad i c t s the countable axiom o f choice: see ch.1 5 2.5. The existence o f u i s then proved: c a l l i t uo . I t e r a t i n g t h i s , we ob ta in an analogous element

do no t contain u1 and uncountably many ones which conta in u1 bu t no t uo . And

an analogous element which i s isomorphic w i t h Q . 0

ul< uo (mod A) w i t h uncountably many i n i t i a l in te rv ,a ls which

u2 7 uo , and so f o r t h : t h i s y i e l d s a r e s t r i c t i o n o f A

1.5. Given an i n f i n i t e sca t te red chain A , the chain o f i nc lus ion o f the i n i t i a l i n t e r v a l s o f A wel l -orderable).

0 L e t B be the chain o f the i n i t i a l i n t e r v a l s o f A , and suppose t h a t the chain Q o f r a t i o n a l s i s embeddable i n B . We sha l l prove f i r s t t h a t there

e x i s t s an element u o f the base I A I , f o r which the chain o f those i n i t i a l i n t e r v a l s conta in ing u , and the chain o f those i n i t i a l i n t e r v a l s no t con ta in ing

u , both admit an embedding o f Q . It s u f f i c e s t o s t a r t w i t h a r e s t r i c t i o n o f

B which i s isomorphic w i t h Q , and t o p a r t i t i o n i t i n t o a sum o f three terms, each isomorphic w i t h Q , then t o take two elements, i .e . two i n i t i a l i n t e r v a l s o f A , i n the middle term; and f i n a l l y t o take an element u i n the d i f f e rence

o f these two i n i t i a l i n t e r v a l s o f A Having proved the existence o f u , c a l l i t uo . I t e r a t i n g t h i s , we ob ta in an analogous element ul< uo (mod A) and u 2 7 uo (mod A) , and so f o r t h (here we use dependent choice; ZF s u f f i c i e n t i f A i s countable o r has wel l -orderable base). This y i e l d s a r e s t r i c t i o n o f A isomorphic w i t h Q . 0

i s sca t te red (uses dependent choice; ZF su f f i ces i f the base i s

1.6. Le t A be an i n f i n i t e sca t te red chain. Then the se t o f i n i t i a l i n t e r v a l s

of A i s equipotent w i t h A (uses axiom o f choice).

0 L e t the aleph be the card ina l of the base I A I , and suppose t h a t the

Chapter 6 161

s e t o f i n i t i a l i n t e r v a l s o f A has c a r d i n a l i t y >/ W o( +1 . By t h e p reced ing pro- p o s i t i o n 1.5, t h e c h a i n Q o f r a t i o n a l s i s n o t embeddable n t h e c h a i n B o f

these i n i t i a l i n t e r v a l s . By ch.5 § 3.5, and s i n c e l i k e eve ry successor

a leph, i s r e g u l a r (ch.2 5 6.2 u s i n g axiom o f cho ice ) , t h e c a i n B admi ts an embedding o f e i t h e r t h e o r d i n a l 0 o(+l

i s embeddable i n B , hence t h a t t h e r e e x i s t s a s t r i c t l y i n c r e a s i n g o r i t s converse. To f i x ideas, suppose t h a t

w &+l-sequence o f i n i t i a l i n t e r v a l s Hi (i o f A . Take an element

w d+l-sequence o f e lements i n t h e base of A , c o n t r a d i c t i n g t h e hypo thes i s

i n each d i f f e r e n c e o f success ive Hi . We then o b t a i n a s t r i c t l y i n c r e a s i n g

t h a t Card A = fd, . 0

5 2 - DECOMPOSITION OF A CHAIN INTO A SUM OF SCATTERED CHAINS: NEIGHBORHOOD; STRONGLY SCATTERED CHAIN, NEIGHBORHOOD RANK

2.1. Every non -sca t te red c h a i n can be u n i q u e l y decomposed i n t o a dense sum o f s c a t t e r e d cha ins (HAUSDORFF 1914; uniqueness uses t h e axiom o f dependent cho ice ) .

0 L e t two elements x, y o f t h e base be e q u i v a l e n t i f t h e i n t e r v a l (x,y) i s

s c a t t e r e d . T h i s c o n d i t i o n y i e l d s equ iva lence c lasses which a re themselves s c a t t e -

r e d i n t e r v a l s . No two o f these i n t e r v a l s can e v e r be consecut ive; t hus they c o n s t i t u t e a dense chain. I f t h e r e were two d i s t i n c t decomposi t ions o f a c h a i n i n t o a dense sum o f s c a t t e r e d

chains, t hen one o f these s c a t t e r e d chains would admi t an embedding o f a dense sum, hence an embedding o f Q : see ch.5 5 3.1, u s i n g dependent cho ice . 0

2.2. NEIGHBORHOOD

Two e lements x, y o f t h e base o f a cha in A a r e s a i d t o be 0-neighbors (mod A) i f f t h e i n t e r v a l w i t h endpo in ts x and y i s f i n i t e . The equ iva lence c lasses of t h i s r e l a t i o n a r e c a l l e d 0-neiqhborhoods . These a r e i n t e r v a l s which a r e e i t h e r f i n i t e , o r i somorph ic w i t h w o r i t s converse, o r isomorphic w i t h Z . L e t o( be an o r d i n a l . I f o( i s a successor, t h e n x and y a r e s a i d t o be

x and y . I f o( i s a l i m i t o r d i n a l , t hen x and y a r e s a i d t o be o( -neigh-

bo rs i f t h e r e e x i s t s an o r d i n a l f i < d f o r which t h e y a r e /3 -ne ighbors .

The equ iva lence c lasses f o r t h i s equ iva lence r e l a t i o n , a r e i n t e r v a l s and a r e c a l l e d o( -neighborhoods.

o( -ne ighbors i f f t h e r e e x i s t o n l y f i n i t e l y many ( W -1)-neighborhoods between

2.3. STRONGLY SCATTERED CHAIN

We say t h a t a c h a i n A i s s t r o n g l y s c a t t e r e d i f t h e r e e x i s t s no dense s e t o f

non-empty mutually dis joint intervals of A . Here, i f I and J are two dis joint intervals , then we write I < J i f every element of I i s less t h a n (mod A ) every element of J ; t h i s yields the notion of a dense chain of intervals . I f A i s strongly scattered, then no dense chain i s embeddable in A . Hence every strongly scattered chain i s scattered. Every scattered chain i s strongly scat tered, then bo th notions are equivalent, modulo the axiom of dependent choice (ZF suffices for a denumerable chain).

0 Suppose tha t A d is joint intervals. By ch.5 5 3.1 (using dependent choice), there ex is t s a subset formed of intervals which constitutes a chain isomorphic with Q . Then take an element i n each o f these intervals (countable axiom of choice): t h i s yields a res- t r ic t ion of A isomorphic with Q . 0

i s not strongly scattered, and take a dense s e t of mutually

2 .4 . NEIGHBORHOOD R A N K Given a strongly scattered chain, there exis ts an ordinal ordinal on, a l l elements of the base are <-neighbors. Moreover for a l l i 4 j

( j < 4 ) , there ex is t j-neighbors which are n o t i-neighbors. 0 The i-neighborhoods form a non-dense chain; hence there ex is t two consecutive i-neighborhoods. The ordinal LX thus defined i s called the neiqhborhood rank of the chain. I n particular th i s rank i s 0 i f f the chain i s f i n i t e or isomorphic with ei ther w or i t s converse or Z . If A i s a strongly scattered chain and i f B i s embeddable in A , then B js- strongly scattered and the neighborhood rank of B is-at __ most e q u a l t o _____g the neighborhood rank of A . If B i s equimorphic with A , then the neighborhood ranks are equal. The converse of A has the same neighborhood rank as A . 2.5. Let A be a strongly scattered chain having a minimum element, and l e t o( be the neighborhood rank of A . (1) If c% = 0 , then A i s f i n i t e or isomorphic with W .

If o( 3 1 , then e i ther o( i s a successor ordinal and A i s a f i n i t e sum or the sum of an o -sequence o f chains w i t h neighborhood ranks Or cd i s a l imit ordinal; l e t t ing $ = Cof o( , then A i s the sum of a f-x- quence of chains with neighborhood ranks < o( . ( 2 ) If 4 = 0 , then every proper i n i t i a l interval o f A i s f i n i t e ; otherwise, every proper i n i t i a l interval i s a f i n i t e sum of chains w i t h neighborhood ranks

( 3 ) Tf. Cof o( i s in f in i te , then Cof A = Cofw ( t h i s conclusion ( 3 ) uses axiom of choice) . We have the analogous statement i f A has a maximum element: replace i n i t i a l

o( such that from t h a t

.- < o( .

- --- -- -_ - ___ - - - - __ _._ .

< d -

Chapter 6 163

i n t e r v a l by " f i n a l i n t e r v a l o f A " and c o f i n a l i t y by " c o - i n i t i a l i t y o f A " . ( 1 ) L e t a be t h e minimum element o f A . I f IX i s a successor o r d i n a l , then

A i s a f i n i t e sum o r t h e sum o f an cd-sequence o f ( & -1)-neighborhoods. If o( i s a l i m i t o r d i n a l , l e t (3 ( i < = Cof a( ) be a ?f-sequence o f s t r i c t l y i n c r e a s i n g o r d i n a l s whose supremum i s OC . The base

o f t h e d i f f e r e n c e i n t e r v a l s

( 2 ) Immediate consequence o f ( 1 ) . ( 3 ) Us ing t h e axiom o f choice, t ake an element i n each d i f f e r e n c e

1 A l i s t hen t h e un ion o f t h e fSi-neighborhoods o f a . C a l l Ai these neighborhoods: t h e c h a i n A i s t h e sum

Ai+l-Ai .

Ai+l-Ai . 2.6. Every s t r o n g l y s c a t t e r e d c h a i n i s e i t h e r t h e sum o f two s t r i c t l y l e s s e r chains

( w i t h r e s p e c t t o embeddab i l i t y ) ; or t h e sum o f s t r i c t l y l e s s e r cha ins a long an o r d i n a l t h e converse o f a r e g u l a r a leph .

E i t h e r A i s t h e sum o f two s t r i c t l y l e s s e r cha ins . O r , t a k i n g an element a i n

t h e base, A i s equimorphic w i t h t h e f i n a l i n t e r v a l w i t h minimum a , o r w i t h t h e

i n i t i a l i n t e r v a l w i t h maximum a . Now a p p l y t h e p reced ing p r o p o s i t i o n , and no te t h a t , i n t h e case o f a f i n i t e sum o f i n t e r v a l s w i t h neighborhood ranks s t r i c t l y

l e s s than t h a t o f A , we can s t i l l decompose A i n t o two i n t e r v a l s , n e i t h e r o f which admi ts an embedding o f A . 0

We s h a l l g i v e ano the r p r o o f , which does n o t use neighborhood ranks and assumes t h a t A i s s c a t t e r e d , y e t n o t n e c e s s a r i l y s t r o n g l y s c a t t e r e d . T h i s p r o o f uses t h e axiom o f choice, which however i s t o o much: i ndeed dependent cho ice s u f f i c e s t o i d e n t i f y

s c a t t e r e d w i t h s t r o n g l y s c a t t e r e d cha ins . 0 Suppose t h a t A i s n o t decomposable i n t o a sum o f two s t r i c t l y l e s s e r chains. We then say t h a t a c u t i n A i s a l e f t c u t i f A i s embeddable above t h a t c u t , and a r i g h t c u t i f A i s embeddable b e f o r e it. Every c u t i s e i t h e r l e f t o r r i g h t . No c u t i s b o t h a l e f t and r i g h t c u t , f o r o t h e r w i s e A+A would be embeddable i n A

and thus t h e c h a i n Q o f r a t i o n a l s would be embeddable i n A : see ch.5 5 3.3. Every l e f t c u t i s s i t u a t e d b e f o r e eve ry r i g h t c u t . There e x i s t s e i t h e r a maximum l e f t c u t o r a minimum r i g h t cu t : see Dedekind ch.4 5 1.1. L e t A = B+C be t h e decomposi t ion o f A d e f i n e d by t h i s c u t . F i x i deas by assuming A 6 B . Then

A = B and C i s empty; f o r o the rw ise , embedding B+C i n B , we would o b t a i n a r i g h t c u t be fo re o u r Dedekind c u t : c o n t r a d i c t i o n . F i n a l l y eve ry c u t o t h e r than i s a l e f t c u t . Take a s t r i c t l y i n - c r e a s i n g o r d i n a l sequence which i s c o f i n a l i n A and has minimum l e n g t h equal t o

Cof A (axiom o f choice, ch.2 5 5.4) . Then A becomes t h e sum o f an o r d i n a l sequence o f i n t e r v a l s , each o f which i s s t r i c t l y l e s s

which i s a r e g u l a r aleph, o r t h e sum o f s t r i c t l y l e s s e r cha ins a long

( A , empty s e t )

t han A . 0

2.7. INDUCTION SCHEME FOR STRONGLY SCATTERED CHAINS (HAUSDDRFF 1914)

Suppose t h a t a cond i t ion chain. I f , supposing t h a t holds f o r each chain Ai ( i < 8 = an i n tege r o r an

i n f i n i t e regu la r aleph), then f holds f o r the sum o f the the converse o f t h i s o rd ina l . Then Consequence o f 2.5, by induc t ion on the neighborhood rank.

Ip holds f o r the empty chain and f o r every s ing le ton

Ai a long 6' and along

holds f o r every s t rong ly sca t te red chain.

5 3 - INDECOMPOSABLE CHAIN, RIGHT OR LEFT INDECOMPOSABLE CHAIN

We say t h a t a chain A i s indecomposable i ff f o r every decomposition A = A ' + A "

where A ' i s an i n i t i a l i n t e r v a l and A" a f i n a l i n t e r v a l , A i s embeddable e i t h e r i n A ' o r A " ; so t h a t A i s equimorphic w i t h e i t h e r A ' o r A" . A chain i s , ,sa id t o be decom osable o thep i i se . Some authors say " a d d i t i v e l y in&- composable or "additi&sable . 3.1. RIGHT OR LEFT INDECOMPOSABLE CHAIN, STRICTLY OR NOT

We say t h a t A i s r i g h t indecomposable i f A i s embeddable i n every non-empty f i n a l i n t e r v a l . We say t h a t A i s s t r i c t l y r i g h t indecomposable i f , f o r every

decomposition of A i n t o an i n i t i a l and a non-empty f i n a l i n t e r v a l , A i s embed-

dable i n the f i n a l i n t e r v a l bu t no t i n the i n i t i a l i n t e r v a l . Analogously, we def ine l e f t indecomposable and s t r i c t l y l e f t indecomposaB chain.

Every r i g h t indecomposable o r l e f t indecomposable chain i s indecomposable. However, s t a r t i n g w i t h the chain Q o f ra t i ona ls , the sum 1 + Q t 1 i s indecomposable y e t ne i the r r i g h t nor l e f t indecomposable. There ex i s t s , though, an i n t e r -

val of t h i s sum, namely Q , which i s equimorphic w i t h the e n t i r e chain and which i s both r i g h t and l e f t indecomposable: see 3.3 below.

( 1 ) I f A i s an i n f i n i t e chain, both r i g h t and l e f t indecomposable, then A & equimorphic t o

( 2 ) I f A i s i n f i n i t e and equimorphic w i t h A.2 , then the chain Q o f the r a t i o n a l s i s embeddable i n A (uses dependent choice, ZF s u f f i c e s i f A i s denumerable). This i s another form o f ch.5 5 3.3. (3) Every non-empty, scattered, r i g h t indecomposable i s s t r i c t l y ____-- r i g h t inde- -- composable; s i m i l a r l y w i t h " l e f t " (uses dependent choice; consequence o f (1), ( 2 ) ) . However Q. 0 i s s t r i c t l y r i g h t indecomposable y e t no t scattered.

3.2. THEOREM OF THE MAXIMUM RIGHT INDECOMPOSABLE INITIAL INTERVAL (1) I f A i s a s t r i c t l y r i g h t indecomposable chain, then every chain which

i s equimorphic w i t h A i s s t r i c t l y r i g h t indecomposable; s i m i l a r l y w i t h " l e f t " . Le t B be equimorphic w i t h A . Decompose B = B'tB" w i t h 6" non-empty. Then

A = A ' t A " w i t h A',< B' and A " 6 B" and A " non-empty, f o r otherwise A would

be embeddable i n a proper i n i t i a l i n t e r v a l . Then B 6 A,< At',< B" and B 4 B'.

A.2 = A+A .

Chapter 6 165

(2 ) Given a chain A which i s n o t r i g h t indecomposable, there e x i s t s a decomposi- t i o n A = A'+A" w i t h A" non-empty - and A ' * A" (axiom o f choice). 0 Set H = Cof A ; s t a r t w i t h a decomposition A = z Ai (i < 4) and note tha t , i f

our conclusion i s fa lse , then each i n t e r v a l o f A . 0

- -

Ai i s non-cof inal ly embeddable i n any f i n a l

(3 ) Every chain sable i n i t i a l i n t e r v a l I , poss ib ly empty. Moreover i f I # 0 and A = I+B ,

A admits a maximum (w i th respect t o i nc lus ion ) r i g h t indecompo-

then B i s the maximum f i n a l i n t e r v a l s t r i c t l y less than A . BY DEDEKIND (ch.4 tj 1.11, there e x i s t s a maximum cu t ( 1 8 ) of A such t h a t A

i s embeddable i n each f i n a l i n t e r v a l which i s s t r k t l y l a r g e r than

then necessar i l y I+B i n t o X+B would map I i n t o X . F i n a l l y l e t J be a f i n a l i n t e r v a l o f I : i f J < I and J non-empty, then J+B 5 A and J+B < A : con t rad i c t i on proving t h a t I i s r i g h t indecomposable. Moreover I i s maximum since B < A . 0

B . I f I # 0 I

B < A . L e t X < I : then X + B C A , since an embedding from

3.3. UNIQUE DECOMPOSITION THEOREM FOR AN INDECOMPOSABLE CHAIN (HAGENDORF 1977) L e t A be an indecomposable chain. Then e i t h e r A i s s t r i c t l y right,indecompo-- sable, o r s t r i c t l y l e f t indecomposable, o r A i s uniquely decomposable i n t o

a sum A = B+H+C where B < A and C < ' A and H i s equimorphic w i t h A and both r i g h t and l e f t indecomposable. These th ree cases are mutua l l y exclusive; i n the t h i r d case only, we have A equimorphic w i t h A.2 . 0 We say t h a t a c u t i n A i s a r i g h t c u t if A i s embeddable before i t but no t a f te r ; l e f t i f A i s embeddable a f t e r i t but no t before; b i l a t e r a l i f A i s e w beddable both before and a f t e r . Suppose f i r s t t h a t there on ly e x i s t r i g h t and l e f t cuts. By DEDEKIND (ch.4 5 l . l) , there e x i s t s a maximum l e f t cu t o r a minimum r i g h t cut . I n the f i r s t case, A i s embeddable i n a f i n a l i n t e r v a l which has on ly r i g h t cuts, so A i s s t r i c t l y l e f t indecomposable. Analogous conclusion i n the second case. Now consider the case where there e x i s t b i l a t e r a l cuts. Then each b i l a t e r a l c u t has o ther b i l a t e r a l cu ts both before and a f t e r it. Hence there can be ne i the r a minimum nor a maximum b i l a t e r a l cu t . Thus there e x i s t s a maximum l e f t c u t ( o r no l e f t cu t )

and a minimum r i g h t c u t ( o r none o f them). So t h a t A i s decomposable i n t o a sum B + H + C , w i t h B < A and C C A and H equimorphic w i t h A ; the i n t e r v a l s B and C may be empty. Every c u t i n H i s b i l a t e r a l (mod A), so i s again b i l a -

t e r a l (mod H) . Hence H i s both r i g h t and l e f t indecomposable. F i n a l l y the ex is - tence of b i l a t e r a l cuts gives A equimorphic w i t h A.2 . The uniqueness of the decomposition fo l lows from the f a c t t h a t a lengthening o f B t o the r i g h t would g ive B a A ; s i m i l a r l y f o r a lengthening o f C t o the l e f t ; a lengthening o f H t o the l e f t o r r i g h t would destroy i t s b i l a t e r a l indecomposa- b i l i t y . 0

Coro l l a r i es . (1) For every indecomposable chain A , there e x i s t s an i n t e r v a l which i s l e f t o r r i g h t indecomposable ( o r both), and equimorphic w i t h A .

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( 2 ) If A i s i n f i n i t e and equimorphic w i t h A.2 , then A i s equimorphic w i t h a chain both r i g h t and l e f t indecomposable. (3) Every non-empty, sca t te red and indecomposable- chain i s s t r i c t l y r i g h t o r s t r i c t l y l e f t indecomposable: re inforcement o f 3.1.(3), uses dep. choice (JULLIEN 1969).

3.4. A chain A i s indecomposable i f f , f o r every X < 0 1 1 A ,w i th respect t o embeddabil ity, we have X.2 < Lr I A ; HAGENDORF 1977 proves the s t ronger r e s u l t :

i f A # 2 i t su f f i ces t h a t X < or 1A y i e l d s X.2 6 o r ) A ( t h . 1 .12) .

0 Suppose t h a t A i s indecomposable and t h a t X.2 > / A : then X>, A . Conversely, suppose t h a t A i s decomposable, so A = B + C w i t h B < A and C C A , y e t A s a t i s f i e s our conclusion. Then 8.2 3 A = B+C so C & B ; s i m i l a r - l y B $ C . Moreover (C+B) .2 = C+B+C+B 2 A so C+B >, A = B+C and thus B ,< C

o r C & B : cont rad ic t ion . 0 Notice t h a t the chain Z o f the in tegers i s decomposable, y e t v e r i f i e s the condi- t i o n t h a t every X < Z y i e l d s X.2 < o r L 2 . The p ropos i t i on i n ch.5 5 4 about o rd ina ls , does no t extend t o chains. Indeed A = w - . i s indecomposable and w < A bu t W.2 I A , and no t c A . I f a chain A i s indecomposable, then every i n i t i a l i n t e r v a l X < A s a t i s f i e s X.2 < A . However, the re t ro -o rd ina l i n t e r v a l X < A i s isomorphic w i t h 0- hence X.2 < A . S i m i l a r l y A = Q+ cJ1

i s decomposable and y e t v e r i f i e s our cond i t ion . S i m i l a r l y A = ( w + 1 ) - . - Problem. Le t A be a chain. I f every chain X < A y i e l d s X.2<A , then i s A an indecomposable chain. A f f i rma t i ve response f o r sca t te red A : HAGENDORF i n 1982, unpublished; see proo f i n ch.8 5 4.6.

2 ( 0 + W ) - i s decomposable and every i n i t i a l

3.5. (1) I f A < A.2 - and A+B> B then A.2 t B > A t B . 0 Suppose A.2 + B 6 A+B ; then A.3 + B S A.2 + B,(A+B . Thus e i t h e r A.Z,(A or AtBG B ; cont rad ic t ion . 0

(2) Le t A be a chain. I f every chain X < A s a t i s f i e s X + A S A , then A - i s indecomposable. Moreover, there e x i s t s a r i g h t indecomposable chain equimorphic w i t h A (HAGENDORF 1976). The converse i s fa lse : the chain A = Z. (J

b u t W2tA 3 A . 0 We can suppose t h a t A i s i n f i n i t e . It su f f i ces t o prove t h a t A i s indecomposable; the r e s t o f the conclusion fo l lows from 3.3, s ince the case o f A s t r i c t l y l e f t indecomposable i s excluded: indeed 1 < A so 1+A< A hence A i s embedda- b l e i n one o f i t s proper f i n a l i n te rva l s . We argue ad absurdum, supposing t h a t A = B+C w i t h B < A and C < A . Then by hypothesis BtAG A so 8.2 + C 6 B+C = A > C . By the preceding (l), we have 8.2 equimorphic w i t h B , so B i s indecomposable.

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2 i s r i g h t indecomposable w i t h w < A

Chapter 6 167

Furthermore C < A imp l ies by hypothesis t h a t C+A4 A so C+B+Cd B+C . Le t us embed C+B+C i n B+C . Then e i t h e r C+B i s embedded i n B . O r , s ince B+C i s not embeddable i n C , the on ly remaining p o s s i b i l i t y i s t h a t C admits an embedding o f C i t s e l f preceded by a f i n a l i n t e r v a l o f B which i s < B . B u t B i s indecomposable, and so i s embeddable i n the complementary i n i t i a l i n t e r v a l . Hence i n a l l cases we have C+B* B and so A = B+C&B+C+B,( B . 2 4 B : contra- d i c t i o n prov ing t h a t A i s indecomposable. 0 (3) Le t A be a denumerable chain. I f every X < A which i s an i n i t i a l i n t e r v a l o f A s a t i s f i e s X+A,(A , then A i s equimorphic w i t h Q (and indecomposable). 0 Repeat the preceding proo f t o ob ta in B.2 equimorphic w i t h B . Moreover B i s not empty s ince i t s complement C < A : so B i s necessar i l y denumerable. Then apply 3.1.(2) (denumerable case where ZF su f f i ces ) . 0

This p ropos i t i on no longer holds i n the uncountable case: take A = Q + q . 3.6. Le t A be an i n f i n i t e r i g h t indecomposable chain, B an i n f i n i t e l e f t inde-- composable chain, and C an a r b i t r a r y chain. I f C > A and C 3 B , then e i t h e r C 2 A+B or C >/ B+A (JULLIEN 1969).

0 L e t f be an isomorphism o f A onto a r e s t r i c t i o n o f C , and l e t CA be the i n i t i a l i n t e r v a l o f C formed by those elements less than (mod C) o r equal t o the images f x as x runs through I A I . Then A i s embeddable i n every non-empty f i n a l i n t e r v a l o f CA . S i m i l a r l y l e t CB be a f i n a l i n t e r v a l o f C such t h a t B i s embeddable i n every non-empty i n i t i a l i n t e r v a l o f CB . Then e i t h e r CA and CB are d i s j o i n t , and hence C b A + B . O r there e x i s t s an element u common t o the bases o f CA and CB . Then B i s embeddable before u i n the i n te rsec t i on CAn CB , and A i s embeddable a f t e r u : hence C>,B+A .

-

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3.7. The chain w + W , which i s embeddable ne i the r i n A = W-. W nor i n B = W . , i s however embeddable i n every common extension o f A a d B (JULLIEN 1969). 0 The chain A i s r i g h t indecomposable and B l e f t indecomposable. Thus i f a chain X 2 A and B , then e i t h e r X 3 A+B >/ k) + c3- o r X >/ B+A >/ &J + W-.O

3.8. Problem posed by HAGENDORF 1977.. Le t A be a s t r i c t l y r i g h t indecomposable chain. I f f o r every chain X < A , we have X + l d A , then i s A i t s e l f a we l l - order ing. A p o s i t i v e response f o r sca t te red chains i s due t o LARSON 1978.

3.9. L e t A be a s t rong ly sca t te red r i g h t indecomposable chain, and o( i t s neighborhood rank. If OC. = 0 , then every proper i n i t i a l i n t e r v a l o f A i s f i n i t e . I f o( >, 1 , then every proper i n i t i a l i n t e r v a l i s a f i n i t e sum o f chains w i t h neighborhood ranks s t r i c t l y less than g .


Consequently we find again 3.1.(3) : A i s s t r i c t l y right indecomposable. Moreover in the case where o( & 1 : i f a proper i n i t i a l interval of A composable, then i t has rank s t r i c t l y less than- cx . 0 I f A has a minimum, then we are in the case of 2.5. Otherwise, take an element U of the base. Let B be the i n i t i a l interval of elements s t r i c t l y less than u (mod A ) and C the final interval beginning with u . By hypothesis A i s equimorphic with C , so the neighborhood rank of C i s o( . The i n i t i a l interval B i s embeddable in a proper i n i t i a l interval of C , hence by 2 .5 i f o( = 0 then B i s f i n i t e , and i f o( >, 1 then B i s embeddable in a f i n i t e sum of chains with neighborhood ranks s t r i c t l y less t h a n o( . Thus B i t s e l f i s such a f i n i t e sum. 0

i s inde-

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5 4 - U N I O N A N D INTERSECTION OF INDECOMPOSABLE CHAINS, COVERING BY INDECOMPOSABLE CHAINS OR BY DOUBLETS

4.1. Let A be a chain which i s the u_nion of an initicallngrval and a final interval, both having a t l eas t one common element and bo th of which are right indecomposable. Then A i s right indecomposable. Same statement for " l e f t " .

0 Let B be the i n i t i a l interval , C the final interval and 0 the i r intersection. Then B i s embeddable in D . Either 0 = C so that A = B i s right indecomposable. Or C has the form O+E and so A = B+E i s embeddable in C = D+E , hence A i s again right indecomposable. 0

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4.2. Consider a chain which i s the union of a r igh t indecomposable i n i t i a l interval- B and a l e f t indecomposable final interval C , b o t h in f in i te and having a t l eas t one common element. Then the intersection BnC i s both l e f t and r ight indecomposable and admits an embedding of the chain Q of rationals (uses dependent choice; th i s i s a strengthening of 3 .1 . (1) and ( 2 ) ) . .-

4.3. C O V E R I N G BY RIGHT OR LEFT INDECOMPOSABLE CHAINS Let A be a chain. We say that two elements u , v of the base are equivalent with respect t o right indecomposable chains i f there exis ts an interval o f A which i s r ight indecomposable and contains the elements u and v . The condition thus defined i s reflexive and symmetric. Moreover by 4.1 i t i s t ransi t ive. Analogously we define the equivalence relation with respect t o l e f t indecomposable chains; we cal l these covering by r ight or l e f t indecomposable chains. There can be inf ini te ly many equivalence classes of th i s covering relation. For instance, take the converse 0- of w : the equivalence classes for covering by right indecomposable chains are singletons. An equivalence class for covering by r ight indecomposable chains i s not necessa- r i l y a right indecomposable chain: take the chain Z of the rational integers.

Chapter 6 169

4.4. DOUBLET OF INDECOMPOSABLE CHAINS A doub le t i s a c h a i n which i s t h e un ion o f a l e f t indecomposable i n t e r v a l and a r i g h t indecomposable i n t e r v a l , b o t h hav ing a t l e a s t one common element.

For example, t h e c h a i n Z o f t h e i n t e g e r s i s a doub le t , be ing t h e un ion o f t h e

f i n a l i n t e r v a l w and t h e i n i t i a l i n t e r v a l d d - , which can be choosen t o have one

o r seve ra l common e lements. On t h e o t h e r hand, a l t hough t h e p roduc t & - . W i s

r i g h t indecomposable and i t s converse W . W - i s l e f t indecomposable, t h e sum W . Lc)- t Z t Csr-. W

The l a t t e r example i s as w e l l a c h a i n which i s t h e un ion o f W - . W and i t s con-

verse: i f we decompose Z i n t o CJ- and o and then a t t a c h w- t o &-. W , we have a sum i somorph ic w i t h W - . i3 , and s i m i l a r l y f o r t h e converse chains;

y e t t h e cons ide red indecomposable chains a r e no l o n g e r i n t e r v a l s o f t h e f i n a l cons t ruc ted sum.

Every r i g h t o r l e f t indecomposable c h a i n i s a p a r t i c u l a r k i n d o f doub le t , i n which

one o f t h e i n t e r v a l s reduces t o a s i n g l e t o n . Note t h a t i t i s n o t r e q u i r e d t h a t o u r indecomoosable cha ins be i n i t i a l o r f i n a l i n t e r v a l s : one o f them may be a m idd le i n t e r v a l .

I n t h e case o f a s c a t t e r e d c h a i n which i s a doub le t , e i t h e r one o f t h e indecompo-

sable cha ins i s a r e s t r i c t i o n o f t h e o the r , o r t h e l e f t indecomposable c h a i n i s an i n i t i a l i n t e r v a l and t h e r i g h t indecomposable c h a i n i s a f i n a l i n t e r v a l o f t h e doublet . T h i s f o l l o w s f rom 4.2 (uses dependent c h o i c e ) .

On t h e o t h e r hand 1tQt1 , where Q i s t h e cha in o f r a t i o n a l s , i s a non -sca t te red doub le t hav ing a decomposi t ion i n t o 1tQ , a r i g h t indecomposable i n i t i a l i n t e r v a l , and Qt1 , a l e f t indecomposable f i n a l i n t e r v a l .

c3. (.d - t W-. bJ i s n o t a doub le t ; n o r even i s t h e sum

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4.5. EQUIVALENCE RELATION FOR COVERING BY DOUBLETS

Consider a s c a t t e r e d c h a i n and l e t u, v be two elements i n t h e base. We say t h a t u and v a r e e q u i v a l e n t w i t h r e s p e c t t o doub le ts i f t h e r e e x i s t s a d o u b l e t o f indecomposable cha ins c o v e r i n g u and v . T h i s c o n d i t i o n i s r e f l e x i v e and sym-

m e t r i c . We s h a l l show t h a t i t i s t r a n s i t i v e (uses dependent choice) ; t h e conside r e d r e l a t i o n w i l l be c a l l e d c o v e r i n q by doub le ts .

0 Take t h r e e elements u < v < w (modulo t h e cha in ) . I f u and v , on t h e one hand, v and w on t h e o t h e r hand, a r e covered by two r i g h t o r two l e f t inde-

composable chains, t h e n app ly 4.1. I f u, v a r e covered by a r i g h t indecomposable chain, n e c e s s a r i l y i n f i n i t e and

hav ing no maximum, and i f v, w a r e covered by a l e f t indecomposable chain, i n f i n i t e and hav ing no minimum, then e i t h e r one o f these indecomposable cha ins covers u and w , o r t h e i r i n t e r s e c t i o n i s i n f i n i t e and b o t h r i g h t and l e f t

indecomposable, hence admi ts an embedding o f Q (see 4.2, dependent cho ice ) : c o n t r a d i c t i o n .


I f U, v are covered by a doublet and v , w by a r ight indecomposable chain, then by 4.1 we are in the case of a doublet covering u and w . If u, v are covered by a doublet and i f the second term of the doublet i s in f in i te , then we obtain a contradiction again using 4.2. If the second term reduces to a singleton, then by 4 .1 we have a l e f t indecomposable chain covering u and w . Now assume that u, v are covered by a doublet formed of A , a l e f t indecomposable, and B , a right indecomposable chain; and similarly v , w are covered by a doublet formed of C , a l e f t indecomposable, and D , a r ight indecomposable chain. Then e i ther the cut situated t o the r ight of B f a l l s in or a f te r 0 , and so by 4.1 we have a doublet covering u and w . Or the cut situated t o the l e f t of C f a l l s in or before A , and we have the same conclusion. Or f ina l ly Q i s embeddable i n the intersection of B and C : contradiction.

Transitivity no longer holds for a chain in which 0 Take A = 1 + Q + W1. W and l e t u be the minimum of A ; l e t v be the minimum of the final interval W1. G) ; l e t w be separated from v by the f i r s t component O1 . Then u and v are covered by the doublet 1+Qt1 formed of a r ight indecomposable 1tQ and a l e f t indecomposable chain Qt1 , and v and w are covered by the indecomposable chain al. W . However no common doublet covers u and w , since 1tQt1 i s neither l e f t nor right indecomposable. 0

Even i f we would reinforce the definition of the doublet, by requiring t h a t the l e f t indecomposable chain always be an i n i t i a l interval , and the right indecomposable chain be a final interval of the doublet, both with a common element, then consider A = 1+Q+1 and l e t u be the minimum, w the maximum of A , and v an element o f the middle interval Q . Then u i s equivalent t o v by a r ight indecomposable chain, v equivalent t o w by a l e f t indecomposable chain, yet u and w are no longer equivalent elements since

v , w by a l e f t indecomposable chain, necessarily i n f i n i t e , and

Q i s embeddable.

1+Q+1 i s no longer a doublet.

4.6. For a scattered chain, the equivalence relation of covering by doublets, i s the union of the equivalence relation for ,right indecomposable chains and the equivalence relat ion for l e f t indecomoosable chains. In other words, x and y are equivalent with respect t o doublets i f f there exis ts a f i n i t e sequence of elements, from x until y , where two consecutive terms are equivalent with respect t o right o r l e f t indecomposable chains. I n ch.8 5 4 . 5 we shall see tha t , for a scattered chain, there are only f in i te ly many equivalence classes with respect t o covering by doublets; moreover these classes are themselves doublets, possibly being right or l e f t indecomposable chains.

Chapter 6 171

s 5 - HEREDITARILY INDECOMPOSABLE CHAIN

Here i s f i r s t an i n t u i t i v e d e f i n i t i o n o f h e r e d i t a r i l y indecomposable chains, o r

h-indecomposable cha ins . The empty chain, eve ry s i n g l e t o n chain, eve ry c h a i n isomorphic w i t h a r e g u l a r aleph, as w e l l as t h e converse o f such, i s h-indecomposable. I f o( i s a r e g u l a r i n f i n i t e a leph and a r e h-indecomposable cha ins such t h a t , f o r each i , t h e s e t of i n d i c e s j (i c j C. % ) f o r which Ai i s embeddable i n A i s c o f i n a l i n o( , then t h e sum o f t h e Ai a long o( o r a long i t s converse, a r e

h-indecomposable. Moreover t h e o n l y h-indecomposable chains, a r e those chains

which can be so cons t ruc ted . The p reced ing " d e f i n i t i o n " i s easy t o use, and i n f a c t we s h a l l u s u a l l y t a k e i t . But i t i s i n c o r r e c t f rom a l o g i c a l p o i n t o f view. It i s a d e f i n i t i o n scheme, i n t h e same sense as t h e axiom schemes o f s e p a r a t i o n o r s u b s t i t u t i o n . Indeed t h e l a s t sentence o f t h e p reced ing paragraph ( t h e o n l y h-indecomposable chains. . .) c o u l d be e x p l i c i t e d t h u s l y : i f an a r b i t r a r y c o n d i t i o n \p ho lds f o r eve ry r e g u l a r a leph and i t s converse, i f i s p rese rved i n t a k i n g any sum o f cha ins Ai a long a

r e g u l a r a leph o r i t s converse, assuming t h a t each Ai be embeddable i n A f o r

a c o f i n a l s e t o f i n d i c e s j ; then f ho lds f o r eve ry h-indecomposable chain.

Th is t y p e o f procedure i s a d m i s s i b l e as an axiom scheme, which rep resen ts i n f i n i t e -

l y many axioms, each f o r a g i v e n c o n d i t i o n b l e as a d e f i n i t i o n , which must be of f i n i t e l e n g t h .

We propose t h e f o l l o w i n g f i n i t e (and admiss ib le ) d e f i n i t i o n , which i s e q u i v a l e n t t o t h e p reced ing scheme and so w i l l p e r m i t us t o use t h e scheme, i n p r a c t i c e . L e t A be a c h a i n and I a s e t o f i n t e r v a l s o f A . Suppose t h a t I c o n t a i n s t h e empty i n t e r v a l and eve ry s i n g l e t o n i n t e r v a l i n A . Suppose t h a t , g i v e n a r e g u l a r i n f i n i t e a leph o( , i f I c o n t a i n s t h e i n t e r v a l s Ai ( i < W ) which a r e m u t u a l l y d i s j o i n t and s i t u a t e d i n A w i t h o u t i n t e r m e d i a t e i n t e r v a l s , i n t h e o r d e r o f i n c r e a s i n g i o r o f decreas ing i , and such t h a t f o r each i t h e s e t o f i n d i c e s j f o r which Ai i s embeddable i n A . i s c o f i n a l i n o( , then I c o n t a i n s t h e un ion i n t e r v a l o f t h e Ails (which i s an i n t e r v a l of A ) . Now we say t h a t A i s h-indecomposable, o r h e r e d i t a r i l y indecomposable, i f t h e c h a i n A i t s e l f be longs t o any s e t I s a t i s f y i n g t h e p reced ing .

2 F o r example &, & , 0 . W - , Z . id and t h e i r converses, a re h-indecompo-

sab le . Fo r eve ry h-indecomposable cha in , i t s converse i s h-indecomposable.

Note t h a t t h e r e g u l a r a leph o( , such t h a t A i s t h e sum o f t h e Ai a l o n g d i n t h e above c o n s t r u c t i o n , i s t h e c o f i n a l i t y o f A ; t h e c o - i n i t i a l i t y i n t h e case of a sum a long t h e converse o f o( .

Ai ( i < o( )

j

j

. Obv ious l y t h i s i s no more admiss i -

____--.__ -.

J

5.1. Every h-indecomposable chain i s r i g h t o r l e f t indecomposable, according t o whether i t i s a sum along a regu la r aleph o r along i t s converse.

5.2. An ord ina l i s h-indecomposable i f f i t i s indecomposable, i n the sense o f ch.1 5 3.6. Proof by induc t ion on the o rd ina l i t s e l f ( s t a r t w i t h a c o f i n a l r e s t r i c t i o n o f minimum length, i . e . isomorphic w i t h the c o f i n a l i t y ) .

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5.3. Every h-indecomposable chain i s s t rong ly scattered ____ (see d e f i n i t i o n i n 2.3). Moreover, i n the decomposition o f anh-indecomposable chain

Ai ( i < Cof A) ,each - o f A . Use f o r instance the d e f i n i t i o n scheme, tak ing f o r " i s s t rong ly scattered" . F i n a l l y use 3.9.

We sha l l see l a t e r t h a t the h-indecomposable chains are exac t l y the sca t te red indecomposable chains: see below, ch.8 5 4.3 using the axiom o f choice. Recal l

tha t , modulo the axiom o f dependent choice, the s t rong ly sca t te red chains are exac t ly the sca t te red ones ( 2.3 above).

A i n t o the i n t e r v a l s

Ai has a neighborhood rank s t r i c t l y less than the rank the cond i t ion

5.4. THEOREMS ON HEREDITARILY INDECOMPOSABLE CHAINS

A form a we l l quasi -order ing w i t h respect t o embeddabil ity. Then A i s a f i n i t e

L e t A be a sca t te red chain. Suppose t h a t the h-indecomposable r e s t r i c t i o n s f

sum o f h-indecomposable chains (LAVER 1968, uses dependent choice). For the we l l

pre-order ing o r we l l quasi-ordering, see ch.4 5 3.2

0 The chain A i s s t rong ly sca t te red (modulo the axiom o f dependent choice), hence has a neighborhood rank. Suppose the statement i s f a l se , and l e t o( be the l e a s t

o rd ina l f o r which there e x i s t s an A o f neighborhood rank o( , whose h-indecomposable r e s t r i c t i o n s fo rm a we l lqas i -o rder ing w i t h respect t o embeddabil ity, y e t A i s no t a f i n i t e sum o f h-indecomposable chains. We can always assume t h a t A has

a minimum o r a maximum element. Indeed, take an element a o f the base, and re-

place A by the i n i t i a l o r f i n a l i n t e r v a l w i t h endpoint a : the neighborhood rank remains a t most equal t o H and the h-indecomposable r e s t r i c t i o n s s t i l l form

a we l l quasi-ordering. To f i x ideas, suppose t h a t A has a minimum element, and apply 2.5. Then e i t h e r

A i s a f i n i t e sum o f chains w i t h neighborhood ranks s t r i c t l y l ess than o( . I n which case, by hypothesis, each o f these chains i s a f i n i t e sum o f h-indecomposa- b le chains, so t h a t A i s as we l l . O r there e x i s t s a regu la r i n f i n i t e aleph 8 such t h a t A i s the sum o f a 8-sequence of chains Ai ( i < )' ) w i t h neigh-

borhood ranks < o( . I n which case, by hypothesis, each h-indecomposable chains: from t h i s p o i n t on, l e t the Ai themselves . We sha l l prove t h a t there e x i s t s a f i n a l i n t e r v a l o f which y i e l d s an h-indecom-

posable f i n a l i n t e r v a l on A : the induced i n t e r v a l being the sum o f the Ai

Ai i s a f i n i t e sum o f designate these chains

Chapter 6 173

whose i n d i c e s i belong t o t h e f i n a l i n t e r v a l o f 8 . T h i s w i l l s u f f i c e , s i n c e by 2.5.(2) t h e complementary i n i t i a l i n t e r v a l i s a f i n i t e sum o f cha ins w i t h ne ighbor-

hood ranks s t r i c t l y l e s s than d ; hence by hypo thes i s , a f i n i t e sum o f h-indecomposable cha ins . Thus we s h a l l prove t h a t f rom some index on , f o r each i , t h e s e t

o f j ( i < j % r ) f o r which Ai i s embeddable i n A i s c o f i n a l i n g . Suppose t h e c o n t r a r y . There e x i s t s an i n d e x i ( 0 ) and an i ' ( 0 ) 7 i ( 0 ) such t h a t f o r a l l

an i(1) 7 i ' ( 0 ) and an i'(l)> i(1) such t h a t f o r a l l i >/ i ' (1) , n e i t h e r Ai(l) no r , by t h e preceding, A , i s embeddable i n Ai . I t e r a t i n g t h i s , we

have an w -sequence o f cha ins which i s bad w i t h r e s p e c t t o embeddab i l i t y . Hence t h e chains Ai

d i c t i o n . 0

j

i 3 i ' ( 0 ) , t h e c h a i n A i(o) i s n o t embeddable i n Ai . Then t h e r e e x i s t s

i (0)

do n o t c o n s t i t u t e a we l f i uas i -o rde r ing : see ch.4 5 3.2; con t ra -

5.5. L e t A be a s c a t t e r e d indecomposable cha in . Suppose t h a t t h e h-indecomposable

r e s t r i c t i o n s o f A A

f o rm a w e l l quas iq rde r ing w i t h r e s p e c t t o embeddab i l i t y . Then i s h-indecomposable (LAVER 1968, uses axiom o f cho ice ) .

0 By 3.3 c o r o l l a r y ( 3 ) , t h e c h a i n A i s s t r i c t l y r i g h t o r l e f t indecomposable; t o f i x ideas, suppose i t i s s t r i c t l y r i g h t indecomposable. Take a c o f i n a l r e s t r i c t i o n o f A which i s isomorphic t o t h e r e g u l a r i n f i n i t e a leph Cof A (ax iom o f choice) .

L e t o( denote t h i s r e s t r i c t i o n , which we i d e n t i f y w i t h t h e o r d i n a l Cof A . For each i CIA l e t Ai be t h e i n t e r v a l o f A between t h e element i ( i n c l u s i v e ) and t h e element i+l ( e x c l u s i v e ) . Thus A i s t h e sum o f t h e Ai a long & . D e f i n e as f o l l o w s t h e i n t e r v a l s Bi ( i < 4 ) , so t h a t A i s t h e sum a long u o f t h e Bi , and eve ry Bi i s embeddable i n eve ry B w i t h j > i . Moreover we

r e q u i r e t h a t no Bi i s a f i n a l i n t e r v a l o f A , so t h a t Bi< A w i t h respec t t o

embeddab i l i t y . L e t Bo = A. . L e t u be an o r d i n a l s t r i c t l y l e s s t h a n 6( , and suppose t h a t we have d e f i n e d t h e Bi ( i < u) . L e t A ' be t h e sum o f these Bi . Then A ' i s d i f f e r e n t f rom A , s i n c e o the rw ise e i t h e r u would be c o f i n a l i n

o( , o r u would have an immediate predecessor u-1 w i t h Bu-l a f i n a l i n t e r - v a l o f A , thus equimorphic w i th A , c o n t r a d i c t i n g o u r hypotheses. L e t A" be t h e f i n a l i n t e r v a l o f A , complement o f A ' . Since A i s r i g h t inde-

composable, A" i s equimorphic w i t h A . Se t Bu t o be a p roper i n i t i a l i n t e r v a l o f A" (so Bu< A) , i n which Buml i s embeddable ( i n t h e case t h a t u i s a

successor o r d i n a l ) , o r i n which t h e sum o f t h e Bi ( i < u) i s embeddable ( i n t h e case t h a t u i s a l i m i t o r d i n a l ) . F i n a l l y , e n l a r g e t h i s i n t e r v a l Bu , i f necessary, so t h a t t h e sum o f t h e Bi (i u) i n c l u d e s t h e corresponding sum o f t h e Ai :

thus we a r e ensu red t o exhaus t A by t h e sum o f a l l t h e Bi (i < 4 ) . By t h e p reced ing 5.4, each Bi i s a f i n i t e sum o f h-indecomposable chains, which we aga in des igna te by Bi . F o r each o rde red p a i r ( i , j ) w i t h i< j < 4 , t h e

j


c h a i n Bi B . + Bj+l + ... , hence i n a c h a i n B o f i ndex g r e a t e r t han o r equal t o j . Thus t h e c h a i n A i s h-indecomposable. 0

i s indecomposable and i s embeddable i n a f i n i t e sum o f t h e fo rm

J

§ 6 - I N D I V I S I B L E RELATION OR CHAIN

We say t h a t a r e l a t i o n R wi th i n f i n i t e base E i s i n d i v i s i b l e i f f o r eve ry

p a r t i t i o n o f E i n t o two complementary subsets C and D = E-C , e i t h e r R Q R / C

o r R 6 R/D . Some au tho rs say " indecomposable" i n s t e a d o f " i n d i v i s i b l e " .

Fo r example, t h e c h a i n G, o f n a t u r a l i n t e g e r s i s i n d i v i s i b l e . Every i n d i v i s i b l e cha in i s a f o r t i o r i indecomposable. However t h e c h a i n 2. W , where Z i s t h e c h a i n o f i n t e g e r s , i s indecomposable y e t n o t i n d i v i s i b l e : i f we

2 p a r t i t i o n 2 i n t o LL: and i t s converse w- , then we have t h e r e s t r i c t i o n s Lc)

and W - . Lc: , n e i t h e r o f which admi ts an embeddinq o f Z. .

6.1. Every r e g u l a r a leph i s i n d i v i s i b l e . 0 I f I i s a r e g u l a r a leph and we p a r t i t i o n I i n t o two complementary subsets ,

t hen one o f these subsets must be c o f i n a l i n

isomorphic w i t h I . 0 I , and thus y i e l d s a r e s t r i c t i o n

6 . 2 . F o r an o r d i n a l , i n d i v i s i b i l i t y i s i d e n t i c a l w i t h i ndecomposab i l i t y .

0

t h a t eve ry power o f w , say WU , i s i n d i v i s i b l e ( i n v iew o f ch.1 9 3 .6 ) . Proceed by i n d u c t i o n on t h e exponent u . Suppose f i r s t l y t h a t u i s a successor o r d i n a l : i s a sum a long dc:, o f i n d i v i s i b l e o r d i n a l s equal

t o civ . Hence a p a r t i t i o n o f uu i n t o two complementary subsets , g i v e s one o f t h e subsets as a sum a long w o f components wv . Now suppose t h a t t h e o r d i n a l u i s a l i m i t o r d i n a l ; t hen u i s t h e supremum o f

a sequence o f o r d i n a l s v ( i ) ; we can suppose t h a t t h e i n d e x i runs th rough t h e

r e g u l a r a leph an i n d i v i s i b l e i n i t i a l i n t e r v a l o f w' , t h e un ion o f which i s w' . The p a r t i - t i o n o f t he base y i e l d s , f o r each i , a t l e a s t one r e s t r i c t i o n i somorph ic w i t h

subsets i n o u r p a r t i t i o n y i e l d s an i - m a j o r i t y r e s t r i c t i o n f o r a s e t o f i n d i c e s

i , which i s c o f i n a l i n I . Thus we o b t a i n a t l e a s t a r e s t r i c t i o n o f wu which i s isomorphic w i t h t h e union, o r supremum o f t h e , hence i somorph ic w i t h t h e g i v e n l i m i t o r d i n a l W u . 0

It s u f f i c e s t o show t h a t eve ry indecomposable o r d i n a l i s i n d i v i s i b l e , hence

u = v + 1 . Then wU

I = Cof u . By t h e i n d u c t i o n hypo thes i s , each power f ~ ~ ( ~ ) i s

0 v(i) : c a l l t h i s t h e i - m a j o r i t y r e s t r i c t i o n . Then a t l e a s t one o f t h e two

Chapter 6 175

6.3. Given an i n d i v i s i b l e c h a i n A w i t h i n f i n i t e base E , t h e r e e x i s t s a p a r t i -

t i o n o f E i n t o two complementary subsets C and D = E-C yitJ A E A/C and A A/D (HAGENDORF 1975; uses axiom o f cho ice ) .

0 The i n d i v i s i b l e c h a i n A i s a f o r t i o r i indecomposable. By 3.3, e i t h e r A i s equimorphic w i t h A.2 , and so t h e d e s i r e d p a r t i t i o n e x i s t s . O r A i s s t r i c t l y

r i g h t o r l e f t indecomposable: t o f i x ideas, suppose t h a t i t i s r i g h t indecomposa- b l e . C a l l I t h e r e g u l a r a leph, c o f i n a l i t y o f A ; and decompose A i n t o a sum along I o f i n t e r v a l s Ai ( i runn ing th rough I ; axiom o f cho ice ) . Wi th t h e i n t e r v a l

i s embeddable, and i n t h i s f i n a l i n t e r v a l t a k e a p r o p e r i n i t i a l i n t e r v a l which A. i s embeddable; t hen c a l l A6 = A. and A; t h e two cop ies o f A. . A f t e r A! i t remains a f i n a l i n t e r v a l equimorphic w i t h A : i t e r a t e t h i s by embed- d ing, i n t h i s f i n a l i n t e r v a l , two consecu t i ve cop ies A i and A: o f A1 , t h e

i n t e r v a l be ing embeddable i n each copy and t h e rema in ing f i n a l i n t e r v a l be ing

equimorphic w i t h A . T h i s procedure can be i t e r a t e d f o r eve ry i ndex i be long ing t o I . Indeed f o r each o r d i n a l i ndex i , t h e i-sequence o f t h e sums

A ! + A'! ( j < i) o f cop ies cannot exhaust A ; s i n c e o the rw ise e i t h e r i would

be c o f i n a l i n I ( c o n t r a d i c t i n g t h e r e g u l a r i t y o f I ) , o r i would have an imme- d i a t e predecessor i-1 wi th A;-1 o r AS-1 a f i n a l i n t e r v a l o f A , which i s

exc luded.

F i n a l l y , t h e un ion o f t h e A; and t h e un ion o f t h e A; f o r a l l i i n I , y i e l d the d e s i r e d p a r t i t i o n . 0

It i s proved by POUZET 1977 (unpub l i shed) t h a t , g i v e n an i n d i v i s i b l e r e l a t i o n w i t h denumerable base E , t h e r e e x i s t s a p a r t i t i o n o f E i n t o two complementary subsets C and D w i t h R equimorphic t o R / C and t o R/D . Problem: t h e case o f an uncountable i n d i v i s i b l e r e l a t i o n .

A. , a s s o c i a t e t h e complementary f i n a l i n t e r v a l , i n which A

A 6 i n

A1

J J

R

EXERCISE 1 - Given a denumerable cha in A , t h e r e e x i s t s an isomorphism o f A

on to a p roper r e s t r i c t i o n o f A

E i t h e r t h e c h a i n Q o f r a t i o n a l s i s embeddable i n A , and as A i s embeddable i n Q , t h e s tatement i s obv ious. O r A i s s c a t t e r e d , and even s t r o n g l y s c a t t e r e d

(see 2.3, denumerable case), so t h a t t h e r e e x i s t two consecu t i ve 0-neighborhoods

o f A . The f i r s t o f these two consecu t i ve 0-neighborhoods admi ts an embedding o f w o r t h e second admi ts an embedding o f w- . P o s s i b l y A reduces t o a unique 0-neighborhood, i somorph ic w i t h k, o r i t s converse o r Z . I n a l l cases t h e s t a t e -

EXERCISE 2 - IMMEDIATE EXTENSIONS OF A SCATTERED C H A I N

1 - L e t A be a s c a t t e r e d , r i g h t indecomposable cha in . Then A < A + 1 < A+2 < ... < A t i < ... f o r each p o s i t i v e i n t e g e r i . Indeed A i s n o t embeddable i n any p roper i n i t i a l i n t e r v a l , t hus A+1 i s n o t embeddable i n A ( 3 . 1 . ( 3 ) u s i n g depen-

dent cho ice i n o r d e r t h a t A be s t r o n g l y s c a t t e r e d ) . Moreover i f cx i s t h e neighborhood rank o f A , then each A + i has neighborhood

rank o(+l . Indeed l e t u be an element o f A : i t s u f f i c e s t o see t h a t t h e maximum element v o f A + l i s n o t an q - n e i g h b o r o f u . Assume t h e c o n t r a r y . E i t h e r

% = /j+l i s a successor o r d i n a l ; t hen t a k e t h e l a s t / 3 -ne ighborhood o f A + l , which con ta ins v . I f t h i s ne ighborhood reduces t o t h e s i n g l e t o n o f v , t a k e t h e n e x t t o l a s t 0 - n e i g h b o r h o o d , i n which A i s embeddable. I n b o t h cases A has rank a t most equal t o (?I : c o n t r a d i c t i o n . O r o( i s a l i m i t o r d i n a l ; t hen t h e r e e x i s t s c o( such t h a t u and v a r e & -ne ighbor , and A i s embeddable

i n a 1/ -neighborhood: c o n t r a d i c t i o n . 2 - More g e n e r a l l y t a k e an i n f i n i t e s c a t t e r e d cha in A . A n t i c i p a t i n g on ch.8 g 4.5, decompose A i n t o a f i n i t e sum o f d i s j o i n t , indecomposable i n t e r v a l s . Each i n t e r v a l i s s t r i c t l y r i g h t o r s t r i c t l y l e f t indecomposable; moreover we can assume

t h a t no i n t e r v a l i s absorbed by any o f i t s two ad jacen t i n t e r v a l s . Then e i t h e r

we have a t l e a s t one l e f t i n t e r v a l . t hen d e f i n e Ai by adding i elements j u s t

b e f o r e t h e f i r s t l e f t i n t e r v a l . Otherwise d e f i n e Ai by adding i elements a t

t h e end o f A . Then u s i n g t h e p reced ing 0 1, e a s i l y see t h a t A < A1 < A 2 < ...

EXERCISE 3 - THEOREM OF THE INTERMEDIATE CHAIN BETWEEN A AND A.2

Given an i n f i n i t e c h a i n A , e i t h e r A.2 i s equimorphic w i t h A o r t h e r e e x i s t s a t l e a s t one c h a i n which i s s t r i c t l y i n t e r m e d i a t e between A a& A.2 w i t h r e s p e c t t o embeddab i l i t y (HAGENDORF 1984 ). Decompose A i n t o B + L , where L i s t h e maximum f i n a l i n t e r v a l which i s l e f t indecomposable (see 3 . 2 . ( 3 ) ) . I f L = A then e i t h e r A.2 equimorphic w i t h A o r

l + A i s s t r i c t l y i n t e r m e d i a t e between A and A.2 . If L i s empty, t h e n each i n i t i a l i n t e r v a l o f A i s s t r i c t l y l e s s than A , so t h a t A+1 i s s t r i c t l y i n t e r - mediate. I n t h e f o l l o w i n g we assume t h a t L # A and L non-empty. Consider t h e sum A1 + A2 where A1 and A2 a r e b o t h isomorphic w i t h A , and decompose t h i s sum, o b t a i n i n g C + L' where L ' i s t h e maximum l e f t indecompo-

sab le i n t e r v a l . E i t h e r L ' i s s t r i c t l y g r e a t e r t han A2 : then A.2 i s equimor- p h i c w i t h A . O r L ' = L and C = A+B K 'A .2 by 3.2. (3) . E i t h e r C i s s t r i c t l y i n t e r m e d i a t e between A and A.2 . O r C i s equimorphic w i t h A . Then e i t h e r L.2 i s equimorphic w i t h L : then A = B+L E B + L.2 = A+L E C+L = A.2 . O r f i n a l l y L i s s t r i c t l y l e f t indecomposable. S ince A = B+L : B+L+B and B < A by 3.2. (3) , t hen L+B i s embeddable i n L , so t h a t 1 + B B L and B < L . Thus L+B+L i s s t r i c t l y i n t e r m e d i a t e between A = B+L and A.2 = B+L+B+L .

CHAPTER 7

USE OF SCATTERED CHAINS FOR THE STUDY OF FINITELY FREE AND WELL PARTIAL ORDERINGS

§ 1 - REINFORCEMENT OF A PARTIAL ORDERING BY SCATTERED CHAINS

1.1. Let A be a pa r t i a l ordering; then the following four conditions a re equivalent ( B O N N E T , POUZET 1969; uses axiom of choice) : (1) A i s f i n i t e l y f r e e and does not admit an embedding of the chain Q of r a t iona l s : ( 2 ) every t o t a l l y ordered reinforcement of A i s s ca t t e r ed ; ( 3 ) the chain Q i s not embeddable i n the p a r t i a l ordering of inclusion f o r the i n i t i a l i n t e rva l s of A ; (4) f o r every function f mapping the r a t iona l s i n t o the base I A l , there e x i s t two r a t iona l s x , y with x < y (mod Q ) and f x 2 fy (mod A ) . 0 Condition ( 2 ) implies (1). Indeed i f Q i s embeddable in A , then Q i s embeddable i n every reinforcement of A . On the o ther hand, i f A has an i n f i n i t e , hence a denumerable f r e e subse t , then take a chain B isomorphic w i t h Q on this subse t , and then by ch.2 5 4.1 take a p a r t i a l l y ordered reinforcement of A which extends red reinforcement of A i n which Q i s embeddable. 0

0 Conditions ( 2 ) and ( 3 ) a r e equiva len t . First suppose t h a t t he re e x i s t s a t o t a l l y ordered reinforcement C of A i n which Q i s embeddable. Then the chain of inclusion f o r i n i t i a l i n t e rva l s of C i s a r e s t r i c t i o n of the p a r t i a l ordering of inc lus ion f o r i n i t i a l i n t e r v a l s of A , and admits an embedding of Q . Conversely, suppose t h a t Q i s embeddable i n the p a r t i a l ordering of inclusion f o r i n i t i a l i n t e rva l s of A . Take a chain U , i n which Q i s embeddable, and which i s maximal w i t h respect t o inc lus ion , and formed of i n i t i a l i n t e rva l s of A

(see ch.2 5 2 . 7 , axiom of choice) . Then a ce r t a in t o t a l l y ordered reinforcement V of A : see ch.4 5 1.3. Moreover by ch.6 5 1.5, the chain Q i s embeddable i n V ( th i s uses dependent choice) . 0

0 Conditions ( 2 ) and ( 4 ) a r e equiva len t . Suppose t h a t there e x i s t s a t o t a l l y ordered reinforcement of A in which Q i s embeddable. Then there e x i s t s a b i jec t ion f of Q onto a r e s t r i c t i o n of A , which maps any two ra t iona l s x < y (mod Q ) i n t o fx < or I f y (mod A ) . Conversely, i f there e x i s t s such a (necessa r i ly i n j e c t i v e ) func t ion , then the image f " ( Q ) i s a reinforcement of the r e s t r i c t i o n A/ f " ( lQ l ) ( r e s t r i c t i o n of A

7

B . By the reinforcement axiom i n ch.2 5 4.2, there exists a t o t a l l y orde-

U i s the chain of i n i t i a l i n t e rva l s of

t o the s e t of images of ra t ionals) . By ch.2 5 4.1 and the reinforcement axiom, there exis ts a to ta l ly ordered reinforcement of A in which Q i s embeddable. 0

0 Condition (1) implies ( 4 ) . Suppose t h a t A i s f i n i t e l y free and t h a t Q i s not embeddable in A , yet there ex is t s a function f from Q into A , such that for x < y (mod Q ) we have fx < or I fy (mod A ) : hence f i s inject ive. Then the restr ic t ion A/f" ( \QI) i s a denumerable f i n i t e l y f ree par t ia l ordering in which Q i s n o t embeddable. By applying RAMSEY (ch.3 5 1 . 2 ) , there exis ts a denumerable, maximal, to ta l ly ordered and scattered restr ic t ion of A/fo(lQl) . Thus there ex is t two rationals xo and yo > x o (mod Q) with f (yo) an immediate successor of f (xo) (mod A/f" ( IQ() . I terate t h i s , by replacing Q by i t s res t r ic t ion t o the open interval (xo,yo) . We obtain, for each integer i , an interval ( x i , y i ) with yi > x i (mod Q ) and each interval ( X ~ + ~ , Y ~ + ~ ) properly included in ( x i , y i )

(mod Q) . For each f ( x i ) < f ( y i ) (mod A ) , without any element of f " ( l Q I ) between f ( x i ) and f ( y i ) (mod A ) : i t suffices t o consider a rational x and a l l possible positions of fx , ei ther for x between xi - l and yi-l (mod Q ) , o r for x elsewhere in the chain Q . Again using RAMSEY, we see that there ex is t denumerably many integers i with a l l values f ( x i ) and f ( y i ) mutually comparable (mod A ) . Hence there ex is t i and j 7 i with, for instance, f ( x . ) situated between f ( x i ) and f ( y i ) (mod A ) : contradiction. 0

i we have

J

1 . 2 . Recall that a partial ordering A i s said t o be s t ra t i f ied (ch.2 5 5.2) i f incomparability (mod A ) together with ident i ty , forms an equivalence relation. The equivalence classes form a chain, called the princioal chain. A s t ra t i f ied par t ia l ordering i s said t o be scattered i f i t s principal chain i s scattered.

Let A be a partial ordering. There exis ts an i n i t i a l interval of A which i s maximal with respect to inclusion, among those i n i t i a l intervals having a scattered s t ra t i f ied reinforcement. Analogous statement for final intervals (uses axiom of choice).

0 Consider those i n i t i a l intervals of A having a scattered s t r a t i f i e d reinforcement. There ex is t such, for example the empty interval. The s e t of these intervals i s ordered by inclusion; so l e t C be a maximal chain of inclusion (see ch.2 5 2 . 7 , axiom of choice). Let X be the i n i t i a l intervals of A which are elements of C , and l e t I be the i n i t i a l interval of A ' which i s the union of the X . No in i t ia l interval of A which i s a proper extension of I has a scattered s t r a t i f i e d rein-

Chapter 7 179

re in fo rcemen t Y+ of Y i s s a i d t o be f i n e r than a s c a t t e r e d s t r a t i f i e d r e i n f o r c e - ment X+ o f X , i f t h e base o f X i s i n c l u d e d i n t h e base o f Y , and each incom-

p a r a b i l i t y c l a s s o f X+ i s an i n c o m p a r a b i l i t y c l a s s o f Y , and f i n a l l y i f t h e p r i n c i p a l c h a i n o f X+ i s an i n i t i a l i n t e r v a l o f t h e p r i n c i p a l c h a i n o f Y+ . I f X, Y be long t o C and i f t h e base 1x1 i s i n c l u d e d i n I Y J , then f o r each s c a t t e r e d s t r a t i f i e d r e i n f o r c e m e n t X+ o f X , t h e r e e x i s t s a s c a t t e r e d s t r a t i f i e d

re in fo rcemen t o f Y which i s f i n e r t han X+ . Indeed, t a k e an a r b i t r a r y s c a t t e r e d

s t r a t i f i e d re in fo rcemen t Z o f Y . Keep t h e i n c o m p a r a b i l i t y c lasses o f X+ and r e p l a c e each c l a s s o f 2 by i t s i n t e r s e c t i o n w i t h ( IY l - 1x1) ( p r o v i d e d t h i s i n -

t e r s e c t i o n i s non-empty). Then w h i l e p r e s e r v i n g t h e i r o rde r , p l a c e these c lasses coming f rom Z a f t e r a l l t h e c lasses o f X+ . Th is i s p o s s i b l e , s i n c e e v e r y e l e -

ment o f I Y I - 1 x 1 i s g r e a t e r t han o r incomparable (mod A) w i t h eve ry element o f t h e base X . The s e t o f s c a t t e r e d s t r a t i f i e d re in fo rcemen ts of t h e e lements o f C

ordered by t h e comparison " f i n e r than" . Take a maximal cha in , aga in u s i n g ch.2

5 2.7, hence axiom o f cho ice . It s u f f i c e s t o p rove t h a t t h i s maximal cha in t e r m i - nates w i t h a s c a t t e r e d s t r a t i f i e d r e i n f o r c e m e n t o f I . F o r t h i s , n o t e t h a t t h i s

maximal c h a i n cannot t e r m i n a t e w i t h a re in fo rcemen t X+ o f an X i n C , d i s t i n c t f rom I . Indeed, take a Y i n C , a p roper e x t e n s i o n o f X . The p reced ing para-

graph shows t h e e x i s t e n c e o f a s c a t t e r e d s t r a t i f i e d re in fo rcemen t o f Y which i s

f i n e r t han X+ : c o n t r a d i c t i o n . Fur thermore, t h i s c h a i n n e c e s s a r i l y has a maximum

element, which i s t h e re in fo rcemen t o f an element o f C . Indeed, g i v e n a s e t

composed of re in forcements X+ o f e lements X o f C , w i t h o u t any X maximum, and t o t a l l y o rde red by t h e comparison " f i n e r than" , t ake t h e i n i t i a l i n t e r v a l J formed 'by t h e un ion o f t h e X b e i n g considered, and then t h e re in fo rcemen t o f J which i s t h e common e x t e n s i o n o f t h e X+ . T h i s re in fo rcemen t i s f i n e r t han each X+ , and i s s c a t t e r e d , s i n c e eve ry c h a i n which i s t h e un ion o f s c a t t e r e d p roper

i n i t i a l i n t e r v a l s i s s c a t t e r e d : see ch.6 5 1.2.

+

i s p a r t i a l l y

1.3. L e t A be a p a r t i a l o rde r ing ; t hen t h e f o l l o w i n g t h r e e c o n d i t i o n s a r e equ i - v a l e n t (BONNET, POUZET 1969; uses axiom o f cho ice ) : (1) t h e c h a i n 9 i s n o t embeddable i n A ; (2 ) t h e r e e x i s t s a s c a t t e r e d t o t a l l y o r d e r e d re in fo rcemen t o f A ; (3) t h e r e e x i s t s a s c a t t e r e d s t r a t i f i e d re in fo rcemen t o f A .

C o n d i t i o n ( 2 ) o b v i o u s l y i m p l i e s (1) and ( 3 ) . C o n d i t i o n (3 ) i m p l i e s ( 2 ) . Indeed, s t a r t i n g w i t h a s t r a t i f i e d re in forcement o f A w i t h a s c a t t e r e d p r i n c i p a l chain, i t s u f f i c e s t o choose a w e l l - o r d e r i n g (hence a

s c a t t e r e d cha in ) based on each i n c o m p a r a b i l i t y c l a s s o f t h e s t r a t i f i e d p a r t i a l o r -

d e r i n g (ax iom o f choice) ; and then t o t a k e t h e o r d i n a l sum o f these we l l -o rde r ings a long t h e p r i n c i p a l cha in .

_I_.

We now show t h a t (1) i m p l i e s ( 3 ) . Fo r t h i s , suppose t h a t eve ry s t r a t i f i e d r e i n f o r c e - ment o f A admi ts an embedding o f t h e c h a i n Q . We s h a l l show t h a t Q i s embed-

dable i n A . App ly ing t h e p reced ing 1.2, l e t I be an i n i t i a l i n t e r v a l o f A which i s maximal w i t h r e s p e c t t o i n c l u s i o n , among those i n i t i a l i n t e r v a l s hav ing a s c a t t e r e d s t r a t i f i e d re in fo rcemen t . L e t J be a f i n a l i n t e r v a l , s i m i l a r l y d e f i - ned. L e t E be t h e base o f A ; t h e d i f f e r e n c e E - ( I I l u l J I ) i s non-empty. Since i f no t , t hen t a k e s c a t t e r e d s t r a t i f i e d re in fo rcemen ts If o f I and J' o f J . Take as new i n c o m p a r a b i l i t y c lasses t h e c lasses o f I+ f o l l o w e d by, i n o rde r , t h e c l a s s o f non-empty i n t e r s e c t i o n s o f t h e c lasses o f J+ w i t h T h i s then y i e l d s a s t r a t i f i e d re in fo rcemen t o f A whose p r i n c i p a l c h a i n i s t h e

sum o f t h e p r i n c i p a l c h a i n o f hence a s c a t t e r e d chain: t h i s c o n t r a d i c t s o u r assumDtion. L e t u be an e lement i n t h e d i f f e r e n c e E - (111 u I J I ) . We s h a l l p rove t h a t t h e

r e s t r i c t i o n o f A t o those elements u , on t h e o t h e r hand, have o n l y re in fo rcemen ts i n which Q i s embeddable. N o t i c e f i r s t t h a t t h e r e e x i s t elements < u which do n o t be long t o I11 . F o r i f n o t , t hen I augmented by u o n l y , would be an i n i t i a l i n t e r v a l o f A and a p r o p e r e x t e n s i o n o f I , y e t i t has a s c a t t e r e d s t r a t i f i e d r e i n f o r c e -

ment, ob ta ined by t a k i n g a s c a t t e r e d s t r a t i f i e d re in fo rcemen t o f I , completed

by a l a s t c l a s s formed by t h e s i n g l e t o n o f u : t h i s c o n t r a d i c t s m a x i m a l i t y o f I . Suppose t h a t t h e i n i t i a l i n t e r v a l o f those elements < u has a s c a t t e r e d s t r a t i - f i e d re in fo rcemen t , say

o f I . Then on t h e un ion I I l u l U l , t a k e t h e i n c o m p a r a b i l i t y c lasses o f I , i n t h e i r o rde r , f o l l o w e d by, aga in i n t h e i r o rde r , t h e non-empty i n t e r s e c t i o n s o f c lasses o f U w i t h l U l - ( l U l n 111) . We thus o b t a i n a s c a t t e r e d s t r a t i f i e d r e i n -

forcement o f A/( IIIu I U l ) , c o n t r a d i c t i n g t h e m a x i m a l i t y o f I . Analogous p r o o f f o r t h e f i n a l i n t e r v a l J and t h e f i n a l i n t e r v a l o f those elements > u (mod A) . Now l e t uo be u . Since t h e i n i t i a l i n t e r v a l < u has o n l y s t r a t i f i e d r e i n f o r - cements i n which Q i s embeddable, t h e r e e x i s t s an element u l < uo such t h a t

each o f t h e two i n t e r v a l s , t hose elements < u1 and those elements s t r i c t l y be t - ween u1 and uo , has o n l y s t r a t i f i e d re in fo rcemen ts i n which Q i s embeddable.

S i m i l a r l y , t h e r e e x i s t s an element 'u2 > uo w i t h t h e same p r o p e r t y f o r t h e f i n a l i n t e r v a l > u2 and t h e i n t e r v a l o f those elements s t r i c t l y between uo and u2 . I t e r a t i n g t h i s and u s i n g dependent cho ice , we o b t a i n a denumerable s e t o f e lements

ui (i i n t e g e r ) g i v i n g a r e s t r i c t i o n o f A i somorph ic w i t h Q . 0

Problem, suggested by t h e lemma f o r p a r t i t i o n i n s l i c e s , o r t h e min imal s t r a t i f i e d re in fo rcemen t i n t h e sense o f ch.4 5 1 . 5 . Does t h e c o n d i t i o n ( 3 ) remain e q u i v a l e n t w i t h (1) and ( 2 ) i f we s t reng then i t by r e q u i r i n g t h a t , i n t h e s c a t t e r e d s t r a t i -

f i e d re in fo rcemen t o f A , f o r e v e r y o rde red p a i r o f elements a, b ( a s b modulo t h e re in forcement) , t h e r e e x i s t a ' and b ' such t h a t a 6 a ' (mod A) and

E - I I I .

I' and a r e s t r i c t i o n o f t h e p r i n c i p a l c h a i n o f J',

U . Take aga in a s c a t t e r e d s t r a t i f i e d re in fo rcemen t I+ +

Chapter 7 181

b ' g b (mod A) and e i t h e r a ' = b ' o r a ' I b ' modulo the reinforcement.

5 2 - SUPPLEMENTARY RESULTS ABOUT FINITELY FREE PARTIAL ORDERINGS; TOPOLOGY ON I N I T I A L INTERVALS

2.1. Le t A

comparable idea ls o f A uses axiom o f choice).

0 Suppose the existence o f an uncountable se t o f mutual ly incomparable i dea ls of

A , hence a se t o f c a r d i n a l i t y a1 o f such idea ls Hi ( i countable o rd ina l ) .

To each Hi we associate a p a r t i a l o rder ing Ci which i s a well-founded co f ina l d i rec ted r e s t r i c t i o n o f Hi : see ch.4 5 5.4 (axiom o f choice). Since A i s f i n i t e -

l y f ree , so i s each Given two d i s t i n c t countable o rd ina l s respect t o i nc lus ion o f the bases o f Hi and H , def ine an element ui,j belonging t o I CiI element o f H

For each countable o rd ina l

values j . More prec ise ly , l e t Ai be the corresponding r e s t r i c t i o n o f A : we

have Card Ai < (J . I n general, f o r each se t U o f countable o rd ina ls and each countable o rd ina l i , l e t Ai(U) be the se t o f the ui,j f o r a l l j # i and j E U . More p rec i se l y Ai(U) w i l l be the corresponding r e s t r i c t i o n o f A . Each Ai o r each Ai(U) i s a well p a r t i a l order ing, as it i s a r e s t r i c t i o n o f the we l l p a r t i a l o rder ing Suppose f i r s t the existence o f a se t U o f countable o rd ina ls , which i s co f ina l

i n O 1 and such t h a t f o r each i o f U , the r e s t r i c t i o n Ai(U) i s countable.

Le t Uo designate t h i s U and l e t i ( 0 ) be i t s l e a s t element. P a r t i t i o n the

elements j # i ( 0 ) o f U i n t o classes, by p u t t i n g j and j ' i n the same

class i f u i(o),j = ui(o),j, . There are countably many such classes, s ince

Ai(o)(U) these classes, say U1 , has c a r d i n a l i t y w 1 . Le t k,, = ui(o) f o r the j

i n U1 , and note t h a t ko 7 o r I (mod A) w i t h a l l elements o f every f o r j e U1 . I t e r a t e t h i s by l e t t i n g i ( 1 ) be the l e a s t o rd ina l index belonging t o U1 . P a r t i t i o n the elements j o f U1 ( j # i ( 1 ) ) i n t o classes def ined by the

. There are countably many such classes, s ince e q u a l i t y u

Ai(l)(U1) i s included i n A (U) and the l a t t e r i s countable. A t l e a s t one o f these classes, say U2 , has c a r d i n a l i t y . Temporarily l e t kl = ui(l)

f o r the j i n U2 . By the preceding, ko > or I kl (mod A) .

be a f i n i t e l y f r e e p a r t i a l order ing. Then every s e t o f mutual ly i n - (w i th respect t o i nc lus ion ) , i s countable (BONNET 1975,

Ci , and hence each Ci i, j,

i s a we l l p a r t i a l order ing.

by using the incomparab i l i t y w i t h

j but no t t o I H . I . Thus ui ,j > o r I (mod A) w i t h every J

j ' i , l e t Ai be the se t o f the ui,; f o r a l l countable

Ci .

i s countable. Since the c a r d i n a l i t y o f U i s W 1 , a t l e a s t one o f

J "j

i ( l ) , j = ' i ( l ) , j l

i ( 1 ) ,j


t ha t ko and kl be incomparable (mod A) . For t h i s , no t i ce t h a t i s an element o f A . , and 1 7 o r I (mod A) w i t h a l l elements

(0 ) 1 ( 1 )

hence

i f necessary by a l y , note t h a t the

1 > or I ko (mod A) . Since C i(l) i s d i rec ted , replace kl common upper bound o f kl and 1 , belonging t o Ci(l) . F ina l -

o l d value, hence a lso the new value o f kl , i s > o r ! (mod A) w i t h a l l elements o f every H f o r j e Up . I t e r a t i n g t h i s y i e l d s an i n f i n i t e

sequence o f elements kr ( r i n tege r ) , which are mutua l l y incomparable (mod A) :

t h i s cont rad ic ts our hypothesis t h a t A i s f i n i t e l y f ree .

Now consider the case where, f o r each se t U c o f i n a l i n w1 , there e x i s t s an i i n U w i t h Ai(U) o f c a r d i n a l i t y W 1 . S t a r t i n g w i t h the se t Uo o f a l l countable o rd ina ls , take i ( 0 ) such t h a t A. has c a r d i n a l i t y U1 . Since

A iio) i s a we l l p a r t i a l order ing, by ch.4 5 3 .1 there e x i s t s a wel l -ordered res-

t r i c t i o n Bo o f Ai(o) , isomorphic w i t h c3 . be1 ongs Le t U1 be the se t o f countable ind ices j > i ( 0 ) , such t h a t u

t o the base I B O l . This U1 has c a r d i n a l i t y cJ1 , j u s t as Bo . I n the case now considered, we can i t e r a t e t h i s , as fo l lows. Take an element i ( 1 ) i n U1 , such

t h a t Ai(l)(U1)

j

1 ( 0 )

i (0) ,j

has c a r d i n a l i t y U1 . Then take a wel l -ordered r e s t r i c t i o n B1

o f Ai(l)(U1) , isomorphic w i t h cJ1 . Le t U2 be the se t o f countable ind ices

belongs t o the base [ B 1 l , and so f o r t h . We obtain, j 7 i ( 1 ) , such t h a t u

f o r each in tege r r , a se t Ur , a countable o rd ina l i ( r ) and a we l l -o rder ing Br isomorphic w i t h w1 . For each in tege r r , consider the se t o f those u

values i ( r + l ) , Take the l e a s t upper bound vr i n the we l l -o rder ing Br o f t h i s subset. Since

> o r I (mod A) w i t h each element o f H , i t fo l lows t h a t vr > o r

i ( l ) , j

where j takes the i ( r t 2 ) , . . . . This i s a countable subset o f the base I Br \ . i ( r ) , j

j U .

1 ( r ) ,j I (mod A) w i t h each element o f and o f and so f o r t h . I n p a r t i -

cu la r , we have vr > o r I v ~ + ~ , It remains t o replace each vr by an element wr o f H t h a t the wr are mutual ly incomparable (mod A) . For t h i s , l e t wo = vo . Take an element xb i n H which i s > o r I w i t h a l l elements o f Hi(o) , hence

v1 and xo . Then take i n H

v ~ + ~ , . . . (mod A) . i n such a manner

i ( r )

i ( 1 ) which i s > o r I wo . Replace v1 by a common upper bound w1 in H i ( l ! o f

an element, again c a l l e d xo , which i s > o r i (2)

\ w i t h a l l elements o f Hi(o) , and take an element x1 in H i ( 2 ) which i s > o r I w i t h a l l elements o f Hi(ll , and replace

o f v2, xo, x1 ; and so f o r t h . This now cont rad ic ts our hypothesis t h a t A i s f i n i t e l y f ree . 0

v2 by a common upper bound w2

A p a r t i a l o rder ing can be denumerable and f i n i t e l y f ree , and can have continuum many mutual ly incomparable i n i t i a l i n t e r v a l s .

Chapter 7 183

0 Take two cha ins , each i somorph ic w i t h t h e c h a i n o f r a t i o n a l s , eve ry e lement o f

t he f i r s t c h a i n be ing incomparable w i t h eve ry e lement o f t h e second. Then t o each

r e a l number x , a s s o c i a t e i n t h e f i r s t c h a i n t h e i n i t i a l i n t e r v a l I ( x ) which represents x as a c u t . S i m i l a r l y i n t h e second chain, a s s o c i a t e t o x t h e i n i t i a l i n t e r v a l which rep resen ts t h e oppos i te number -x as a c u t , say J ( - x ) . F i n a l l y

i n t h e g i v e n p a r t i a l o r d e r i n g , t h e un ion I ( x ) u J ( - x ) y i e l d s , when x runs through t h e r e a l s , cont inuum many incomparable i n i t i a l i n t e r v a l s (example due t o BONNET 1975). 0

2.2. TOPOLOGY ON IN IT IAL INTERVALS; BASIC CLOPEN SET; OPEN SET AND CLOSURE

L e t A be a p a r t i a l o rde r ing ; we s h a l l ex tend t o t h e s e t 3 ( A ) o f i n i t i a l i n t e r - va l s o f A , t h e topo logy a l r e a d y i n t r o d u c e d i n ch.1 § 8 f o r t h e s e t o f a l l s e t s o f n a t u r a l i n t e g e r s , i . e . f o r g ( N ) where N reduces t o t h e i d e n t i t y , o r f r e e

p a r t i a l o r d e r i n g on n a t u r a l i n t e g e r s . Le t F, G be two f i n i t e subsets o f t h e base I A I . L e t UF be t h e s e t o f those i n i t i a l i n t e r v a l s o f A which i n c l u d e F and a r e d i s j o i n t f rom G . For F and

G empty, we o b t a i n t h e e n t i r e s e t 5 (A) . Note t h a t UF i s non-empty i f f each element i n F i s < o r I (mod A) w i t h each element i n G . The i n t e r s e c t i o n UF n UF, i s UFuF, . Consequently, d e f i n i n g an open s e t t o be

any un ion o f U se ts , t hen t h e i n t e r s e c t i o n o f any two open s e t s i s s t i l l an open se t : we have a topo logy on 'j ( A ) . The complement o f a U s e t i s a un ion o f U se ts , t hus an open s e t ; so t h a t o u r U s e t s a r e c lopen se ts : b o t h open and c losed. We c a l l t hese U t h e b a s i c c lopen s e t s . T h i s topo logy i s Hausdor f f : see ch.1 8.

Consider an o r d i n a l sequence o f i n i t i a l i n t e r v a l s Hi , where i < d and d i s an o r d i n a l , which we can assume t o be a r e g u l a r a leph. We say t h a t an i n i t i a l i n t e r -

va l H i s t h e l i m i t o f t h e sequence Hi , i f f o r e v e r y e lement x i n t h e base, e i t h e r x belongs t o H , and then x be longs t o Hi f r om some p o i n t on ; o r

x be longs t o t h e complement o f H , and then x be longs t o t h e complement o f Hi f rom some p o i n t on. Then t h e c l o s u r e o f a s e t S o f i n i t i a l i n t e r v a l s , i s d e f i n e d as t h e s e t o f a l l

l i m i t e lements f o r a l l convergent o r d i n a l sequences o f elements o f S . We l e t i t t o t h e reader t o v e r i f y , as i n ch.1 5 8 , t h a t t h i s topo logy i s compact. So t h a t t h e c lopen s e t s a r e e x a c t l y t h e f i n i t e un ions o f b a s i c c lopen s e t s .

G

G

G G ' Gu G '

2.3. ISOLATED ELEMENT, SPARSE SET OF INITIAL INTERVALS

L e t S be a s e t o f i n i t i a l i n t e r v a l s o f A . An element X o f S i s s a i d t o be

i s o l a t e d (mod S ) , i f t h e r e e x i s t s an open se t , and consequent ly a b a s i c c lopen s e t U such t h a t X i s t h e o n l y e lement i n t h e i n t e r s e c t i o n SnU . A s e t S o f i n i t i a l i n t e r v a l s i s s a i d t o be sparse, i f e v e r y subset S ' o f S

c o n t a i n s a t l e a s t an i s o l a t e d e lement (mod S ' ) .

Now we are able t o complete by topo log ica l considerat ions, our p ropos i t i on 1.1. Le t A be a p a r t i a l order ing; then the fo l l ow ing cond i t i on i s equ iva len t t o any o f

o f cond i t ions (1 ) t o (4) i n 1.1 :

(T) the se t 3 ( A )

sets (POUZET, unpublished; uses axiom o f choice).

0 Condit ion (T) imp l ies 1 .1 . (3 ) . Indeed suppose t h a t t h i s ( 3 ) i s no t v e r i f i e d , i . e .

t h a t the chain Q o f r a t i o n a l s i s embeddable i n J ( A ) . Ca l l H those i n i t i a l i n t e r v a l s o f A which form a chain isomorphic w i t h Q under embeddabil ity; then

c a l l K any i n i t i a l i n t e r v a l o f A which i s the union o f a s t r i c t l y inc reas ing

W-sequence (under embeddabi l i ty) o f i n t e r v a l s H . The se t o f these K i s non-

empty (obvious) and has no i s o l a t e d element. Indeed each K i s the topo log ica l l i m i t o f an o-sequence o f H , and between any two o f these H , we can take an o -sequence o f o ther i n t e r v a l s H , whose union i s a K . So t h a t cond i t i on (T) i s f a l s i f i e d .

Conversely 1.1.(4) imp l ies (T) . Indeed suppose t h a t (T) i s f a l se , i . e . t h a t there e x i s t s a se t o f i n i t i a l i n t e r v a l s which does no t contains any i s o l a t e d element. Ca l l t h i s se t , which i s obviously i n f i n i t e . We c la im t h a t there e x i s t s an e le -

ment x i n t he base I A I , such t h a t the s e t Sx formed o f those elements i n

S which contain x , and S x , formed o f those elements which do no t con ta in x , are both i n f i n i t e and w i thout any i s o l a t e d element. Indeed take i n S two d i s t i n c t elements and x belonging t o one o f them and no t

t o the other. Then consider the basic clopen sets o f a l l i n i t i a l i n t e r v a l s which

contain x and a l l those which do no t contain x . The existence o f an i so la ted element i n Sx o r i n S x , would imply the existence o f an i s o l a t e d element i n S

Now def ine as fo l lows the func t i on f . Take f ( x ) = 0 f o r the preceding x . Then

take f o r Sx an element y p lay ing the same r o l e as x i n S ; and s i m i l a r l y an

element z p lay ing the same r o l e f o r Sx , ; and then l e t f ( y ) = 1 and f ( z ) =

= -1 ; and so fo r th , us ing successively the fou r i n t e r v a l s before -1, between -1 and 0, between 0 and 1, a f t e r 1 ; and so exhausting a l l the se t o f ra t i ona ls . Then we ob ta in our func t i on f - l mapping the r a t i o n a l s i n t o the base I A l , w i t h r < s

(mod Q) always g i v i n g (image of r) < o r I (image o f s) (mod A) . 0

o f i n i t i a l i n t e r v a l s o f A , i s sparse, w i t h respect t o the

S

2.4. Let A be an i n f i n i t e , f i n i t e l y f r e e p a r t i a l o rder ing i n which Q i s no t

embeddable. Then the se t o f i n i t i a l i n t e r v a l s o f A BONNET 1975; uses axiom o f choice; generalizes ch.6 5 1.6.We g i ve POUZET's proof.

0 Le t E be the base o f A . By the preceding 2.3, the se t 5 ( A ) o f i n i t i a l

Chapter 7 185

i n te rva l s i s sparse w i t h respect t o the given toDology. Use as fo l lows the Cantor- Bendixson procedure (us ing choice axiom). F i r s t take an ordinal-indexed sequence

of a l l i s o l a t e d elements Hi o f I) (A) . To each Hi l e t us associate a basic

clopen se t UG(!) , where F ( i ) and G ( i ) are two f i n i t e subsets o f E , such t h a t Hi i s the on ly element o f 3 ( A ) which belongs t o t h i s basic clopen se t . Removing from 3 (A) a l l these i s o l a t e d elements, i t then remains a sparse subset

denoted by ( 1 ( A ) ) ' , the f i r s t d e r i v a t i v e o f 'j (A) . I t e r a t i n g the procedure, l e t us denote by H a l l the i s o l a t e d elements i n t h i s f i r s t de r i va t i ve ; j i s

an o rd ina l index, s t r i c t l y g rea ter than a l l the preceding ind ices i . Again we

ob ta in f o r each j a basic clopen se t UG( j ) such t h a t H i s the on ly element i n the f i r s t de r i va t i ve , belonging t o t h i s clopen se t . I t e r a t i n g t h i s t o successive der iva t ives , which are def ined e i t h e r by removing a l l i so la ted elements, o r i n the case o f a l i m i t o rd ina l rank, by tak ing the i n te rsec t i on o f preceding der iva t ives , we f i n a l l y reach the empty se t . Now no t i ce t h a t dur ing the procedure, a given basic clopen se t UF can occur

only once. Indeed the f i r s t t ime t h a t t h i s clopen se t i s used i n order t o remove G an element Hk o f 'j (A) , then t h i s UF can conta in some Hi w i t h i < k , but

cannot contain as an element any o f the fo l l ow ing H w i t h j > k . Consequently the c a r d i n a l i t y o f 3 (A) cannot exceed the c a r d i n a l i t y o f the se t o f a l l ordered

pa i r s (F,G) o f f i n i t e subsets o f E ; hence cannot exceed Card E . 0

F(1)

j

F(J) j

G

j

2.5. L e t A be a denumerable p a r t i a l order ing; then the fo l l ow ing two condi t ions

are equivalent:

(1) A i s f i n i t e l y f r e e and does n o t admit an embedding o f Q ; (2) the s e t o f i n i t i a l i n t e r v a l s o f A i s denumerable.

We use axiom o f choice t o deduce ( 2 ) from (1). 0 I f A has a denumerable f ree 'subset o r i f Q i s embeddable i n A , then there

are continuum many i n i t i a l i n t e r v a l s : hence ( 2 ) imp l ies (1). The converse i s a

p a r t i c u l a r case o f the preceding p ropos i t i on 2.4. 0

2.6. Le t A be an i n f i n i t e p a r t i a l ordering; then the fo l l ow ing two condi t ions are equ iva len t (uses axiom o f choice): (1) A i s f i n i t e l y f ree and does no t admit an - e m b e d d m . -- Q ;

( 2 ) t o t a l l y ordered reinforcement o f A i s embeddable.

0 The cond i t i on ( 2 ) imp l ies t h a t every t o t a l l y ordered reinforcement o f A i s

scattered, hence ( 2 ) imp l ies (1) by 1.1 above (cond i t ions (1) and ( 2 ) ) . Conversely, i f A s a t i s f i e s our ( l ) , then again by 1.1, cond i t ions (1) and ( 3 ) , the p a r t i a l o rder ing o f i n i t i a l i n t e r v a l s o f A does no t admit an embedding o f Q .

there e x i s t s a sca t te red chain o f the same c a r d i n a l i t y as A , i n which every

Hence by 1.3, there e x i s t s a t o t a l l y ordered reinforcement B o f t h i s p a r t i a l o rder ing o f inc lus ion , i n which Q i s no t embeddable. On the one hand, by 2.4,

B has the same card ina l as A . On the o ther hand, f o r every t o t a l l y ordered reinforcement A ' o f A , the chain o f i n i t i a l i n t e r v a l s o f A ' i s a r e s t r i c t i o n o f the p a r t i a l o rder ing o f i nc lus ion f o r i n i t i a l i n t e r v a l s o f A , hence a res-

t r i c t i o n o f B . F i n a l l y A ' i s embeddable i n B .

5 3 - EVERY DIRECTED WELL PARTIAL ORDERING HAS A COFINAL RESTRICTION ISOMORPHIC WITH A DIRECT PRODUCT OF REGULAR ALEPHS (POUZET)

3.1. Le t A be a d i rec ted we l l p a r t i a l order ing. L e t 8 = Cof A and l e t F be a co f i na l subset o f l e a s t c a r d i n a l i t y , hence o f c a r d i n a l i t y . Then under these

condi t ions and m3dulo the axiom of choice: (1) there e x i s t s a -sequence bi ( i < E ( ) wi thout r e p e t i t i o n , o f elements forming a co f i na l subset o f F ;moreover t h i s sequence i s bad w i t h respect t o the

converse o f A ; i .e . i f i< j < 1 then (2) f o r each element c i n F there e x i

quence w i t h values i n F , beginning w i t h c , and f o r which noe lement o f A ~

an upper bound; we say t h a t t h i s sequence i s unbounded;

( 3 ) given two unbounded s t r i c t l y inc reas ing 8-sequences i n F , there e x i s t s a

t h i r d s t r i c t l y inc reas ing r -sequence i n which the f i r s t two are embeddable, i n the sense o f ch.4 5 2; (4) the i dea ls generated by the unbounded s t r i c t l y inc reas ing r-sequences i n F

cons t i t u te a d i rec ted p a r t i a l o rder ing w i t h respect t o inc lus ion ; moreover t?.

union o f these idea ls includes F . 0 (1) T o t a l l y order F according t o i t s c a r d i n a l i t y . In the thus obtained 8-se - quence o f the ai ( i < 7/ ) , f o r each index i , remove those a. f o r which

j > i and a . < ai (mod A ) . This ex t rac ted sequence y i e l d s a co f i na l subset

G o f F . Thus G has c a r d i n a l i t y and our ex t rac ted sequence i s a i$-sequence s a t i s f y i n g (1).

(2) Le t bi be the preceding &-sequence s a t i s f y i n g (1). By ch.4 0 3.4 (ex t rac- t i o n theorem), there e x i s t s a s t r i c t l y inc reas ing (mod A) ex t rac ted -sequence.

Suppose t h a t there e x i s t s an upper bound u o f the elements o f t h i s sequence:

we can assume t h a t u belongs t o the c o f i n a l se t G o f (1); hence u i s one o f the bi's : con t rad i c t i on proving t h a t our sequence i s unbounded. Now l e t c be an element o f F , and ui ( i < r ) an unbounded s t r i c t l y increa-

s ing '6-sequence i n F . Since A i s d i rec ted , t o each i we associate an e le - ment vi i n the co f i na l se t F , which i s a comnon upper bound o f c and ui . Moreover, we can requ i re t h a t the vi be d i s t i n c t . For t h i s , l e t vo be a common upper bound o f c and uo . I n general, g iven an index i ( 1 4 i < )

b j > or I bi (mod A ) ; -

.-

J J

Chapter 7 187

and v . ( j< i ) , note t h a t the se t o f these v has c a r d i n a l i t y s t r i c t l y less

than , and so i s no t c o f i n a l i n A . Thus there e x i s t s an x i n F w i t h

x 7 o r I w i t h each v . . Take vi t o be a common upper bound o f c, ui and x . This makes yi

t i o n theorem ch.4 0 3.4, we e x t r a c t a s t r i c t l y inc reas ing

sequence o f the ui i s embeddable i n i t , t h i s s t r i c t l y inc reas ing r-sequence i s unbounded. Furthermore, i t i s formed o f elements g rea ter than c (mod A) . (3 ) Le t ui and vi be two unbounded s t r i c t l y inc reas ing &--sequences. Since A

i s d i rec ted , t o each i associate a common upper bound wi o f ui and vi . By

the procedure i n the preceding (2 ) , we can ensure t h a t the ly, e x t r a c t a s t r i c t l y inc reas ing r -sequence from the sequence o f the

(4) Follows immediately from (2) and (3 ) . 0

J j

J d i s t i n c t from the v . ( j < i ) . F i n a l l y , again using the extrac-

J Y-sequence. Since the

wi are d i s t i n c t . F ina l -

wi .

3.2. The fo l l ow ing propos i t ions u n t i l 3.9 are due t o POUZET 1979, unpublished. Le t A be a d i rec ted p a r t i a l o rde r ing and l e t LJ, = Cof A . There e x i s t idea ls Ai ( i < W, ) o f A w i t h Ai included i n A f o r i< j < ad , and such t h a t the union o f the Ai - i s A ; f i n a l l y Cof Ai < Cof A f o r each i (uses axiom o f choice).

0 By the axiom o f choice, associate t o each f i n i t e subset X o f the base o f A , a common upper bound o f the elements o f X , which we designate by u(X) . In the case t h a t X i s the s ing le ton o f the element x , we requ i re t h a t u(X) = x . For each subset D o f the base, we de f ine D ( l ) = the se t o f the u(X) f o r a l l f i n i t e subsets X o f D . Then f o r each in tege r h , l e t O(h+l) = the se t o f the

u(X) f o r a l l f i n i t e subsets X o f D(h) . F i n a l l y l e t D+ be the union o f the D(h) , which we sha l l c a l l the c losure o f D w i t h respect t o the func t ion u . Note t h a t D+ includes D and t h a t Dc D ' imp l ies D+c D" ; and f i n a l l y t h a t

D+ i s equipotent w i t h D . S t a r t i n g now w i t h a c o f i n a l (mod A) subset C o f minimum c a r d i n a l i t y cd& = Cof A , l e t ci ( i < C.dM ) be a we l l -o rder ing o f C . To each i < W4 , i t su f f i ces t o associate Di = the se t o f the c . ( j < i ) , then the closure se t Di+ , and

t o take fo r Ai

j -

J the i dea l , o r d i rec ted i n i t i a l i n t e r v a l generated by Di+ .

3.3. L e t A be a d i rec ted p a r t i a l o rder ing f o r which there e x i s t s a c o f i n a l (mod A) r e s t r i c t i o n isomorphic w i t h the d i r e c t product (see ch.4 5 7) o f a f i n i t e number

o f wel l -order ings. Then we can always suppose:

( i ) t h a t each o f these wel l -order ings i s isomorphic w i t h a regu la r aleph; ( i i ) t h a t the c a r d i n a l i t i e s o f two d i s t i n c t wel l -order ings are d i s t i n c t ; (iii) t h a t the l a rges t o f these alephs i s the c o f i n a l i t y o f A (ZF su f f i ces ,

provided t h a t Cof A e x i s t ) .

0 By hypothesis, each element representable by a sequence o f n coordinates: t = (tl, ..., t n ) (n i n tege r ) , w i t h t $ t ' (mod A) i f f tid ti f o r each i = 1, ..., n . Note t h a t the wel l -order ings, o r chains, numbered from 1 t o n , are no t themselves r e s t r i c t i o n s o f A . We obta in a wel l -ordered r e s t r i c t i o n o f A only by tak ing a l l the coordinates cons-

t a n t , except f o r one which var ies . S t a r t w i t h a co f i na l (mod A) se t C o f minimum c a r d i n a l i t y , hence o f c a r d i n a l i t y

equal t o Cof A . Replace each x o f C by an upper bound t ( x ) , where t i s a choice func t ion whose values are some o f the preceding sequences o f n coordi-

nates. We obta in again a se t which i s c o f i n a l and has c a r d i n a l i t y Cof A . Moreover, we can replace each o f the n chains by a co f i na l r e s t r i c t i o n , hence

by a we l l -o rder ing whose order type i s a regu la r aleph. And i f two chains, say the f i r s t and the second, have the same c a r d i n a l i t y , hence are isomorphic, then we

again ob ta in a c o f i n a l s e t by only tak ing those t f o r which t2 i s equal t o tl : t h i s amounts t o decrease the number n o f chains by 1 . F i n a l l y , note t h a t one o f the chains necessar i l y has c a r d i n a l i t y a l l the o ther chains have s t r i c t l y smal ler c a r d i n a l i t y . The c a r d i n a l i t y o f the d i r e c t product i s the l a r g e s t c a r d i n a l i t y of these var ious chains. 0

t o f the c o f i n a l subset under considerat ion i s

Cof A , whi le

3.4. Le t be a d i rec ted p a r t i a l order ing, having a c o f i n a l r e s t r i c t i o n which i s

the d i r e c t product o f a f i n i t e p o s i t i v e number n o f i n f i n i t e wel l -order ings. Le t B be an i dea l i n A w i t h Cof B < Cof A . Then there e x i s t s an intermediate i dea l C such t h a t B 5 C c A w i t h Cof C < Cof B . Moreover C has a c o f i n a l r e s t r i c r t i o n which e i t h e r i s the d i r e c t product o f n-1 wel l -order ings ( i f n & 2) , which has a maximum element ( i f n = 1) (uses axiom o f choice, i n order t h a t c o f i -

n a l i t i e s always e x i s t ) .

0 The proo f i s obvious i f Cof A = W thus Cof B = 1 , so t h a t B has a maximum

element. I n the fo l low ing , we s h a l l assume t h a t Cof A 3 Let B ' be a co f i na l (mod B ) r e s t r i c t i o n o f l e a s t c a r d i n a l i t y , hence o f cardina- l i t y Cof B . To each x i n B ' associate a unique upper bound t ( x ) belon- g ing t o the d i r e c t product mentioned i n our hypotheses. The se t o f these t ( x ) has c a r d i n a l i t y a t most say the f i r s t chain, has c a r d i n a l i t y a t l e a s t product i s c o f i n a l i n A . Each wel l -order ing, o r chain, can be reolaced by a c o f i n a l r e s t r i c t i o n whose order type i s a regu la r aleph. Since Cof B < Cof A , there e x i s t s an element u i n the f i r s t chain, which i s g rea ter than a l l values

i n B ' the element (u, t2 ,..., t n ) where the ti ( i = 2 ,..., n) are the coordinates o f t ( x ) o ther than the f i r s t one. It now su f f i ces t o take C t o be the i n i t i a l i n t e r v a l generated by these new elements, there being ,< Cof B many such.

A

-

W1 .

Cof B . One o f the wel l -order ings o f our d i r e c t product, Cof A >Co f B , since our d i r e c t

tl(x) (modulo t h i s f i r s t chain). Associate t o each x

Chapter 7 189

More prec ise ly , i n order f o r C t o be d i rec ted , we complete the previous by

associat ing t o each f i n i t e subset X o f C a common upper bound o f the elements o f X , and by i t e r a t i n g t h i s . To achieve the i t e r a t i o n , note t h a t our f i r s t chain has length a t l e a s t equal t o Cof A >, Lu , so t h a t there always ex i s t s an upper bound f o r denumerably many successive

3.5. Consider a regu la r aleph 8 w 1 . Le t A be a d i rec ted p a r t i a l order ing and Ai ( i < ) be i dea ls i n A with Ai c A f o r i c j ; moreover the union

Ai i s A . Suppose t h a t Cof Ai < Cof A f o r each index i < 8 , and

t h a t each Ai number o f wel l -order ings. Under these condi t ions, Cof A = 8 and hence Cof A i s a regu la r aleph (uses axiom o f choice).

0 Since i s regu la r , we can assume t h a t there i s a constant integer,

say a co f i na l r e s t r i c t i o n o f Ai ( i < 3 ) . More p rec i se l y we choose n t o be the minimum poss ib le i n tege r corresponding t o any c o f i n a l sequence of i dea ls

t i s f y i n g our hypotheses. We have Cof A >/ ; indeed since Cof Ai< Cof A , a c o f i n a l se t i n A necessa-

r i l y has elements no t belonging t o Cof A i s less than o r equal t o the sum o f the Cof Ai . Suppose f i r s t t h a t Cof Ai se t o f ind ices i c o n s t i t u t i n g a if-sequence, c o f i n a l i n the o r i g i n a l sequence. I n t h i s case Cof A 2 / r d = Max(r ,d) . But d i s s t r i c t l y less than Cof A

by hypothesis, thus d < 2( : i f not, we would have d < Cof A 4 d . Hence we have

Cof A .4< 8 , and by the previous discussion Suppose now t h a t Cof Ai i s s t r i c t l y inc reas ing i n i , t h i s being always possi- b le by ex t rac t i ng a c o f i n a l sequence o f such i . Now apply the preceding 3.4, rep lac ing A by Ai+l and B by Ai . Thus there e x i s t s an intermediate i dea l

Ci having a co f i na l r e s t r i c t i o n which i s the d i r e c t product o f n-1 well-orde-

r i ngs . Replacing the Ai by Ci , we again s a t i s f y a l l the hypotheses o f our p ropos i t ion , i nc lud ing Cof Ci 6 Cof Ai < Cof A , bu t w i t h n-1 ins tead o f n , con t rad i c t i ng the m in ima l i t y o f n . Hence we have n = 0 , so each Ai has a

maximum element mi , and then the s e t o f the mi i s a c o f i n a l subset o f A having card ina l 8 . 0

u ' s i n the f i r s t chain. 0

j -

has a c o f i n a l r e s t r i c t i o n which i s the d i r e c t product o f a f i n i t e

3 w 1 n , f o r the f i n i t e number o f wel l -order ing5 whose d i r e c t product cons t i tu tes

Ai sa-

Ai , f o r each i 4 8 . On the o ther hand,

i s constant and equal t o a c e r t a i n aleph d , f o r a

Cof A = 8 .

3.6. L e t A be a d i r e c t e d p a r t i a l o r d e r i n q and o( a non-zero o r d i n a l w i t h d6 r e g u l a r a1 eph. Then t h e q u a s i - o r d e r i n g o f e m b e d d a b i l i t y (see ch.4 5 2) f o r t h e s t r i c t l y i n c r e a - s i n g w d - s e q u e n c e s i n A , i s f i n i t e l y f r e e (uses denumerable subse t axiom). -

0 Suppose on t h e c o n t r a r y t h a t t h e r e e x i s t denumerably many s t r i c t l y i n c r e a s i n g

ul,. . . , ui ,. . (i i n t e g e r ) which a r e m u t u a l l y incomparable under embeddab i l i t y . To each ui and t o each i n t e g e r j # i , a s s o c i a t e an e lement

a. o f ui such t h a t no t e r m o f u . i s g r e a t e r t han o r equal t o ai,j (mod A) . Since wd i s r e g u l a r and s t r i c t l y g r e a t e r t han 0 , t a k e an e lement which i s g r e a t e r t han (mod A) a l l t h e a . . (i f i x e d , j a r b i t r a r y i n t e g e r # i). The bi a r e m u t u a l l y incomparable, hence A i s n o t f i n i t e l y f r e e : c o n t r a d i c t i o n . 0

c3& -sequences

1 ,j J bi i n ui

1 ,J

3.7. L e t A be a p a r t i a l o r d e r i n g and a non-zero o r d i n a l such t h a t ad i s a r e g u l a r a leph. Then f o r eve ry i n c r e a s i n g ( w i t h r e s p e c t t o embeddab i l i t y )

s t r i c t l y i n c r e a s i n g W, -sequence i n A i n which each i s embeddable.

L e t ui be t h e g i v e n sequences and ai L e t bo = ao,O ; l e t bl be t h e f i r s t t e r m i n u1 which i s a common upper bound

o f bo , aO,l and al,l . I n aenera l f o r each i < Wo( , l e t bi be t h e f i r s t t e rm i n ui which i s a common upper bound o f t h e b . ( j < i) and t h e a

( j 4 i) . Then t h e wd -sequence o f t h e bi s a t i s f i e s t h e conc lus ion . 0

W, -sequence o f s t r i c t l y i n c r e a s i n g W&-sequences i n A , t h e r e e x i s t s a

- be t h e jth t e r m i n ui (i, j < Wa).

,j

J i ,j

3.8. L e t A be a d i r e c t e d w e l l p a r t i a l o r d e r i n e w h o s e c o f i n a l i t y i s a r e g u l a r

a leph w4 w i t h o< # 0 . L e t F be a c o f i n a l subse t o f l e a s t c a r d i n a l i t y , hence o f c a r d i n a l i t y L+ . Consider t h e i d e a l s which a re generated, each o f

them by an unbounded s t r i c t l y i n c r e a s i n g L3* -sequence w i t h va lues i n F (see ch.4 5 1.8). Then t h e r e e x i s t s a s e t o f a t most c30( many i d e a l s which, under

i n c l u s i o n , c o n s t i t u t e a d i r e c t e d w e l l p a r t i a l o r d e r i n g o f c o f i n a l i t y s t r i c t l y l e s s than c.+ ; moreover t h e un ion o f t h e bases o f t hese i d e a l s i n c l u d e s F

(uses axiom o f cho ice ) . By 3.6, t h e i d e a l s cons ide red i n t h i s p r o p o s i t i o n fo rm a f i n i t e l y f r e e p a r t i a l

o r d e r i n g under i n c l u s i o n , which fu r the rmore i s d i r e c t e d by 3 . 1 ( c o n d i t i o n ( 3 ) ) . I n t h i s p a r t i a l o r d e r i n g , t a k e a we l l - f ounded c o f i n a l r e s t r i c t i o n , hence a d i r e c t e d w e l l p a r t i a l o r d e r i n g : see ch.2 5 5.1. There a r e a t most wOc many i d e a l s

Chapter 7 191

under consideration: see 2.4 above. Every element of F belongs t o an unbounded

s t r i c t l y increasing mM -sequence, by 3.1, condition ( 2 ) . Thus the union of our ideals includes F , and th is holds as well when taking the s e t of ideals in a cofinal s e t . The cofinal i ty of the partial ordering under inclusion, for our ideals, i s a t most the cardinal of th i s s e t , hence a t most ad. . Yet th i s cofinality cannot be equal t o then ex is t unbounded s t r i c t l y increasing W & -sequences of ideals , contradicting 3.7. Thus the cofinal i ty i s s t r i c t l y less t h a n w+ . 0

W d , since by 3.1, condition ( 2 ) , there would

3.9. THEOREM O N THE COFINAL RESTRICTION OF A D I R E C T E D WELL PARTIAL ORDERING (POUZET 1979, unpubl ished) Let A be a directed well partial ordering. Recall that Cof A i s a regular aleph (ch.4 5 10) . There exis ts a cofinal res t r ic t ion of A which i s isomorphic with the direct product of f in i te ly many d is t inc t regular alephs, the largest of which i s (uses axiom of choice) . 0 We already know tha t a directed well par t ia l ordering of countable cof inal i ty , e i ther has a maximum element and so i t s cofinality i s 1 ; or has a cofinal res t r ic t ion isomorphic with W : see ch.4 5 5.5. Proceed by induction. Let wd ( o( non-zero) be an inf in i te regular aleph, and assume tha t , for each regular aleph k s t r i c t l y less t h a n W* and each directed well partial ordering X with cofinal i ty k , there exis ts a cofinal res- t r ic t ion of alephs, the largest of which i s k . Apply the preceding 3.8. Let by the ideals in 3.8. By the induction hypothesis, the cofinality of B , which i s s t r i c t l y less than wd and regular, i s e i ther equal t o 1 or t o w a , with an ordinal f i s t r i c t l y less t h a n q . If Cof B = 1 , then there ex is t s a s t r i c t l y increasing ud -sequence which generates a cofinal subset of A , and we are finished. Suppose now that regular and there ex is t s an integer n and a cofinal res t r ic t ion C o f B which i s isomorphic with a direct product of n which i s wf i . Consequently, t o each ideal in base \ C l , bijectively associate an n-tuple of coordinates ( t l,...,tn) , each of which runs through a d is t inc t regular aleph, the f i r s t coordinate running through W R . For each n-tuple, l e t I ( t l . . . . , tn) designate an unbounded, s t r i c t l y increasing

Cof A

X which i s a direct product of f in i te ly many d is t inc t regular

B be the directed well par t ia l ordering formed

Cof B = Wn . Then by our induction hypothesis, C d f i i s

regular alephs, the largest of A which i s an element of the

tl

W , -sequence i n A which generates the i dea l whose coordinates are tl, ..., tn

(obviously I Given two n-tuples, the i n e q u a l i t y product, i . e . the se t o f i n e q u a l i t i e s

the cond i t ion t h a t the sequence I ( t l,...,tn) i s embeddable i n I ( t i ,..., t,!,)

(mod A) , i n the sense o f ch.4 5 2 . We are now going t o replace each U&-sequence under considerat ion by an ex t rac ted

sequence o f the same length, w i t h the aim i n mind o f b i j e c t i v e l y assoc ia t ing t o

each (n t1 ) - tup le ( i ,tl,. . . ,tn) where i c a, and the t are as prev ious ly , an element o f the base \ A I designated by a ( i ,tl,. . . ,tn) , which more p rec i se l y

w i l l be a term i n the sequence

the i nequa l i t y

i . e . the se t o f i n e q u a l i t i e s

t o the i n e q u a l i t y a ( i , t l ,..., t n ) 6 a ( i ' , t i ,..., t,!, ) modulo A . F i r s t o f a l l , i t i s easy t o ensure t h a t our elements

are U,, many n-tuples ( t l , . . . ,tn) and wR < Q~ . We f i r s t take the

a(0, t l,...,tn) ensure t h a t the cho sen values a(u, t l,...,tn) are a l l d i s t i n c t , as we l l as being d i s t i n c t from the already def ined values a(i,tl, ... , tn) f o r i < u . We do t h i s wh i le respect ing the cond i t ion t h a t a(u,tl, ..., t n ) sequence values a(i,tl, ..., t n ) f o r i < u . A l l t h a t i s easy because cdd i s regu la r , hence f o r each u < wd , no u-sequence ex t rac ted from the sequence I i s

co f i na l . Secondly, we s h a l l ensure tha t , f o r two incomparable n-tuples (tl,.. . ,tn) and

(ul ,..., un) , every value o f a term i n w i t h every value o f a term i n

( t l,...,tn) , designate by ( u l,...,un) any and a l l n- tuples which are incomparab le w i t h ( t l ,..., t n ) . Thus the sequence I ( t l ,..., t n ) i s incomparable, w i t h respect t o embeddabil ity (mod A) , w i t h each sequence I(ul, ..., un) . Take a term

o f the f i r s t , f o r which no term o f the second i s g rea ter (mod A) than i t , and designate i t by b(t l ...., tn;ul ,..., u,) . Then, f i x i n g

(ul, ..., un) vary, and replace the sequence I( t l , ..., t n ) by an ex t rac ted se-

quence o f the same length od , bu t beginning w i t h an upper bound o f a l l the preceding b . This i s poss ib le because there i s a t most many b . Th i rd ly , we sha l l ensure tha t , f o r two d i s t i n c t n-tuples s a t i s f y i n g

i s a choice func t ion , several sequences y i e l d i n g a same i d e a l ) .

n \ (tl, ..., t ) < ( t i , ..., t,',) modulo the d i r e c t i s equ iva len t t o tl 6 t i ,. . . , tn 6 t; ,

I ( t l....,tn) . We sha l l do t h i s i n a manner t h a t

( i , t l ,... ,tn)Q ( i ' , t i ,... ,tA) modulo the new d i r e c t product,

i s i ' , tlb t i , ... , tnd t,', , i s equ iva len t

a a r e d i s t i n c t , s ince there

t o be a l l d i s t i n c t . I n general, g iven an ord ina l u < 0, we

belongs t o the

I(t l, ..., t n ) and i s s t r i c t l y g rea ter (mod A) than the already defined

I ( t l ,..., t n ) i s incomparable (mod A ) I(ul,. . . ,un) . To do t h i s , f o r each n- tup le

( t l ,..., t n ) , l e t

(tl, ... ,tn) < (ul ,..., un) , there do n o t e x i s t values a i n I ( t l ,..., t n ) and b i n

I(ul, ...,un) s a t i s f y i n g the opposite i n e q u a l i t y a 7 b (mod A ) : the on ly

Chapter 7 193

possibi l i t ies being incomparability To do t h i s , for each n-tuple (tl ,..., tn) l e t (ul ,..., un) designate any and a l l d i s t inc t lesser n-tuples: t h u s the sequence in any the second i s greater, and continue as in the preceding paragraph. Fourthly and f ina l ly , we shall define by induction on i the elements a ( i , t l , . . . , tn) t e the beginning term of the sequence n-tuple and suppose that (ul ,..., un) < ( t l ,..., t n ) . Then s e t a ( O , t l ,..., t n ) t o be the f i r s t term of

I ( t l , ..., tn) which i s s t r i c t l y greater (mod A) than a l l a(O,ul, ..., u n ) fo r

(ul ,..., u n ) < ( t l ,..., tn) . These values of a for f i r s t coordinate 0 shall be kept in the end. To the (n+l)-tuple ( l , O , ..., 0 ) associate the f i r s t term of I ( 0 , ..., 0 ) which i s s t r i c t l y greater t h a n than any value for any n-tuple since there i s no element in the base I A I which i s greater (mod A) t h a n every element of the sequence I ( 0 , . .. , O ) . Note that by the paragraph "thirdly", the value a ( l , O , ..., 0) will be incomparable (mod A) with a l l those a with f i r s t coordinate 0 followed by a non-zero n-tuple ( u1 ,. . . , u n ) . Indeed by construction a(l,O, ..., 0 ) i s n o t s t r i c t l y less (mod A) than these values; nor i s i t strictly greater, since otherwise the inequality o f two terms would be i n t h e opposite sense o f the inequality o f the n-tuples Now l e t (tl ,..., tn) be an n-tuple, and suppose that a( l ,ul ,..., un) has alrea-

dy been defined for a l l n-tuples

a ( l , t l ,..., t n ) take the f i r s t term o f I ( t l ,..., t,) which i s s t r i c t l y greater

(mod A ) than a(O,tl ,..., tn) and which i s s t r i c t l y greater t h a n a ( l , u l ,..., u n )

fo r a l l n-tuples (ul, ..., u,) < (t l , ..., tn) ; and f ina l ly which i s n o t s t r i c t l y

lesser (mod A J t h a n a(O,vl, ..., v n ) fo r any n-tuple ( v l , ..., v n ) . This l a s t condition is possible by the f a c t tha t no element in I A I i s an upper bound of the sequence I ( t l , ... , tn) . Note t h a t , by the paragraph "secondly", th i s value

a ( l , t l , ..., tn) will be incomparable (mod A) with a l l the values a w i t h f i r s t coordinate 0 or 1 followed by an n-tuple incomparable with ( t l , ..., t,) . Our value a ( 1 , t l , . . . , tn) will also be incomparable with the a with f i r s t coordinate 0 followed by an n-tuple n o t s t r i c t l y lesser (mod A) by construction, nor i s i t s t r i c t l y greater, since otherwise the inequality of the terms would be in the opposite sense o f the

a I b (mod A) or the inequality a < b (mod A ) .

i s not embeddable (mod A) I ( t l , . . . , tn) I(ul, ..., u n ) . Take a term of the f i r s t sequence, for which no term of

as previously announced. To the (n+l)-tuple ( O , O , . . . ,0) associa- I (0 ,..., 0 ) . Let ( t l ,..., tn) be an

a(O,ul,. . . , u n ) has already been defined for a l l n-tuples

a(O,O,. . . ,0) and which i s not s t r i c t l y less (mod A)

a(O,ul ,..., u n ) (ul ,..., u n ) . This i s possible,

(0 ,..., 0) < (ul ,..., u n ) .

( u l ,..., u n ) < (tl ,... ,tn) . Then for

( v l ,..., v,) 7 (tl ,..., t,) . Indeed i t i s

inequality of the n-tuples. I n general, l e t k be an ordinal s t r i c t l y lesser t h a n , and suppose t h a t a l l values of a with f i r s t coordinate < k have been defined. To the ( n + l ) - tuple ( k , O , ..., 0) associate the f i r s t term of I ( 0 , ..., 0) which i s s t r i c t l y greater than a(i,O, ..., 0) for a l l i < k , and which i s n o t s t r i c t l y lesser (mod A) than any a( i ,u l ,..., u n ) . The value a ( k , O ,..., 0 ) will be incomparable (mod A) with a l l values a having f i r s t coordinate i < k followed by a non- zero n-tuple. Now l e t ( t l ,..., tn) be an n-tuple and suppose t h a t the values a(k,ul ,..., u n ) have been defined for a l l n-tuples

a ( k , t l ,..., tn)

t h a n a ( i , t l , ..., tn) for a l l i < k and s t r i c t l y greater than a(k,u l , . . . . u n )

for a l l n-tuples

less (mod A) than a ( i ,v l ,..., v n ) fo r any i < k and any n-tuple ( v l ,..., v n ) . The reader can verify the desired incomparabilities. 0

(ul ,..., t in) C (tl ,..., tn) . Then take

t o be the f i r s t term of I ( t l ,..., tn) which i s s t r i c t l y greater

(ul ,..., u n ) < (t l ,..., tn) , and f inal ly which i s n o t s t r i c t l y

3.10. Recall that fo r every well-founded par t ia l ordering A , we have

Cof H t AS Cof A : see ch.2 0 7 . 2 . S t r i c t inequality Cof H t A c Cof A can be obtained, even in the case of a directed well par t ia l ordering A . 0 Consider the ordinal product W 1. 4

A = W x 1. . Then H t A = ol. (h, so that Cof H t A = c3 . Yet

Note t h a t , in agreement with the preceding theorem, we have the direct product

and then the direct product

1 Cof A = a, . 0

W l x k J as a cofinal res t r ic t ion of A .

Chapter 7 195

5 4 - SZPILRAJN CHAIN, DENUMERABLY SZPILRAJN CHAIN

We say t h a t a chain in which C ly ordered reinforcement of A in which C i s n o t embeddable. If C i s a Szpilrajn chain, then so i s every chain equimorphic with C , as Well as with the converse C- . Among the f i n i t e chains, only the empty chain and the singleton chain are Szpi7- r a j n chains. Indeed, for each integer p z / 2 , the f ree partial ordering on p elements, when reinforced, necessarily yields a chain of cardinality p .

C i s a Szpilrajn chain i f , fo r every par t ia l ordering A

i s not embeddable, there ex is t s , modulo the axiom of choice, a total-

4 .1 . I f a chain C does n o t admit an embedding of C + l , then for each pa 2 , the chain C+p i s n o t a Szpilrajn chain. For example W+2 , ~ + 3 ,... are n o t Szpilrajn chains. 0 Star t with the par t ia l ordering formed by the chain C followed by p mutual- l y incomparable elements. 0

4.2. The chain W of the natural integers i s a Szpilrajn chain. 0 We shall show t h i s , equivalently, for the converse a- . A partial ordering A , in which 12- i s n o t embeddable, i s well-founded: see ch.2 5 2 .4 , using dependent choice. Then there exis ts a well-ordered reinforcement of A : see ch.2 5 4 .4 (axiom of choice).

4.3. The chain Q of the rationals i s a Szpilrajn chain. This i s another s ta tement of the equivalence of conditions (1) and ( 2 ) of 1 .3 (modulo the axiom of choice).

4.4. We say t h a t a chain C i s denumerably Szpilrajn i f C i s denumerable and i f , for every denumerable par t ia l ordering A in which C i s n o t embeddable, there ex is t s a to ta l ly ordered reinforcement of A , in which C i s not embeddable. If a chain i s denumerable and Szpilrajn, then i t i s denumerably Szpilrajn. However, we shall see below that + 1 i s denumerably Szpilrajn without being Szpi 1 rajn. We see immediately that o + p , with an integer pa 2 , i s n o t denumerably Szpil rajn.


4 .5 . The chain C = CAJ + a- i s not denumerably Szpilrajn.

0 Star t with A = w -. LJ and B = A- = U . W - . Take the par t ia l ordering formed by A and B with each element of one incomparable with each element of the other. Then C i s not embeddable in th i s partial ordering; yet C i s embeddable in every to ta l ly ordered reinforcement, by ch.6 § 3.7. 0

4.6. Let C , D be two chains. rf C + l .a& l + D are Szpilrajn, then so i s C + l + D ; same statement for "denumerably Szpilrajn" (JULLIEN 1969).

0 Let A be a par t ia l ordering in which C + l + D i s not embeddable. Call H the i n i t i a l interval of those elements x above which there i s a chain in A isomorphic w i t h 1+D ; the element x being the minimum of this chain 1 + D . Then C + l i s n o t embeddable in H . Moreover, the chain l + D i s n o t embeddable in the final interval H ' complementary t o H . Take a to ta l ly ordered reinforcement of A formed by an i n i t i a l interval , which i s a reinforcement of H yet in which C + l i s n o t embeddable; followed by a final interval , which i s a reinforcement of H ' yet in which l + D i s n o t embeddable. Then C + l + D i s n o t embeddable in such a chain. 0

I n particular, the chain Z of the positive and negative integers i s Szpilrajn, since (13- = d - + 1 and LJ = 1 + W are Szpilrajn.

4 .7 . The ordinal w+1 i s denumerably Szpilrajn, b u t i s n o t Szpilrajn: see JULLIEN 1969.

0 We shall prove tha t w+1 i s denumerably Szpilrajn. S ta r t with a denumerable partial ordering A in which c3+1 i s n o t embeddable. I f A has a maximum element, then even LJ i s n o t embeddable in A , and so by the above 4.2, there exis ts a to ta l ly ordered reinforcement of A , i n which w i s not embeddable. If A has no maximum element, then take an w-sequence of elements ai ( i integer) , forming a cofinal s e t in A with the condition a . > o r I ai (mod A ) fo r a l l i and j 7 i . Possibly we can have only a f i n i t e cofinal s e t of maximal elements, thus a f i n i t e sequence. Associate t o each i the rest r ic t ion Ai of A t o those elements which are

6 a i are mutually dis joint and their union i s the base of A . None of the admit an embedding of w , since otherwise, taking in account the maximum ai , the ordinal w+1 would be embeddable. By 4 . 2 , there exis ts a to ta l ly ordered reinforcement Bi of Ai , in which W i s not embeddable. Then the sum of the Bi according to increasing i i s a to ta l ly ordered reinforcement of A , in which 0+1 i s not embeddable. 0

J

b u t .$ ao, ... , 6 ai-l . The bases of the Ai Ai

Chapter 7 197

0 Now we prove t h a t w +1 i s n o t Szpilrajn. Take an uncountably inf in i te se t U . To each enumeration f , without repet i t ion, of a denumerable subset of U and to each natural integer i , associate the ordered pair ( f , i ) . Take as base E

the union of U a n d the s e t of these ordered pairs. Set ( f , i ) < ( f , j ) , for the same enumeration f , i f i < j in the usual orderina of integers. Moreover for each ordered pair , s e t ( f , i ) < f ( i ) : th i s second term being an element of U . Hence by t rans i t iv i ty we have ( f , i ) < f ( j ) for a l l j > i . Apart from these cases, two d is t inc t elements of E shall be incomparable. Call A the thus defined partial ordering on E : we easi ly see that w i s embeddable in A , b u t not 0+1 . Let B be a to ta l ly ordered reinforcement of A , based on E . There exis ts a t least one element u of U having a t l eas t denumerably many elements x < u (mod B ) with x E U . Take an enumeration f of such elements. The ordering A , hence B as well, admits as a res t r iot ion the chain of the ( f , i ) where f i s fixed and i = 0,1,2, ... : a chain of order type which has the element u as an upper bound (mod B ) . Thus w+1 i s embeddable in B . 0

COROLLARIES due t o JULLIEN 1969 (1) If a denumerable chain C sa t i s f ies the s t r i c t inequality C + l > C (with respect t o embeddability), then C + 1 i s n o t Szpilrajn.

0 Either C has no maximum element. Then use the preceding argument: replace W

by C ; replace each f by an injection of the base I C l onto a denumerable subset of U ; the inequality i < j being made modulo C. Or C has a maximum element. Denote by C-1 the chain C a f t e r removing i t s maximum: C-1 does n o t admit an embedding of C . Then i t suffices t o consider the partial ordering obtained from C-1 by adding two maximal and mutually incomparable elements. 0

-

( 2 ) The ordinal 0 .2 , and more generally 0 . p (with p integer >/ 2 ) & n o t Szpilrajn.

0 Take an uncountably inf in i te s e t U which i s the union of uncountably many disjoint denumerable subsets U k . On each U k take a chain isomorphic with w . Furthermore any two elements of U which belong t o two d i s t inc t U k , will be hcomparable. Then i t suffices t o terminate as for 0+1 . 0

4.8. I t i s proved by GALVIN and MAC KENZIE in 1969 (unpublished) that w i s the only denumerable Szpilrajn ordinal. I t i s proved by BONNET in 1971 (see BONNET, POUZET 1982) that: The only denumerable Szpilrajn chains ( u p t o equimorphism) are the following: the chain Q of rationals; the scattered chains defined as follows by ordinal products and sums: in general PI = 0 ; P2 = w - . CJ ; P - CJ. W - . W ; 3 -


p. ,+1 . = p i-. (,d f o r each in t ege r i so r ord ina l . Furthermore P, = P i ( i i n t e g e r ) ; more genera l ly , given any denumerable limit ordinal u , take any -sequence of ind ices i forming a cofinal set i n u : then we s e t pU = Z. p i . Fina l ly , for each preceding P i ( i countable o rd ina l ) , the converse Pi i i

Szpi l ra jn ; furthermore each sum P i - + P i i s Szp i l r a jn . Note t h a t

For any two countable ord ina ls i and j > i , we obviously have P i + P Szpi l ra jn ; however t h i s i s already mentioned, s ince the l a t t e r sum i s equimorphic w i t h P. .

, and more generally f o r each countable succes-

-

l + P i = P i , so the l a t t e r sum i s obviously Szpi l ra jn by 4.6 above.

j

J

4.9. Let C be a denumerable chain, I the denumerable i d e n t i t y r e l a t ion ; in o ther words, the denumerable an t icha in . We say t h a t the ordered pa i r un iversa l ly Szpi l ra jn i f , f o r every pa r t i a l ordering A in which ne i ther C nor I A in which C is embeddable (even when u s i n g the axiom of choice) . For example ( Q , I ) , where Q i s the chain of r a t i o n a l s , i s universally Szpilrajn

(C,I) i s

i s embeddable, t he re does not e x i s t any t o t a l l y ordered reinforcement of

by 2.6.

If C i s a denumerable chain, and C' equimorphic with C , and universally Szp i l r a jn , t h e n so i s (C',I) . Given a denumerable sca t t e red chain C , the ordered pair: (C-+C,I universa l ly Szp i l r a jn . Same r e s u l t w i t h C+C- . 0 First not ice t h a t C-+C i s not embeddable i n C : use the sam

(C,I) i s

i s not

argument as as i n ch.5 0 3 . 3 . Then take a pa r t i a l ordering, composed of two chains C and C- w i t h d i s j o i n t bases, every element of the f i r s t being incomparable w i t h every element of the second. I t su f f i ces t o see t h a t C-+C i s a t o t a l l y ordered reinforcement, y e t not embeddable in our pa r t i a l ordering. 0

I t i s proved by BONNET in 1971 ( see BONNET, POUZET 1982) t h a t ;he only denume- rab le chains (up t o equimorphism) which y i e ld with the iden t i ty r e l a t ion a univer- s a l l y Szp i l r a jn p a i r , a r e the chain Q o f r a t iona l s and the preceding products Pi ( i countable o r d i n a l ) ,

4.10. The reinforcement of R being defined f o r an a r b i t r a r y r e l a t ion R , l e t us mention possible genera l iza t ions of t h e notion of Szpi l ra jn chain. Using ch.5 0 2.4 and 2.7, r eca l l t h a t R is a p a r t i a l ordering i f f R does not admit an embedding of: A2 = r e l a t ion always (+) w i t h c a rd ina l i t y 2 ; A3 = re f lex ive binary cycle w i t h c a rd ina l i t y 3 ; A4 = consecut iv i ty w i t h c a rd ina l i t y 3 . Moreover R i s a chain

A1 = the r e l a t ion w i t h c a rd ina l i t y 1 and value (-) ;

Chapter 7 199

i f f R does n o t admit any embedding of A1 or A2 or Ag o r A5 = identity relation with cardinality 2 . Now we can t ranslate as follows the definition of a Szpilrajn chain. An arbitrary binary relation R i s Szpilrajn i f f : ( i ) R does not admit any embedding of A1 or A2 o r A3 or A5 ( t ranslate: R i s a chain); ( i i ) for every relation A , i f A $ R and A does n o t admit any embedding of A1 or A2 or A3 or A4 ( t ranslate: A i s a partial ordering), then there exists a reinforcement of A which does n o t admit any embedding of R or A1

I n the so "translated" def ini t ion, there only occur general notions as embedding and reinforcement, which are defined f o r arbi t rary relations. Hence we have many possible generalizations, for instance by replacing A1 to A5 by an arbitrary f ini te sequence of f i n i t e re la t ions. We do n o t know i f th i s yields interesting problems.

or A2 or A3 o r A5 .

EXERCISE 1 - TUKEY'S THEOREM O N PAIRS OF D I R E C T E D PARTIAL O R D E R I N G S

Let us consider two directed par t ia l orderings A and B . A function f from A into B i s said t o be convergent, i f every cofinal s e t (mod A ) , say X , i s transformed into a cofinal s e t f"(X) , modulo B . Following T U K E Y 1940, we shall prove, modulo the axiom of choice, the equivalence between (1) and ( 2 ) : (1) there exis ts a convergent function from A i n t o B , and similarly a conver- pent function from B into A ; ( 2 ) there exis ts a directed par t ia l ordering C which i s a common extension of A and B (more exactly A and an isomorphic image of B ) , such t h a t both A - and B are cofinal res t r ic t ions modulo C . 1 - To see that ( 2 ) implies ( l ) , i t suffices t o associate, t o each element of A , an element f ( x ) of B with f ( x ) >/ x (mod C ) . 2 - Conversely, suppose that A and B sa t i s fy (1). Denote by f a convergent function from A into B . Consider the directed par t ia l ordering of final intervals of A , ordered by reverse'inclusion, then similarly the directed par- t ia l ordering of f inal intervals of B ; then denote by C the direct product of these two partial orderings, which i s obviously a directed partial ordering. Now define as follows the embedding of A in to C . To each element x in A , associate the ordered pair ( A x , B x ) where Ax i s the final interval of elements 3 x (mod A ) and Bx i s the final interval of the upper bounds (mod B ) of a l l the images f ( t ) where t 3 x (mod A ) .

x

Note t h e f o l l o w i n g easy lemma. Given x i n A , t h e s e t o f those elements t a x (mod A) such t h a t f ( t ) < o r I f ( x ) (mod B ) do n o t c o n s t i t u t e a c o f i n a l s e t i n A . I n o t h e r words, t h e r e e x i s t s an e lement u ( x ) such t h a t e v e r y t i s < o r 1 u(x ) . So t h a t i f v ( x ) i s a common upper bound o f x and u ( x ) (mod A) , then the f i n a l i n t e r v a l >/ v ( x ) (mod A ) i s t rans fo rmed under f " i n t o a subset o f t he

f i n a l i n t e r v a l >, f ( x ) (mod B ) . 3 - From t h e p reced ing lemma, e a s i l y deduce t h a t A , when mapped i n t o C as

p r e v i o u s l y i n d i c a t e d , y i e l d s a c o f i n a l r e s t r i c t i o n i n C . F i n a l l y repea t the

same argument by exchanging A and B and u s i n g t h e convergent f u n c t i o n from

B i n t o A . As a c o r o l l a r y , no te t h a t c o n d i t i o n (1) (and o b v i o u s l y ( 2 ) ) i m p l i e s t h a t Cof A = Cof B .

E X E R C I S E 2 - KRASNER'S LEMMA ON DIRECTED PARTIAL OROERINGS

L e t A be a d i r e c t e d p a r t i a l o r d e r i n g , and l e t u = Cof A . F o l l o w i n g KRASNER

1939, l e t us prove, modulo t h e axiom o f choice, t h a t , g i v e n a c o f i n a l r e s t r i c -

t i o n A ' w i t h minimum c a r d i n a l i t y Card A ' = u , t h e r e e x i s t s an i n j e c t i v e ,

i n c r e a s i n g f u n c t i o n whose domain i s t h e s e t o f a l l f i n i t e subsets i n u , ordered

under i n c l u s i o n , and whose range i s A ' . We can always suppose t h a t A ' = A , so t h a t u = Card A = Cof A . 1 - As a p r e l i m i n a r y e x e r c i s e , cons ide r t h e case where know by ch.4 5 5.5 t h a t t h e r e e x i s t s a c o f i n a l r e s t r i c t i o n o f A which i s isomor- p h i c t o W ( o r t o 1 , i n which case a l l i s obv ious ) .

Then u = 0 = s e t o f i n t e g e r s . Enumerate t h e base o f A as an u - s e q u e n c e

ao, ..., ai,.. (i i n t e g e r ) . Then d e f i n e as f o l l o w s t h e d e s i r e d f u n c t i o n Take t h e i n t e g e r s 0 and 1 o n t o f ( 0 ) = a. and f(1) = al , and no te t h a t

f ( i ) = ai, w i t h i' g r e a t e r t han o r equal t o i . Take t h e p a i r (0 , l ) onto

t h e ai w i t h t h e l e a s t i ndex i , p r o v i d e d t h a t ai i s an upper bound (mod A)

o f a. = f ( 0 ) and al = f ( 1 ) . Then t a k e t h e i n t e g e r 2 o n t o f ( 2 ) = ai where i i s t h e l e a s t i ndex n o t y e t used: so t h a t t h i s i i s g r e a t e r t han o r equal t o 2 . D e f i n e as p r e v i o u s l y t h e images o f t h e p a i r s {0 ,2) and {1,2) . Then t a k e t h e 3-element s e t {0,1,2} o n t o t h e element ai w i t h t h e l e a s t index i , p r o v i d e d t h a t t h i s ai i s n o t y e t used and i s a common upper bound (mod A) o f

t h e images under f o f t h e t h r e e p a i r s { O , l } , {0,2} , {1,2). . Note t h a t a l l elements i n A a r e f i n a l l y used; moreover t h e procedure always works, s ince f o r

any f i n i t e s e t o f elements a i n A , t h e r e e x i s t denumerably many common upper bounds.

2 - Now ex tend t h e same p rocedure t o t h e genera l case. Consider u = Cof A =

A i s denumerable. We

f .

Card A as an ordinal-indexed sequence, or u-sequence o f terms i < u .

Chapter I 20 1

Well-order the base of A as a u-sequence ai . Then repeat the previous procedure for ordinals 0 and 1 . I n general, l e t i be an ordinal index s t r i c t l y less t h a n u ; suppose t h a t f i s already defined for a l l f i n i t e sets of indices j < i . Moreover suppose t h a t f ( j ) = a . with j' greater t h a n or equal t o j , for every such j . Take i onto f ( i ) = a i , where i ' a i i s the leas t index not yet used. Then take every pair i i , j j where j -z i , onto the element a with the least possible index, provided tha t th i s a i s n o t yet used, and i s a common upper bound (mod A ) of the images f ( i ) and f ( j ) . Then take every 3-element set \ i , j , k } where j , k < i , onto the element a with the leas t possible index, provided that a i s n o t yet used, and i s a common upper bound of the images under f of the three pairs included in { i , j , k ) ; and so on. The procedure always works, since for any f i n i t e s e t of elements a in A , on one hand there ex is t u many comnon upper bounds; on the other hand the number of f ini te sets of indices less than or equal t o i < u i s i t s e l f < u . - I n connection with the two preceding exercises, say that two partial orderings A, B have same convergence type i f f condition ( l ) , or equivalently ( 2 ) in exercise 1 holds. Then the reader may be interested t o know tha t , in the case of cofinality w tha t there ex is t ( 2 t o the power ol) many different convergence types; or that there exist only three possible convergence types, namely: ( i ) the case of an increasing cofinal wl-sequence; ( i i ) the type of the direct product W x W1 ; ( i i i ) the type of the ordered s e t (under inclusion) of a l l f i n i t e subsets of

J'

, then i t i s consistent with the axioms of ZF t o suppose ei ther

W 1 ; this resul t i s due t o TODORCEVIC 1984.

EXERCISE 3 - THE POSSIBLE COFINAL RESTRICTIONS OF A DENUMERABLE PARTIAL O R D E R I N G

I n the particular case of a denumerable, directed par t ia l ordering, we know by ch.4 5 5.5 that there exis ts e i ther a maximum, or a cofinal res t r ic t ion which i s isomorphic with I n the general denumerable case, show that the possible cofinal res t r ic t ions are the following: (1) an antichain (with f i n i t e or denumerable cardinal i ty) ; (2 ) the union of components each of which i s isomorphic with w , with mutual incomparability for elements belonging t o different components; ( 3 ) the W-tomic t r e e , i . e . the t ree with denumerably many edges from each vertex; ( 4 ) the union of 2 or 3 among the preceding components, with mutual incomparabi- l i ty .

EXERCISE 4 - O N SIERPINSKI'S PARTIAL ORDERING

In connection with Sierpinski's counterexample (ch.3 5 3.1), a par t ia l ordering is called a Sierpinski par t ia l ordering, i f we are i n the framework of ZF plus choice plus continuum hypothesis, so that the chain of real numbers i s equipotent

t o w . Take a chain R o f rea l s , such t h a t each f i n a l i n t e r v a l o f R has conti-

nuum many elements; a lso take an a r b i t r a r y we l l -o rder ing U on the base I R I , iso - morphic w i t h the ord ina l w . Then the S ie rp insk i ( p a r t i a l order ing) A associated w i t h the b icha in (R,U)

x < y (mod A) i f x i s s t r i c t l y less than y both modulo R and U , and x l y (mod A) i f x, y are ordered i n opposite senses modulo R and U . 1 - We see t h a t every S ie rp insk i A i s f ree i n te rp re tab le i n the associated b i - chain; y e t the converse i s fa lse : take the t ranspos i t i on exchanging t w o incompa-

rab le elements (mod A) . We see t h a t every chain o r an t icha in i n A i s countable:

compare w i t h ch.4 5 1.6. Show t h a t A i s d i rec ted : given two elements x, y there e x i s t s z > x,y modulo

U and modulo R . Show t h a t Cof A = w = continuum; indeed given a countable se t D , there are on ly countably many elements having an upper bound i n D . More prec i se l y a s e t C i s c o f i n a l (mod A) i f f C has continuum many elements i n each f i n a l i n t e r v a l o f R . 2 - Show t h a t i f f i s a l oca l automorphism o f A (ch.9 5 1.7) and i f Dom f has continuum c a r d i n a l i t y , then there e x i s t s a r e s t r i c t i o n o f f t o continuum many elements which i s a l oca l automorphism o f the associated b icha in (R,U) . To see it, show t h a t (1 ) given a countable subset H o f Dom f , there always ex i s t s an element u i n (Dom f ) - H , such t h a t u i s an upper bound o f H

modulo U , and f ( u ) i s an upper bound o f the image se t f " (H) modulo U ; and ( 2 ) every common l o c a l automorphism o f U and A i s s t i l l a loca l automorphism o f R ( i t su f f i ces t o prove t h i s f o r an automorphism on 2 elements).

3 - By an easy refinement o f the argument i n ch.1 exerc. 4, show t h a t there e x i s t ( 2 t o the continuum power) many co f i na l sets C (mod A) having pairwise

only a countable i n te rsec t i on . Now take f o r R a r i g i d chain o f rea l s (BONNET 1978 p. 7 ) , i n t h i s sense that

every l oca l automorphism f o f R moves on ly countably many elements x

( f ( x ) # x) . Assume wi thout p roo f t h a t R ex i s t s and can have continuum many elements i n each rea l i n t e r v a l (non-singleton). Consequently, given two cof inal

sets C, C ' i n the previous fami ly , every l oca l isomorphism from A/C i n t o A / C '

By completion, ob ta in a fami ly of ( 2 t o the continuum power) co f i na l sets C such tha t , f o r any two such sets C, C ' , there ex i s t s i n R a f i n a l in te rva l Fc,cl y i e l d i n g a countable i n te rsec t i on CnC'n FC,!, . Consequently, a local isomorphism from A / C i n t o A / C ' cannot have a co f i na l domain. F i n a l l y f o r every co f i na l se t X (mod A) , there e x i s t s a t l e a s t one C i n the family, such t h a t the i n te rsec t i on C n X be co f i na l (mod A) (comnunicated by POUZET).

i s def ined on the same base by the condi t ions that

has on ly a countable domain.

203

CHAPTER 8

BARRIER, BARRIER SEQUENCE, FORERUNNING, EMBEDDABILITY THEOREM FOR SCATTERED CHAINS, BETTER PARTIAL ORDERING

5 1 - BARRIER, B A R R I E R PARTITION THEOREM, SUCCESSIVE ELEMENTS, SQUARE OF A BARRIER

Let E be a denuneyable set of natural numbers , and U a set of finite non- empty subsets o f E , whose union i s E . The s e t U i s a b a r r i e r i f f : (1) the elements o f U are mutual ly non-inclusive; ( 2 ) f o r every i n f i n i t e subset X o f E , there e x i s t s a f i n i t e i n i t i a l i n t e r v a l

o f X t o u . Examples. For E take the se t o f a l l na tura l sumbers , t h i s case being the most f requent ly encountered. Then the se t o f s ing le tons o f the in tegers i s a b a r r i e r .

S im i la r l y , the se t o f unordered p a i r s o f the i n tege rs i s a ba r r i e r , as we l l as

the s e t o f p-element subsets o f the integers, f o r any given p o s i t i v e i n tege r p . Another example o f a b a r r i e r : the union o f the s e t o f pa i r s whose minimum i s 0 , and the se t o f 3-element subsets w i t h minimum 1, ... , and f o r each in teger i a 2 , the se t o f i-element subsets w i t h minimum i - 2 .

( i n i t i a l w i t h respect t o the usual o rder ing o f the in tegers ) which belongs

1.1. L e t U be a b a r r i e r and P an i n f i n i t e subset o f the union U U . Then

the s e t o f those elements o f U which are subsets o f P i s b b a r r i e r . Moreover,

every b a r r i e r included i n U i s thus obtained, i . e . by tak ing an i n f i n i t e se t

o f in tegers and r e s t r i c t i n g U t o those elements which are subsets o f t h i s i n f i - n i t e se t .

The elements o f U which are subsets o f P are mutua l l y incomparable w i t h

respect t o i nc lus ion . Moreover, f o r every i n f i n i t e subset X o f P , there ex i s t s a f i n i t e i n i t i a l i n t e r v a l o f X which i s an element o f U . For the second assert ion, l e t V be a b a r r i e r included i n U , and P be the union u V . For each element r o f U which i s a subset o f P , take an i n - f i n i t e subset R o f P which begins w i t h r : there e x i s t s an element o f V

which i s an i n i t i a l i n t e r v a l o f R , and t h i s can on ly be r . 0

Note tha t , i f U i s a b a r r i e r and P an i n f i n i t e subset o f u U , then the se t o f i n te rsec t i ons o f P w i t h the elements o f U does no t necessar i l y form a ba r r i e r . For instance, s t a r t i n g w i t h the b a r r i e r U o f p a i r s o f integers, l e t P be the se t o f s t r i c t l y p o s i t i v e in tegers (we remove zero). Then we ob ta in


b o t h the singleton of {1,2} in which the singleton i s included.

1 , coming from the pair { 0 , l ) , and for example the pair

1.2. Let U be a barrier and r a f i n i t e se t of natural integers which i s a- per i n i t i a l interval of an element of U . Then the se t of elementsof the form x - r , where x i s any element of U beginning with the i n i t i a l interval r , i s a barrier.

0 Let V be the se t of our difference se t s x - r . The union of the elements of U beginning with the i n i t i a l interval r i s in f in i te ; so u V i s in f in i te . Any two of these elements are incomparable with respect t o inclusion: th i s subsists when removing the i n i t i a l interval r . Finally, for every inf in i te subset P of the union u V , which necessarily begins with an integer s t r i c t l y greater t h a n Max r , there ex is t s a f i n i t e i n i t i a l interval y of P , such that the union r L J ~ belongs t o U , so y belongs t o V . 0

1.3. Every barr ier i s lexicographically well-ordered: th i s i s a particular case o f ch.3 0 2 . 1 . In other words, the se t of the elements of a barr ier , when ordered lexicographically s tar t ing with the usual ordering of the integers, forms a denumerable well-ordering.

Every barrier U order type o f the well-ordering of the elements of U when ordered lexicographical ly. for example, the barr ier o f the singletons has rank W . The barr ier of the p-element subsets ( p = positive integer) has rank u p . The barr ier formed of the pairs w i t h minimum 0 , and the 3-element subsets w i t h minimum integer i 3 2 , the i-element subsets with minimum i - 2 , has rank 0 . for each denumerable indecomposable ordinal, i .e. each power of a, say there ex is t s a barrier of rank 8 The lexicographic ranks of barr iers are exactly a l l the ordinals up ( p = posi- t ive integer) and ( w'( ) . p , where o( i s a denumerable ordinal and p a posi- t ive integer (ASSOUS 1974).

t h u s has a lexicographic rank, in the sense of ch.3 5 2 .1 : the

1 , and for each w

, (POUZET 1972"). More generally:

1 .4 . B A R R I E R PARTITION THEOREM Let U be a barr ier and E be the union of U . Partition the elements of U

into two complementary se t s U ' and U" . Then there ex is t s an inf in i te subset H f E such that the elements of U which are subsets of H , a l l belong to U ' or a l l belong to U" (NASH-WILLIAMS 1968). Note tha t these elements form a barr ier by 1.1. I n par t icular , we obtain RAMSEY's theorem by considering a positive integer p and taking for U the s e t of a l l

Chapter 8 205

p-element subsets o f E . 0 Given two d i s t i n c t e lements o f U , one i s never i n c l u d e d i n t h e o t h e r , hence one i s never an i n i t i a l i n t e r v a l o f t h e o t h e r : t h e theorem now f o l l o w s f rom NASH-WILLIAM'S theorem i n ch.3 5 2.4. 0

0 We g i v e ano the r more d i r e c t p r o o f , due t o COROMINAS i n 1 9 7 0 , unpubl ished.

Assoc ia te t o each b a r r i e r U i t s l e x i c o g r a p h i c rank c( ( U ) . A b a r r i e r has rank o

Now l e t U be a b a r r i e r o f rank M ( U ) > o . Assume t h a t t h e theorem ho lds f o r

eve ry b a r r i e r o f rank s t r i c t l y l e s s e r than o( (U ) : we s h a l l prove t h a t i t ho lds

f o r U . L e t uo be t h e l e a s t i n t e g e r i n E = u U f o r which t h e s i n g l e t o n o f uo does n o t be long t o U . L e t U1 be t h e s e t o f those e lements o f t h e fo rm x - { u o } , where x i s an e lement o f U beg inn ing w i t h t h e i n t e g e r uo : by 1 . 2 above, t h e

s e t U1 i s a b a r r i e r . We have t h e s t r i c t i n e q u a l i t y o ( (U1) < M(U) : indeed

i n t h e l e x i c o g r a p h i c o r d e r i n g , a f t e r t h e e lements o f U which beg in w i t h uo , we s t i l l have p o s t e r i o r e lements o f U which beg in w i t h a s t r i c t l y g r e a t e r

i n t e g e r . The p a r t i t i o n o f U i n t o U ' and U " g i v e s a p a r t i t i o n o f t h e s e t

o f elements o f U beg inn ing w i t h uo , hence a l s o a p a r t i t i o n o f U1 . By t h e

i n d u c t i o n hypo thes i s , s i n c e t h e l e x i c o g r a p h i c rank o f i s s t r i c t l y l e s s than

H ( U ) , t h e r e e x i s t s an i n f i n i t e subset o f t h e un ion w U1 , such t h a t , l e t t i n g V1 be U1 r e s t r i c t e d t o t h e subsets o f HI , then t h i s b a r r i e r V1 e i t h e r i s i n c l u d e d i n U ' o r i n U " (whose e lements have o b v i o u s l y been modi-

f i e d by removing t h e i r minimum i n t e g e r uo ) . Accord ing t o t h e case, we say t h a t uo i s l i n k e d t o U ' o r l i n k e d t o U " . L e t u1 be t h e l e a s t i n t e g e r i n H1 = i~ V1 ; t h e s i n g l e t o n o f u1 does n o t

be long t o U , s i n c e t h e r e e x i s t s an element o f U which begins w i t h uo, u1 . L e t Up be t h e s e t o f t hose e lements o f t h e fo rm x - {ul) , where x i s an

element o f U which begins w i t h u1 and i s a subset o f H1 . Then U2 i s a

b a r r i e r whose rank i s s t r i c t l y l e s s than N ( U ) . The p a r t i t i o n o f U i n t o U' and U" g i v e s a p a r t i t i o n o f t h e s e t o f elements o f U beg inn ing w i t h u1 , hence a l s o a p a r t i t i o n o f U2 . By t h e i n d u c t i o n hypo thes i s , t h e r e e x i s t s an i n - f i n i t e subset H2 o f u U 2 , such t h a t , l e t t i n g V 2 be U2

r e s t r i c t e d t o t h e subsets o f H2 , then t h i s b a r r i e r V2 e i t h e r i s i n c l u d e d i n U ' o r i n U" (whose elements have been m o d i f i e d by removing t h e i r minimum

i n t e g e r u1 ). Accord ing t o t h e case, we say t h a t u1 i s l i n k e d t o U ' o r l i n k e d t o U" . I t e r a t i n g t h i s , we have a s t r i c t l y i n c r e a s i n g &-sequence o f i n t e g e r s each o f which i s l i n k e d w i th e i t h e r U ' o r U" ; and t h e co r respond ing sequence of i n f i n i t e s e t s Hi , which i s dec reas ing w i t h r e s p e c t t o i n c l u s i o n . L e t H be

i f f i t i s a s e t o f s i n g l e t o n s ; i n t h i s case t h e theorem i s obv ious.

U1 H1

H~ -, {,+\ =

ui ,

an i n f i n i t e se t o f these f o r example t o U ' . Then H s a t i s f i e s our conclusion. Indeed, enumerate H as

uo, ul, u2, .. : the renumber the corresponding b a r r i e r s Vi . The elements o f U which are subsets o f

H remain pa i rw ise incomparable w i t h respect t o i nc lus ion . Moreover, given an i n f i - n i t e subset P o f H , t h i s P begins w i t h a c e r t a i n u (p i n tege r ) , corresponding w i t h the b a r r i e r V .And the i n i t i a l i n t e r v a l o f P which i s an element

o f U P

belongs t o the subset U' , t o which u i s l i nked . P The proo f i s f in ished; however i n order t o see t h a t i t on ly needs the axioms ZF , avoiding f o r instance the axiom o f dependent choice when we def ine the i n f i n i t e

sequence o f sets me t h a t Hi-l i s a l ready def ined. Among the i n f i n i t e subsets o f Hi-l which y i e l d f o r instance a b a r r i e r included i n U ' , choose those i n f i n i t e subsets which begin w i t h the l e a s t poss ib le i n tege r , say h t . Then among the i n f i n i t e subsets

1 which begin w i t h hi and y i e l d a b a r r i e r included i n U ' , choose those i n f i n i t e subsets which begin w i t h hi, hi where h: i s the l e a s t poss ib le in teger ; and

so on. F i n a l l y we take f o r Hi the se t o f a l l these h r ( i f i xed , k va r ies ) . 0

ui , a l l o f which are l i n k e d t o the same subset, say

here forming an ex t rac ted sequence o f the o l d ui , and ui

P

P + i i s the union o f the s ing le ton { u 1 and an element o f Vp+,,hence i t

Hi , we make prec ise the cons t ruc t ion o f each Hi . Assu-

1 2

1.5. SUCCESSIVE ELEMENT

Given two f i n i t e sets o f na tu ra l numbers r and s , we say t h a t r precedes s o r t h a t s succeeds r , o r t h a t s i s a successive element o f r , denoted by ra s , i f s i s obtained from r by adding on in tegers which are a l l s t r i c t -

l y g rea ter than Max r and then by removing Min r . For example, given two in tegers a, b, the s ing le ton o f a precedes the s ing le - ton of b i f f a < b . Given th ree in tegers a < b t c , the p a i r {a,b) pre-

cedes the p a i r \b,c) . (1 ) L e t U be a b a r r i e r and r, s two elements of U ; pu t m = Card r . I f Max r < Min s , then there e x i s t s a sequence o f m + l successive elements - o f U g o i n g f r o m r s ,say r = r o a r l Q r 2 Q ...a r , = s . 0 Le t I be an i n f i n i t e se t o f in tegers which begins w i t h the i n i t i a l i n t e r v a l r fo l lowed by s and f i n a l l y by in tegers l a r g e r than Max s , a l l belonging t o the union o f U . S t a r t w i t h ro = r and I,, = I . Then def ine I1 = I. minus

i t s minimum, then l e t r1 be the i n i t i a l i n t e r v a l o f I1 which belongs t o U , so t h a t roq rl . I t e r a t e u n t i l reaching rm = s . 0

(2 ) L e t U be a b a r r i e r and r, s be two a r b i t r a r y elements o f U . Then there

ex i s t s an element t o f U and two f i n i t e sequences o f successive elements o f U , the f i r s t sequence going from r t o t and the second going from s 2 t . 0 Take an a r b i t r a r y element t i n U w i t h Min t s t r i c t l y l a r g e r than both

-

- -

Chapter 8 207

Max r and Max s , then a p p l y p r o p o s i t i o n ( 1 ) above. 0

1.6. L e t U be a b a r r i e r , r be an element o f U and m = Card r . Given an i n t e g e r k l a r g e r than o r equal t o m , t h e r e e x i s t s a sequence o f m + l

success ive elements o f U ro = r 4 rl U . . . <1 rm and a sequence o f k + l success ive e lements of U , b o t h sequences hav ing same f i r s t

and same l a s t terms, say so = r QslQ ... a sk = rm . 0 F i r s t c o n s t r u c t a sequence o f k + l success ive terms beg inn ing w i t h r , say r = s o d sl(l ... Q sk , u s i n g an i n f i n i t e s e t o f i n t e g e r s which begins w i t h t h e

i n i t i a l i n t e r v a l r , a l l t h e i n t e g e r s be long ing t o t h e un ion o f U . Since k i s l a r g e r than o r equal t o m = Card r , t h e minimum o f l y l a r g e r than Max r . Using t h e p rev ious 1.5. (1) , we c o n s t r u c t t h e sequence of m + l success ive e lements o f U go ing f rom r t o rm = sk . 0

beg inn ing w i t h r , say

sk i s an i n t e g e r s t r i c t -

1.7. L e t h be a f u n c t i o n f rom t h e s e t G, o f non-negat ive i n t e g e r s i n t o G, . L e t U be a b a r r i e r and f a f u n c t i o n f rom U i n t o W , such t h a t f o r each

couple of success ive e lements x Q y we have f ( x ) s t r i c t l y l e s s than f(y) . Then t h e r e e x i s t two success ive elements S Q t i n U w i t h h ( f ( s ) ) l e s s than

o r equal t o f ( t ) (POUZET 1977) . - -

. . .

0 Suppose on t h e c o n t r a r y t h a t s at always i m p l i e s t h a t f(t),< h ( f ( s ) ) - 1 . S t a r t w i t h an a r b i t r a r y e lement s o f U , which s h a l l be f i x e d f o r t h e f o l l o - wing d i scuss ion , and l e t m = Card s . We s h a l l prove t h a t t h e r e e x i s t s an i n -

t e g e r k such t h a t , f o r eve ry sequence o f m + l success ive elements o f U

beg inn ing w i t h s , say so = s d s l c \ s2 d... (1 sm , t h e va lue f(s,) i s bounded by

choices f o r sl, ..., s, . Indeed, t h e va lue f(sl) i s bounded by t h e maximum h ( f ( s ) ) - 1 , which we de-

note by kl . The va lue f ( s 2 ) < h ( f ( s l ) ) -1 i s bounded by t h e maximum, denoted by k2 = Max(h (x ) ) -1 , where x runs th rough t h e i n t e r v a l 0,1,2, ..., kl . We con t inue t h u s l y u n t i l f(sm)< h(f(sm-l))-l , bounded by k, = Max(h (x ) ) -1 , where x runs th rough t h e i n t e r v a l 0,1,2,...,km-1 . We then t a k e k = km + 1 ,

We can always assume t h a t t h e bound ka m . Then by t h e p reced ing p r o p o s i t i o n ,

t h e r e e x i s t two sequences, say so = s 4 s l a ...(I sm and to = s Q tld . . . Q t k = sm . The f i r s t sequence r e q u i r e s t h a t f(s,) < k , and t h e second, g i v e n o u r hypotheses, r e q u i r e s t h a t f(s,) - f ( s ) 3 k , hence t h a t f(s,) 2 k : con t ra - d i c t i o n . 0

k , say f(s,)< k ; i n s p i t e o f t h e i n f i n i t e l y many p o s s i b l e

1.8. SQUARE AND POWER OF A BARRIER

Le t U be a b a r r i e r . To each ordered p a i r o f elements s , t o f U such t h a t

sa t , associate the union s u t . Note t h a t t h i s union uniquely determines s as t h a t element o f U which i s an i n i t i a l i n t e r v a l o f the union. S i m i l a r l y t

i s uniquely determined, as being the union w i t h i t s minimum element removed. The unions i n considerat ion are mutua l l y non- inclusive; moreover, every i n f i n i t e

subset o f the union se t o f U has such a union s y t as an i n i t i a l i n t e r v a l . Thus these unions form a b a r r i e r , c a l l e d the square o f U and denoted by U . For each p o s i t i v e i n tege r h , the power Uh i s d e f i n e d by i nduc t i on .

U = U and Uh+' = (Uh)2 . These not ions w i l l be used already i n 0 2 o f t h i s chapter, f o r b a r r i e r sequences.

2

1

§ 2 - BARRIER SEQUENCE, THEOREM ON THE MINIMAL BAD BARRIER SEQUENCE (NASH-W I LLIAMS)

GOOD OR BAD BARRIER SEQUENCE

Given a b a r r i e r U and a p a r t i a l o rder ing A , we def ine a b a r r i e r sequence,

more p rec i se l y a U-bar r ie r sequence i n A o r w i t h values i n A , t o be a

func t i on w i t h domain U such t h a t the range i s included i n the base \ A 1 . We a lso more simply speak o f a U-sequence. A b a r r i e r sequence f o f U i n A i s c a l l e d goodif there e x i s t two elements s , t i n U w i t h s q t and f s g ft (mod A) ; the sequence f i s c a l l e d bad otherwise.

I n the t r i v i a l case o f a b a r r i e r o f s ingletons, r e c a l l t h a t f o r u, v integers, we have {u) 4 i v ) i f f u < v i n the usual o rder ing o f integers.

We have again the no t i on o f good and bad a-sequence, i n the sense o f ch.4 5 2.2. Consequently, if A i s a we l l p a r t i a l order ing, then the b a r r i e r sequen-

ces whose domain i s the b a r r i e r o f s ing le tons are good: see ch.4 5 3.2.(2)

This r e s u l t does no t extend t o a l l ba r r i e rs . Indeed, l e t A be the we l l p a r t i a l o rder ing o f RADO (see ch.4 5 4.2) and U be the b a r r i e r o f a l l p a i r s o f non-negative in tegers . Then the fo l l ow ing func t ion from U A i s a bad b a r r i e r sequence:

f o r a l l in tegers x, y (x< y ) , our func t i on takes the p a i r {x,y) i n t o the couple o f in tegers (x,y-x+l) . 0 Two successive p a i r s are o f the form {x,y) and i y , z \ w i t h three in tegers

x ( y < z . They are taken respec t i ve l y i n t o the couples (x,y-x+l) and

(y,z-y+l) . These two couples have d i s t i n c t f i r s t terms. Therefore the second couple can be grea ter (modulo the we l l p a r t i a l o rder ing o f RADO) than the f i r s t on l y i f y >/ x + (y -x+ l ) = y+l : cont rad ic t ion . 0

non-negative

Chapter 8 209

2 . 1 . RESTRICTION OF A B A R R I E R SEQUENCE, PERFECT B A R R I E R SEQUENCE Let A be a partial ordering and U , V be barriers with V included in U . As with any function, we say tha t , given a U-sequence f , i t s res t r ic t ion t o V , denoted by f/V , i s the V-sequence taking for each element of V the same value as f . Let U be a barr ier and A a par t ia l ordering. A barr ier sequence f which i s a function from U into the base of A i s called perfect if( for each ordered pair of elements s , t in U , the condition s at implies f s 6 f t (mod A ) . Let U be a barr ier and f a function from U into a partial ordering. Then there ex is t s a barrier V included in U , such that the rest r ic t ion f/V ei ther perfect or bad (NASH-WILLIAMS 1965' p . 705) .

0 Consider the square U2 of the barr ier U , which i s formed of unions r = s u t where s , t are successive elements in U . Recall that each element r in U 2 determines s as the i n i t i a l interval of r which belongs t o U , as well as t , which i s r with i t s minimum removed. Partition the elements t i a l ordering) or not. By the barr ier par t i t ion theorem 1 . 4 , there exis ts an in f in i te s e t E of integers in which the elements of U 2 are a l l in the same class; and by 1.1 they constitute a barr ier V . According t o whether the class in consideration corresponds t o the condition or to i t s negation, the rest r ic t ion f/V i s e i ther perfect or bad. 0

r of U2 into two classes, according whether f s$ f t (modulo the par-

f s 6 f t

2 . 2 . INF-RESTRICTION, MINIMAL BAD SEQUENCE Let U, V be two barriers with V included in U , and l e t A be a partial ordering. A V-sequence g with values in A i s called an inf-restriction (mod A ) of the U-sequence f (with values in A ) , i f for each element x of V we have g x d fx (mod A ) .

We say that a barrier sequence f with domain U and with values in A i s minimal bad (mod A ) i f f f i s bad and i f every bad inf-restr ic t ion of f i s a res t r ic t ion. I n other words, for each barrier V included in U , each bad V-sequence g such that g x g fx s a t i s f i e s gx = fx (and not gx < fx ) for every x in V . This generalizes the notion of a minimal bad u-sequence: see ch.4 5 2 . 3 .


I n t h e case where U i s t h e t r i v i a l b a r r i e r o f s i n g l e t o n s , f becomes an a - s e - quence and we f i n d t h e min imal bad O-sequence o f ch.4 § 2.3.

THEOREM ON THE MINIMAL BAD BARRIER SEQUENCE

L e t A be a we l l - f ounded p a r t i a l o r d e r i n q and f a bad b a r r i e r sequence w i t h

va lues i n A . f

and i s minimal bad (mod A) (NASH-WILLIAMS 1965', lemma 24 p . 707; uses dependent

cho ice ) . Th i s genera l i zes t h e theorem on t h e min imal bad sequence i n ch.4 5 2.7.

0 L e t U denote t h e b a r r i e r domain o f f , and l e t ro be an element i n U whose

maximum i n t e g e r a i s t h e l e a s t p o s s i b l e . L e t Uo be a b a r r i e r i n c l u d e d i n U

and which c o n t a i n s t h e e lement ro . L e t fo be a bad f u n c t i o n f rom Uo i n t o A which i s an i n f - r e s t r i c t i o n o f f and f o r which f o ( r o ) i s min imal (mod A) , i n t h e sense t h a t no o t h e r c h o i c e o f Uo and fo g ives a va lue fo(ro) s t r i c t l y l e s s than t h e above va lue. L e t rl be an e lement o f Uo which i s d i s t i n c t f rom ro and whose maximum i n t e -

g e r i s t h e l e a s t p o s s i b l e . L e t U1 be a b a r r i e r i n c l u d e d i n Uo and which con- t a i n s t h e elements ro, r1 . L e t fl be a bad f u n c t i o n f rom U1 i n t o A , which i s an i n f - r e s t r i c t i o n o f fo and f o r which fl(rl) i s min imal . I t e r a t i n g t h i s , we ob ta in , f o r each n a t u r a l number i t a f i n i t e s e t ri o f n a t u r a l numbers,

a b a r r i e r Ui and a f u n c t i o n fi f rom Ui i n t o A (dependent c h o i c e ) . Each

Ui i s t h e s e t o f those e lements o f U which a r e subsets o f t h e un ion u Ui , and each o f these unions i s i n c l u d e d i n t h e p reced ing one. Note t h a t fl(rO) = fo(ro) , and i n genera l f .(r.) = f i ( r i ) f o r a l l n a t u r a l numbers i and j > i . The maximum o f ri inc reases (however n o t always s t r i c t l y ) i n i and tends t o i n f i n i t y , s i n c e t h e r e a r e o n l y f i n i t e l y many se ts

of n a t u r a l numbers hav ing a g i v e n maximum. We s h a l l p rove t h a t t h e s e t V o f t h e ri i s a b a r r i e r .

Indeed, two d i s t i n c t ri a r e m u t u a l l y n o n - i n c l u s i v e , s i n c e they belong t o U . Moreover, eve ry element r i n U which i s i n c l u d e d i n t h e un ion o f t h e ri i s an ri : l e t t i n g a denote t h e maximum o f r , i t s u f f i c e s t o see t h a t t h e r e o n l y e x i s t f i n i t e l y many s e t s o f n a t u r a l numbers w i t h maximum a , hence t h a t r w i l l be one of t h e ri o r o v i d e d t h a t i t i s s imu l taneous ly i n c l u d e d i n u Uo and u U1 and u U 2 ... . Here t h i s i s indeed t h e case, s i n c e r i s i n c l u d e d i n t h e un ion of t h e ri . F i n a l l y t h e b a r r i e r V i s t h e s e t o f t h e ri , as w e l l

as t h e i n t e r s e c t i o n o f t h e Ui , as w e l l as t h e s e t o f those e lements o f U which a r e i n c l u d e d i n t h e un ion o f t h e

t o t h e un ion of t h e ri , as w e l l as t h e i n t e r s e c t i o n o f t h e L e t g denote t h e b a r r i e r sequence w i t h domain V , and which i s d e f i n e d by

g(r i) = fi(ri) f o r i = 0,1,2, ... . Note t h a t g i s a bad i n f - r e s t r i c t i o n o f

f . We s h a l l p rove t h a t g i s min imal bad. Suppose t h e c o n t r a r y : t h e r e e x i s t s

J 1

ri . Moreover t h e un ion u V i s equal

u Ui .

Chapter 8 21 1

a b a r r i e r W i n c l u d e d i n V , and a bad W-sequence h which i s an i n f - r e s t r i c -

t i o n o f g , and an i n t e g e r i f o r which ri E W w i t h h ( r i ) < g(r i ) (mod A) . L e t n be t h e l e a s t i f o r which t h i s ho lds . C a l l W+ t h e s u b - b a r r i e r o f V

d e f i n e d by

ments o f W : hence W c W+s V . L e t h+ be t h e b a r r i e r sequence w i t h domain W+

w i t h h ( r i ) = h ( r i ) f o r r ig W , and h ( r i ) = g( r i ) f o r ri E W -W ; so i n

p a r t i c u l a r h+(rn) = h ( r n ) < g ( r n ) (mod A) . Note t h a t h + ( r ) $ g ( r ) r 6 W+ : hence o u r b a r r i e r sequence h+ i s an i n f - r e s t r i c t i o n o f g . To o b t a i n a c o n t r a d i c t i o n , i t s u f f i c e s t o show t h a t h+ i s bad. Indeed f o r i = O , l , ..., n -1 we have h(r i ) = g( r i ) , and e i t h e r ri E W+-W o r ri E W : i n

b o t h cases we have h ( r . ) = g(r i) . Since h + ( r n ) < g ( r n ) (mod A) , we o b t a i n a c o n t r a d i c t i o n w i t h t h e m i n i m a l i t y o f fn , more p r e c i s e l y t h e m i n i m a l i t y o f t h e

e lement o u r second paragraph. Thus suppose t h a t h+

assumption, t h e r e e x i s t two success ive elements i n t h e b a r r i e r W+ , say s q t , w i t h h + ( s ) s h + ( t ) . Then s CZW , f o r o the rw ise h+(s) = g(s),< h + ( t ) d g ( t )

(mod A ) , so g would be good; c o n t r a d i c t i o n .

From t h e f a c t t h a t s E W , we deduce t h a t h+(s) = h ( s ) . Moreover s i s d i s -

t i n c t f rom each

c o n d i t i o n s : S ince t h e maximum o f ri inc reases i n i , t h e maximum o f s i s g r e a t e r t han

o r equal t o t h e maximum o f belongs t o t h e union

u W , e i t h e r because i t belongs t o s , o r because i t i s s t r i c t l y g r e a t e r t han

t h e e lements o f t h e ri ( i = O , l , ..., n ) . F i n a l l y t 6 W so h ( s ) = h+(s) < h + ( t ) = h ( t ) , so h i s good: c o n t r a d i c t i o n . 0

u W+ = t h e un ion o f ro, o f rl, .. . , o f rn-l and o f a l l t h e e l e -

+ + +

f o r eve ry

+ 1

f n ( r n ) = g ( rn ) : see t h e i t e r a t i v e d e f i n i t i o n o f fo, fl, ... , fn i n

i s good, and o b t a i n a c o n t r a d i c t i o n as f o l l o w s . By o u r

ri (i = O , l , ..., n-1) ; f o r o the rw ise we would have t h e f o l l o w i n g

h+(s) = h ( s ) = g(s),( h + ( t ) $ g ( t ) , thus g would be good.

rn . Hence eve ry element o f t

§ 3 - FORERUNNING BETWEEN BARRIERS AND B A R R I E R SEQUENCES

3.1. FORERUNNER AND SUCCESSOR BARRIER

Given a b a r r i e r U , r e c a l l t h a t t h e m i o n u U i s t h e i n f i n i t e s e t o f those

i n t e g e r s which belong t o elements o f U . We say t h a t a b a r r i e r U i s a fore- r u n n e r o f V o r t h a t U f o r e r u n s V , o r aga in t h a t V i s a successor o f U

w i t h r e s p e c t t o fo re runn ing , i f t h e un ion U V i s i n c l u d e d i n U U and each e lement i n V has a (un ique) i n i t i a l i n t e r v a l be long ing t o U . F o r example, each b a r r i e r i n c l u d e d i n U i s a successor o f U . The b a r r i e r o f

a l l p a i r s o f n a t u r a l numbers i s a successor o f t h e b a r r i e r o f a l l s i n g l e t o n s . The b a r r i e r o f t hose p a i r s which a r e subsets o f an a r b i t r a r y i n f i n i t e s e t o f


integers The relation of forerunning i s reflexive, antisymmetric and t ransi t ive: i t i s a partial ordering on the se t of the barr iers . The notion of forerunning, for barriers and barr ier sequences, i s introduced by NASH-WILLIAMS 1965' p . 714; the following resul ts abou t forerunning are a t l eas t implicitly obtaines by him, b u t some of them are improved and c la r i f ied by L A V E R 1978 and ROSENSTEIN 1982.

i s a successor of the barr ier of a l l singletons.

3.2. COMPLETION OF A SUCCESSOR B A R R I E R Let U be a barr ier and V a successor of U , with respect t o forerunning; assume that V i s n o t simply a sub-barrier of U . Let p be the leas t maximum integer of those elements of U which are subsets of u V yet do not belong t o V . Call W the s e t of those elements of U which are n o t subsets of u V yet which are formed of natural numbers 6 p or of elements of u V . Then V u W i s a barr ier , and furthermore a successor of U . Note t h a t p can be replaced by any smaller n a t u r a l number. I n the particular case where V i s a sub-barrier of U , take p t o be an arbi- t rary natural number. Then the preceding definition o f W i s preserved and our conclusion s t i l l holds with the same following proof.

0 The sets V and W are obviously d is jo in t . Prove f i r s t t h a t any elements s in V a n d t in W are incomparable with respect t o inclusion. Indeed, there exis ts an i n i t i a l interval r of s with r.s U . If s c_t then r s s ~ _ t and, since r and t belong t o U , we have r = s = t . Yet r i s a subset of u V and t i s not: contradiction. Now i f t s s , then t i s a subset of of u V : contradiction. Now l e t X be an inf in i te s e t of integers in u ( V u W ) , thus a s e t of integers

interval of X , which belongs e i ther to V or t o W . Since U i s a barr ier , there exis ts an i n i t i a l interval s of X , which belongs t o U . Then e i ther s i s n o t a subset of yV , and hence s E W and we are finished. Or s i s a subset of u V . I n t h i s case, e i ther s E V s 4 V and then Max s 3 p by the definition of the integer p . Then the elements of X which are s t r i c t l y greater t h a n Max s a l l belong t o W V , as well as the elements of s . Thus there ex is t s an i n i t i a l interval of X which belongs t o v . 0

p o r which belong to u V . We shall prove that there ex is t s an i n i t i a l

and we are finished. Or

3.3. FORERUNNER AND SUCCESSOR B A R R I E R SEQUENCE Let A be a par t ia l ordering and 6 be any function, which t o each element in the base I A I associates an ordinal. Let U be a barr ier and l e t f be a U-sequence w i t h values in A . Similarly l e t V be a successor barr ier of U

Chapter 8 213

and g be a V-sequence w i t h va lues i n A . We say t h a t f i s a f o r e r u n n e r o f

g (mod A, s ) o r t h a t g i s a successor o f f (mod A, s ) w i t h r e s p e c t t o f o r e -

runn ing , i f f o r each e lement t i n V and f o r t h e i n i t i a l i n t e r v a l s o f t

which belongs t o U , e i t h e r we have t h a t s = t and t h e n g t = f s ; o r we have s C t and then g t < f s (mod A) and fu r the rmore S g t < J f s ( i n t h e usual o r d e r i n g o f o r d i n a l s ) .

i t d e f i n e s a p a r t i a l o r d e r i n g on t h e s e t o f b a r r i e r sequences w i t h va lues i n

f o r a g i ven f u n c t i o n S . I n f o l l o w i n g a p p l i c a t i o n s , s p e c i a l l y i n n e x t 9 4 , we t a k e A t o be t h e embedding

r e l a t i o n on a s e t o f cha ins , and

which t o each c h a i n x assoc ia tes t h e neighborhood rank o f x (see ch.6 § 2.4) . F o r t h i s reason we w i l l c a l l s a r a n k i n g f u n c t i o n ; however we o n l y need, i n t h e p resen t theo ry , t h a t $ assoc ia tes t o each e lement o f A

Fo re runn ing f o r b a r r i e r sequences i s r e f l e x i v e , an t i symmet r i c and t r a n s i t i v e :

A ,

$ t h e ( n o n - s t r i c t l y ) i n c r e a s i n g f u n c t i o n

any o r d i n a l va lue.

3.4. COMPLETION OF A SUCCESSOR BARRIER SEQUENCE

(1) L e t A be a p a r t i a l o r d e r i n g and 8 be a r a n k i n g f u n c t i o n on A ( i . e . any

f u n c t i o n w i t h o r d i n a l va lues ) . L e t U be a b a r r i e r and V a successor o f U . L e t f be a U-sequence w i t h va lues i n A , and g be a V-sequence w i t h va lues

i n A ; assume t h a t g i s a successor o f f (mod A, s ) . L e t p be t h e l e a s t

maximum i n t e g e r o f those elements i n U which a r e subsets o f u V y e t do n o t be long t o V . C a l l W t h e s e t o f those elements i n U which a r e n o t subsets of

Then as V u W i s a b a r r i e r by 3.2, t h e b a r r i e r sequence h , d e f i n e d t o be

equal t o f on W and t o g on V , i s a successor o f f (mod A, $ ) . ( 2 ) Moreover if f !I g a r e bad, t hen h i s bad (ROSENSTEIN 1982 ch.10).

As i n 3.2 above, p can be rep laced by any s m a l l e r n a t u r a l number. Moreover i n

t h e p a r t i c u l a r case where V i s a s u b - b a r r i e r o f U , take p t o be an a r b i t r a - ry n a t u r a l number.

0 (1) For eve ry s E W , we have hs = f s . For eve ry t E V and f o r t h e i n i t i a l i n t e r v a l s o f t which belongs t o U , we have h t = g t . Then i n t h e case s = t we have g t = f s by hypo thes i s , hence aga in h t = f s . I n t h e case sc t by hypo thes i s we have g t < f s (mod A) and S g t < S f s ; hence h t ( f s (mod A) and $ h t < S f s . 0

0 (2 ) Now suppose t h a t f and g a r e bad, t hen p rove t h a t h i s bad. F o r t h i s

cons ide r two e lements s ' and s" o f V v W , w i t h S ' Q s " . We d i s t i n g u i s h f o u r cases. I f s ' and s " be long t o V , then h s ' = gs ' and hs" = gs" and by hypo thes i s gs '+ gs" (mod A) , hence h s ' $ hs" . S i m i l a r l y argue i f s ' and s " be long t o W , by r e p l a c i n g g by f . Suppose t h a t s ' E W and s " E V , and l e t s be t h e i n i t i a l i n t e r v a l o f s "

v V , y e t a r e formed o f i n t e g e r s 6 p o r be long ing t o u V .


which belongs t o U , hence s i s n o t included in s ' (since S ' E U ) , hence S ' Q s . Then hs' = f s ' and hs" = g s " 6 f s (mod A) . Since f i s bad, we have f s ' + f s (mod A) : thus Finally suppose t h a t s ' 6 V and s " e W , and l e t s be the i n i t i a l interval of s ' which belongs t o U ; hence s 4 s " . Then e i ther s = s ' , in which case we have hs' = gs' = f s and hs" = f s" ; and by hypothesis f s p f s" (mod A) , so t h a t h s ' 4 hs" (mod A) . Or s c s ' , in which case s $ V , yet s i s a subset of u V , hence Max s > / p . I n th i s case i t follows t h a t s , s ' and s " are subsets of uV , hence s " Q W : contradiction. 0

hs '+ hs" (mod A) .

3.5. MINIMAL BAD BARRIER SEQUENCE WITH RESPECT TO FORERUNNING Let A be a Dartial ordering, and cf be a ranking function with domain I A I . Say that a barr ier sequence f with values in A i s minimal bad (mod A, s ) with respect t o forerunning, i f f i s bad , and i f every bad successor of f reduces t o a res t r ic t ion of f . I n other words, denoting by U the barr ier Dom f , there does n o t ex is t any barrier V successor of U , non-included in U , with a V-sequence which i s a bad successor of f (mod A, s ) .

Let A be a par t ia l ordering, and s a rankinq function with Dom $ = 1 A I . Let U be a barr ier and f a bad U-sequence with values in A . Then t& exis ts a minimal bad barrier sequence (mod A, $ ) , wh&hLs a successor of f ( LAVER 1978 ; uses deoendent choice). Note the analogy with the theorem on the minimal bad barr ier sequence in 2 . 2 above. However here A i s an arbi t rary par t ia l ordering, n o t necessarily well- founded; indeed we shall use the present proposition in order t o prove that certain partial orderings are well-founded, and even are well par t ia l orderings. I n compensation, the present statement uses the ranking function associated with the partial ordering A . 0 Set Vo = U and fo = f . Suppose that there exis ts a barr ier V1 , a successor of U which does n o t reduce t o a sub-barrier of U , and a bad barr ier sequence f l with domain V1 , a successor of f (mod A, 6 ) . Choose V1 and f l such t h a t the leas t maximum integer p,, of those elements of U which are subsets of u V1 yet n o t elements of V1 i s the leas t possible. Using the preceding completion statement, we can assume t h a t integers 6 po which belong t o u U . I terate t h i s (axiom of dependent choice). For each natural integer i , we have a barrier V i and a bad Vi-sequence f i , with f i+l a successor of f i modu- 10 A, 6 . If one of these t i v i t y of forerunning.

u V1 contains a l l those

f i i s minimal, we are finished, by using the t ransi-

Chapter 8 215

Suppose the contrary: for each integer index i s not a sub-barrier of V i . Then to each i i s associated an integer pi , the least possible maximum of those elements of union s e t v V i t l ye t n o t elements of V i t l . By the t rans i t iv i ty of forerunning, we see t h a t pi increases in i , t h o u g h not necessarily s t r i c t l y . Indeed i f we had p i + l < pi , then the barr ier sequence f i+* , or another giving pi+l or giving an integer 6 p i+ l , would appear before f i + l . However pi tends t o in f in i ty , since there are only f in i te ly many se ts of integers with maximum and a given f i n i t e s e t can a t most once be an element of without being an element of the union s e t u V for a l l j i , hence for a l l integers j . Same remark for any element of u V i which i s pi . The union se t s uVi form a decreasing 0 -sequence with respect t o inclusion. Let H be the i r intersection, which i s in f in i te since every p i belongs t o H . For any inf in i te subset X of H and any integer i , there exis ts a (unique) f i n i t e i n i t i a l interval s i of X which belongs t o the barr ier V i . This si increases in i , although not necessarily s t r i c t l y . Because of the decreasing values from which point on s i remains constant. Call s th i s constant s e t , and note t h a t , when X varies, these s form a barr ier V with u V = H . Indeed by construction each inf in i te subset of H admits a certain s as an i n i t i a l interval ; moreover given any two such f i n i t e i e t s s and t , there exis ts an i from which point on s and t are b o t h elements of the barr ier V i : so t h a t they are incomparable with respect t o inclusion. Finally the barr ier V i s a successor of each V i . Define the V-sequence g as follows. For each s E V , take an i from which point on the chosen i . The barr ier sequence g so defined, i s bad, since the f i are bad. Also g i s a successor of each f i , in par t icular a successor of fo = f (mod A, 6 ) . I t remains t o see that g i s minimal. Take an arbi t rary barr ier W successor of V , which i s not simply a sub-barrier of V . Take a W-sequence h which i s a bad successor of g (mod A, s ) . Then there ex is t s an element t of W with an i n i t i a l interval s belonging t o V and d i s t inc t from t , so s c t . Consider the leas t i for which we have simultaneously pi > Max s and s an element o f V i . By t rans i t iv i ty h i s a bad successor of f i . The s e t s belongs t o V i yet not to W , hence h should have been taken instead of f i+ l , because i t leads to Max s pi : contradiction. 0

i , the barr ier V i + l

V i which are subsets of the

pi ,

belongs t o V i

V i + l . By the preceding 3.4, we can assum tha t each p i

j

$ f i ( s i ) , for X fixed, there exis ts an i

s E V i : then we s e t gs = f i ( s ) , th i s value being independent from

( o r t o a lesser integer) instead o f

5 4 - EMBEDDABILITY THEOREM FOR SCATTERED CHAINS (LAVER)

4.1. L e t o rde red under embeddab i l i t y . Assume t h a t & i s c l o s e d w i t h r e s p e c t t o t a k i n g any h-indecomposable i n t e r v a l o f a cha in . L e t s be t h e r a n k i n g f u n c t i o n which

assoc ia tes t o each cha in i t s ne ighborhood rank (ch.6 5 2.4) .

L e t U be a b a r r i e r and f be a bad U-sequence w i t h va lues i n 44 . Then there e x i s t s a b a r r i e r V successor o f U which i s n o t a s u b - b a r r i e r o f U , and a bad V-sequence which i s a successor o f f

0 The cha ins i n 64 which belong t o t h e range o f

wise, e i t h e r t h e empty c h a i n o r a c h a i n c o n s i s t i n g o f a s i n g l e t o n would be long t o Rng f , and so t h e b a r r i e r sequence f would be good. P a r t i t i o n t h e elements o f U

f which we aga in denote by U : see p a r t i t i o n theorem 1.4. To f i x i deas , suppose t h a t a l l t h e cha ins a r e r i g h t indecomposable.

Compose t h e f u n c t i o n f w i t h t h e f u n c t i o n 8 which t o each c h a i n assoc ia tes i t s

c o f i n a l i t y : we o b t a i n a b a r r i e r sequence p r e c i s e l y va lues which a r e r e g u l a r a lephs . It f o l l o w s t h a t 8 .f has no bad res -

t r i c t i o n : t ake an w -sequence of success ive e lements

( i n a t u r a l number); t hen t h e va lues y ,,f(si) cannot be s t r i c t l y decreas ing.

t h a t t h e r e s t r i c t i o n of 2( .f t o t h i s b a r r i e r i s p e r f e c t . L e t us denote t h i s b a r r i e r aga in by U . To each h-indecomposable c h a i n A i n , a s s o c i a t e a decomposi t ion i n t o a sum,

a long t h e o r d i n a l 8 (A) , o f h-indecomposable i n t e r v a l s Ai w i t h t h e c o n d i t i o n t h a t each has neighborhood rank s t r i c t l y l e s s than t h e rank of A , and t h a t f o r each i , t h e r e a r e Z ( A ) many A . ( i C j < (A) ) i n which Ai i s embeddable: see ch.6 5 5. T h i s decomposi t ion o f each A i n t o

t h e Ai i s chosen once f o r a l l and s h a l l be c a l l e d t h e s tandard decomposi t ion (ax iom o f c h o i c e ) .

S ince f i s bad, f o r eve ry s and s ' i n U w i t h s d s ' , we have t h e non- embeddab i l i t y f s d f s ' . Consider t h e s tandard decomposi t ions f s = E Ai and f s ' =

j

It fo l l ows t h a t t h e r e e x i s t s a t l e a s t one i n t e r v a l i n any A ' . The f i r s t o f these Ai s h a l l be assoc ia ted w i t h t h e un ion

t = s v s ' . l h e s e t o f these unions i s a b a r r i e r V . More p r e c i s e l y V square o f a s u b - b a r r i e r o f U , hence a successor o f U .

be a s e t o f h-indecomposable cha ins (see ch.6 § 5 ) , which i s quas i -

(mod & , 6 ) (uses axiom o f c h o i c e ) . - f a r e a l l i n f i n i t e . F o r o t h e r -

i n t o two se ts , acco rd ing t o whether t h e image under i s r i g h t o r l e f t indecomposable. A t l e a s t one o f these s e t s i n c l u d e s a b a r r i e r ,

8 .f t a k i n g o r d i n a l va lues, and more

so 4 s1 4 . . . Q siQ.. . i n u

By 2.1, t h e r e e x i s t s a b a r r i e r i s p e r f e c t . L e t us denote t h i s

( i < x ( A ) ) ;

Ai

J

A j . Since 8 o f i s p e r f e c t , we have r f s < Y f s ' , hence t h e i ndex runs through a r e g u l a r a leph a t l e a s t as l a r g e as t h a t which i runs through.

Ai which i s n o t embeddable

j i s t h e

Chapter 8 217

Now we define the b a r r i e r sequence g w i t h domain V , by g t = Ai . This g i s a successor of f (mod L/z , 6 ) : indeed f o r each t in V and f o r the i n i t i a l in te rva l s of t belonging t o U , we have s c t and g t < f s under embedda- b i l i t y . Furthermore by ch.6 5 5.3, the in te rva l s t r i c t l y l e s s than the rank of A : i n o ther words I t remains t o see t h a t g i s bad. Let t and t ' be elements of V such t h a t t at' . Then t i s a union s u s ' w i t h sas' ; s imi l a r ly t ' i s of the form s ' u s " with s ' q s" and s , s ' , s " elements of U . The chain g t i s an in te rva l Ai of the standard decomposition of A = f s ; s imi l a r ly g t ' i s an in te rva l A' of the standard decomposition of A' = f s ' ; and f i n a l l y Ai has been defined t o be non-embeddable i n any term of the standard decomposition of A ' : so AiCA: , i . e . g t b g t ' . 0

4.2. Every s e t of h-indecomoosable chains forms a well quasi-ordering under embedda- b i l i t y (uses axiom of choice). I t i s even a b e t t e r quasi-ordering, i n the sense of 5 5 and 6 below.

0 Suppose the cont ra ry , t h a t there ? x i s t s an w-sequence taking h-indecomposable chains as values, which i s bad w i t h respect t o embeddability (ch.4 5 3 . 2 . ( 2 ) ) . Consider this w -sequence as a b a r r i e r sequence on s ingle tons . By 3 .5 , there e x i s t s a minimal bad b a r r i e r sequence which i s a successor of the above, modulo the p a r t i a l ordering of embeddability and the ranking function which, t o each chain, assoc ia tes i t s neighborhood rank. B u t the preceding proposit ion 4 .1 (axiom of choice) proves, on the contrary, t h a t every bad b a r r i e r sequence of t h a t kind, has a bad successor b a r r i e r sequence which i s not simply a r e s t r i c t i o n t o a sub-barrier: cont rad ic t ion . 0

Ai has a neighborhood rank s g t 4 S f s .

j

3

4.3. Every sca t t e red chain i s a f i n i t e sum of h-indecomposable chains (uses axiom of choice) . In p a r t i c u l a r , every indecomposable sca t t e red chain i s h-indecomposabie. In other words, f o r a sca t te red chain, the notion of indecomposability coincides with t h a t of h-indecomposability.

0 By the preceding propos i t ion , every set of h-indecomposable chains i s well quasi-ordered under embeddability. Our proposit ion now follows from ch.6 5 5.4 and 5 . 5 . 0

4.4. EMBEDDABILITY THEOREM FOR SCATTERED CHAINS Every s e t of s ca t t e r ed chains forms a well quasi-ordering under embaddability ( L A V E R 1968; see a l s o 1971; uses axiom of choice) .

0 The re l a t ion of embeddability f o r h-indecomposable cha ins , forms a well quasi-

ordering, by the preceding 4.2 (axiom of choice). Now use HIGMAN's theorem on words, ch.4 5 4.4. The relation of embeddability between words, or f i n i t e sequences of h-indecomposable chains, i s also a well quasi-ordering. The same holds for the embeddability among f i n i t e sums of h-indecomposable chains: indeed embeddabi- l i t y on sums i s a reinforcement of embeddability on sequences. By 4.3, every scattered chain i s a f i n i t e sum of h-indecomposable chains: our theorem i s proved. 0

I n L A V E R 1971, th i s theorem i s extended t o countable unions of scattered chains. Unfortunately, the proof i s t o o complicated to be presented here. I n COROMINAS 1984, the theorem i s extended t o countable t rees .

4.5. C O V E R I N G BY A FINITE NUMBER OF DOUBLETS Given a scattered chain, the equivalence relation of covering by doublets of indecomposable chains (see ch.6 5 4.5 and 4.6) has only a f i n i t e number of equivalence classes. Each class i s an indecomposable interval o r a doublet (uses axiom of choice).

0 Using the preceding 4.3, decompose our chain - n t o a f i n i t e number of r ight or l e f t indecomposable intervals . Replace any two contiguous such intervals by the i r union, provided th i s union i s indecomposable. When i t becomes impossible to effect these replacements, then the intervals thus obtained, or the unions of two contiguous intervals , constitute the covering by doublets. The uniqueness of th i s decomposition follows from ch.6 propositions 4.1 and 4.2. 0

For a non-scattered chain, the relation of covering by right or l e f t indecomposable chains, i s s t i l l an equivalence relation: see ch.6 5 4.3. Hence the union of both equivalence relat ions, i s again an equivalence relat ion. However there can be inf in i te ly many equivalence classes for th i s equivalence relation (BONNET 1972, unpublished; uses axiom of choice).

0 Star t with A. = the chain of the reals . By ch.5 5 5.2 (DUSHNIK, M I L L E R ) , we have a s t r i c t l y decreasing (with respect t o embeddability) a-sequence of chains Ai ( i integer) , where each Ai has cardinality of the continuum; moreover we can require that Ai $ . w for each i (same paragraph, propo-

s i t ion ( 3 ) ; uses axiom of choice). On the other hand , we have Q,< Ai for each i ; indeed Ai has a t l eas t W1 many elements, and neither the ordinal W1

nor i t s converse i s embeddable in the rea ls , hence in Ai : use ch.5 5 3.5 in the particular case where o( = 1 . T h u s Ai for any two natural numbers i and h . Let U = U1 + W1 and consider the sum of the O-sequence A. + U t A1 + U + ... + U + A . + U + ... . We shall prove tha t each interval isomorphic with U

i s one of the desired equivalence classes; hence t h a t there ex is t in f in i te ly

Q + A i + l + Q +...+ Q + Ai+h

1

Chapter 8 219

many equiva lence c lasses .

Indeed, take two e lements x and y i n two consecu t i ve components: f o r example x belongs t o U and y belongs t o Ai f o l l o w i n g t h e considered component U . We must j o i n x t o y by f i n i t e l y many i n t e r m e d i a t e elements, such t h a t any two consecut ive elements be e i t h e r r i g h t e q u i v a l e n t ( i . e . covered by a same r i g h t i n - decomposable i n t e r v a l ) o r l e f t e q u i v a l e n t . We can assume t h a t x and y a re them-

se lves consecu t i ve elements; t hen i t s u f f i c e s t o see t h a t t hey a r e n e i t h e r r i g h t

no r l e f t e q u i v a l e n t . F i r s t , a n o n - f i n a l i n t e r v a l I which c o n t a i n s x and y i s o b v i o u s l y decompo-

sable i n t o a f i n i t e sequence o f d i s j o i n t s u b - i n t e r v a l s i n which embedded. Secondly, a f i n a l i n t e r v a l i s o b v i o u s l y n o t l e f t indecomposable; n o r

i s i t r i g h t indecomposable; f o r o the rw ise , i t would be necessary t h a t Ai , f o r example, be embeddable i n a sum o f t h e f o r m U + Ai+l + U + ... + U + . B u t

an i n t e r v a l o f Ai which i s a r e s t r i c t i o n o f U i s coun tab le , s i n c e i t i s i s o - morphic w i t h t h e u n i o n o f a w e l l - o r d e r e d s e t o f r e a l s and t h e converse o f such

a w e l l - o r d e r e d s e t . So i t must be t h a t Ai i s embeddable i n Q + + Q + ... + Q + The p reced ing c h a i n w i l l be cons ide red aga in i n e x e r c i s e 3.

4.6. L e t A be an i n f i n i t e s c a t t e r e d cha in . I f e v e r y c h a i n X < A s a t i s f i e s

X.2 6 A , then unpubl ished) . As a l s o n o t i c e d by HAGENDORF, i t then r e s u l t s f rom LARSON's theorem (ch.6 5 3.8)

t h a t A i s e i t h e r a w e l l - o r d e r i n q o r t h e converse o f a well-ordering.

0 Us ing 4.3 (ax iom o f cho ice ) , decompose indecomposable o r l e f t indecomposable i n t e r v a l s . L e t I be one o f t hese i n t e r -

v a l s , which i s >/ o r I a l l o t h e r s under embeddab i l i t y . We can assume t h a t I i s i n f i n i t e , and r i g h t indecomposable, t o f i x i deas . L e t k be t h e number o f

i n t e r v a l s equimorphic w i t h I , i n t h e cons ide red decomposi t ion. E i t h e r 1.k i s equimorphic wi th A ; then i f k = 1 we a re f i n i s h e d . I f k 3 2 , s e t t i n g X = I ( k - 1 ) + 1 , we have X < A (use ch.5 5 3.3) y e t X . 2 4 A , c o n t r a d i c t i n g ou r hypo thes i s . O r I . k < A ; then I ( k t l ) 4 A (use again ch.5 5 3.3) ; s e t t i n g

X = 1.k , t h i s c o n t r a d i c t s o u r hypo thes i s . 0

I cannot be

, c o n t r a d i c t i n g t h e p r e r i o u s d i s c u s s i o n . 0

A , i s indecomposable (uses axiom o f choice; HAGENDORF 1982,

A i n t o a f i n i t e sum o f e i t h e r r i g h t

4.7. L e t A be an i n f i n i t e s c a t t e r e d cha in . I f e v e r y p r o p e r i n i t i a l i n t e r v a l

X - o f A s a t i s f i e s X . 2 4 A , then A i s r i g h t indecomposable (uses axiom o f

choice; HAGENDORF 1982, unpub l i shed) .

Fo r A non-scat tered, we have t h e counterexample Q + W 1 . I f t h e hypo thes i s

i s weakened by r e q u i r i n g t h a t X be an i n i t i a l i n t e r v a l s t r i c t l y l e s s than A

under embeddab i l i t y , t hen we have t h e counterexample ned i n ch.6 5 3.4.

2 ( W + W)- a l ready ment io-

Decompose A i n t o a f i n i t e sum o f e i t h e r r i g h t o r l e f t indecomposable i n t e r v a l s . We can assume t h a t there are a t l e a s t two such i n t e r v a l s i n the decomposition; denote by D the l a s t such i n t e r v a l . I f D i s i n f i n i t e and l e f t indecomposable, then take an element d i n D and l e t X be the i n i t i a l i n t e r v a l o f A w i t h l a s t element d . Then X i s equimorphic w i t h A and so A.2 equimorphic w i t h A , thus A i s no t scattered, by ch.5 5 3.3. I f D i s i n f i n i t e and r i g h t indecomposable, then again take an element d i n D and l e t X be the i n i t i a l i n t e r v a l o f A w i t h l a s t element d . Then X . 2 6 A and X.2 has a l a s t element. Thus the i n i t i a l i n t e r v a l Y generated by X.2 s a t i s f i e s Y . 2 5 A , and so X.4 < A . Hence e i t h e r X . 2 4 X and then ne i the r X nor A i s scattered. O r XB D y e t i s no t c o f i n a l i n D ; so A equimorphic w i t h X + D y i e l d s A,< D and thus A i t s e l f i s r i g h t indecomposable. F i n a l l y i f D reduces t o a s ing le ton , then l e t t i n g X = A-D :

we have X . 2 6 A and even X.2 + 16 A ( d i s t i n g u i s h the case where X has a maximum); hence X . 2 6 X so t h a t X i s no t scattered. We ob ta in the contradic- t i o n i n the f i r s t and i n the t h i r d cases: so t h a t on ly the second case occurs, and A i s r i g h t indecomposable. 0

§ 5 - INDECOMPOSABLE SEQUENCE, BETTER PARTIAL ORDERING

TAIL OF AN ORDINAL SEQUENCE; INDECOMPOSABLE SEQUENCE Given a p a r t i a l o rder ing A and an o rd ina l sequence u o f leng th o( which takes values i n A , we c a l l a tail, o r f i n a l i n t e r v a l of u , any sequence obtained by tak ing a . f i na l i n t e r v a l (3 o f the length a( , then the r e s t r i c t i o n u / p , then re index ing t h i s r e s t r i c t i o n , s u b s t i t u t i n g fo r (3 the o rd ina l i s o - morphic w i t h p . Recal l t h a t the connected no t i on o f i n i t i a l i n t e r v a l o f a sequence, has already been introduced i n ch.4 5 2.1.

An ord ina l sequence u w i t h values i n A i s sa id t o be indecomposable (mod A) i f u i s embeddable (mod A) i n every non-empty t a i l o f u (embeddabi l i ty among sequences has been introduced i n ch.4 9 2) . This requires t h a t the o rd ina l leng th of the sequence u be i t s e l f an indecomposable o rd ina l , hence a power o f w :

see ch.1 5 3.6.

I n the cont ra ry case, the o rd ina l sequence i s sa id t o be decomposable (mod A ) . Note tha t , w i t h an indecomposable o rd ina l , u f o r example, one can cons t ruc t decomposable sequences o f l eng th U : s t a r t w i t h the f r e e o a r t i a l o rder ing based on two elements a, b taken t o be incomparable, then take the w-sequence b,a,a,a, ... . We can even cons t ruc t W-sequences none o f whose t a i l s are indecomposable: s t a r t w i t h the i d e n t i t y p a r t i a l o rder ing on a denumerable base E and take an &-sequence i n E , wi thout r e p e t i t i o n .

Chapter 8 22 1

5.1. o( -BETTER PARTIAL ORDERING Let o( be an indecomposable ordinal. A par t ia l ordering A i s said t o be an

composable t a i l . Obvious definition of a

Every well partial ordering i s an w -bet ter par t ia l orderinq, and conversely. The "converse" part uses dependent choice; however ZF suffices i f the given par- t i a l ordering i s denumerable.

0 Let A be a well par t ia l ordering, and u be an w-sequence in A , with terms u i ( i natural number ) . Consider those integers h such that there exis t only f in i te ly many integers u i 2 u h (mod A ) . Then there are only f in i - tely many such h . For otherwise, there would ex is t an w -sequence h ( i ) ( i na tu - ral integer) with u h ( j ) < o r I u (mod A) for a l l i and j 7 i , hence a bad d-sequence. The t a i l obtained by beginning with an index s t r i c t l y greater t h a n these values h i s an indecomposable sequence. Conversely, suppose that A i s not a well par t ia l ordering, and l e t u be a bad

W -sequence: see ch.4 5 3 . 2 . ( 2 ) , dependent choice. For such a sequence, no t a i l i s embeddable in a proper t a i l of i t s e l f . 0

BETTER PARTIAL O R D E R I N G A Dartial ordering i s said t o be a bet ter par t ia l ordering, i f i t i s an o( -better partial ordering for every ordinal o( (more precisely, for every indecomoosable ordinal o( ) . By the preceding, every bet ter par t ia l ordering i s a well partial ordering. The only obvious example of a bet ter par t ia l ordering i s the ordering defined on a unique element a . In th i s case, every ordinal sequence repeats the element a ; i f q i s the length of the sequence, then the indecomposable t a i l i s obtained by taking the l a s t term in the Cantor normal form of o( with respect t o base w .

m -better partial orderinq i f every cx -sequence in A has a non-empty inde- ( < H ) or ( 6 a ) - b e t t e r partial ordering.

i with

h(i 1

--

Every restr ic t ion, and every par t ia l ly ordered reinforcement of an o( -better par t ia l ordering, i s an 6 -bet ter par t ia l ordering. Same statement 'with "bet ter par t i a1 orderi ng," . A quasi-ordering i s said t o be an o( -better quasi-ordering, i f the partial ordering of the equivalence classes (each formed o f elements simultaneously greater than and lesser t h a n each other) , i s an o( -bet ter par t ia l ordering. Equivalent- ly , i f every o( -sequence has a non-empty indecomposable t a i l . A quasi-ordering i s said t o be a bet ter quasi-ordering, i f i t i s an o( -better quasi-ordering for every ordinal o( . 5.2 . The well partial ordering of RADO, defined in ch.4 5 4.2, i s an example of a well par t ia l ordering which i s n o t a bet ter par t ia l ordering.

2 More exactly, Rado's well-partial ordering i s not an W -bet ter partial ordering. 0 Recall t h a t th i s partial ordering i s defined on ordered pairs of non-negative integers. Take the lexicographical LJ -sequence (O,O), ( O , l ) , (0 ,2) , ... , ( l , O ) , ( l , l ) , ( 1 , 2 ) , ... . A t a i l a suff ic ient ly large integer p , the ordered pairs ( p , O ) , ( p , l ) , ( p , 2 ) , ... followed by (p+l,O) . If we attempt t o embed U in i t s t a i l which begins with (pf1,O) , then for each integer i the ordered pair ( p , i ) i s not embeddable before (p+i,O) . Hence t o embed a l l these ordered pairs ( p , i ) where p i s fixed and i varies, we exceed a l l the terms of U : i t i s then impossible t o embed the term (p+l,O) . 0

2

U i s necessarily an U2-sequence which contains, for

5.3. (1) Let d be an indecomposable ordinal. If A (6 4 ) -bet ter partial ordering, then the se t of ordinal indexed sequences in A with lengths s t r i c t l y less than o( constitutes a well quasi-ordering with respect t o embedda- b i l i ty . Uses dependent choice; ZF suffices i f A i s countable. In the case where o( = w ordering o f words (ch.4 4.4) in view of the equivalence between well par t ia l ordering and W-bet te r par t ia l ordering, 0 5.1 above.

0 Since A i s a ( 6 % )-better par t ia l ordering, every sequence in A with length less than or equal t o o( i s a f i n i t e sum of indecomposable (mod A) sequences. Indeed, i t suffices to operate regressively, s ta r t ing with the indecomposable t a i l of the given sequence. Now consider a se t of indecomposable (mod A) sequences of lengths s t r i c t l y less t h a n o( . Let B be the quasi-ordering between these sequences, with respect t o embeddability. We shall prove that B i s a well quasi-ordering. Take an w -sequence of these indecomposable sequences s i ( i non-negative integer) .

Then take the i r sum u , whose length i s a t most equal t o o( , since a i s a power of w . By hypothesis, there exis ts a t a i l of u which i s indecomposable; l e t s ( p integer) be i t s f i r s t term. I n par t icular , t h i s t a i l i s embeddable i n i t s proper t a i l which begins with s . Then s i s embeddable in a f i n i t e sum of s i ( i p t l ) ; hence by indecomposability, th i s s i s embeddable in some

s i ( i 7, p+1) . T h u s the sequence of the si i s good, and so B i s a well quasi- ordering (dependent choice, see ch.4 § 3.2 . (2) ) . By HIGMAN's theorem already mentioned, every s e t of words formed of indecomposable (mod A ) sequences of lengths s t r i c t l y less t h a n o< constitutes a well quasi- ordering. The same i s true for the s e t of f i n i t e sums corresponding t o these words, the i r quasi-ordering with respect t o embeddability being a reinforcement o f the quasi-ordering of the words: use ch.4 5 3.5.(1) . Since every sequence in A with length s t r i c t l y less than O( i s such a sum, our proposition i s proved. 0

.--.__I_------

, th i s i s jus t HIGMAN's theorem on the well par t ia l

P P+ 1 P

P

Chapter 8 223

( 2 ) Let o( be an indecomposable ordinal and A a partial ordering. If the s e t of

embeddabi 1 i ty , then A

0 Take an o( -sequence in A , and then the se t of a l l non-empty t a i l s of th i s sequence. By hypothesis one of these t a i l s i s minimal, thus minimum with respect t o embeddability, i .e . indecomposable. 0

( 3 ) Let OC be a denumerable indecomposable ordinal and A a partial ordering. If the s e t of sequences in A with lengths s t r i c t l y less than o< constitutes a well quasi-ordering under embeddability, then A 4 -better Ordering.

0 Suppose that A i s not an w -better ordering. Then there exis ts an o( -sequence in A which, by countability of o( , has a s t r i c t l y decreasing W-sequence of t a i l s si can choose s i+l t o begin with a given term of s i situated as f a r in s i as we desire. Then there ex is t s an i n i t i a l interval to of so and a natural number i from which point on to$ s i . For otherwise, every proper i n i t i a l interval of would be embeddable in some proper i n i t i a l interval of s i for every i ; then by the preceding discussion we could embed a l l of so in s1 for example: contradiction. Let h(0) = 0 and l e t h ( 1 ) be the least integer such t h a t for a l l larger integers i we have to4 si . There ex is t s an i n i t i a l interval tl of

s h ( l ) obtain a bad W-sequence, with respect t o embeddability, of intervals of lengths s t r i c t l y less than o< . Finally the se t of sequences in A of lengths s t r i c t l y less than q , does n o t constitute a well quasi-ordering. 0

From propositions (1) and (Z), i t follows that a necessary and suff ic ient condition for A t o be a bet ter par t ia l ordering, i s tha t every se t of ordinal sequences in A forms a well quasi-ordering under embeddability (necessity uses dependent choice).

4 -sequences in A constitutes a well-foundedquasi-ordering w i t h - , , r r g c t t o o( -better orderi nq.

( i natural number ); where i t i s understood t h a t , fo r each i , we

so

and an i , from which point on we have tl.$ s i . Iterating t h i s , we

ti

5.4. Let o( be a denumerable indecomposable ordinal, l e t A be a (,<o()-better par t ia l ordering and U a barrier o f phic rank less than or equal t o c(. Then every U-sequence in A i s good' (POUZET 1970; uses dependent choice; ZF suffices i f A i s countable). 7

0 By 5.1 , A i s a well par t ia l ordering; so t h a t i f U reduces t o the barrier of the singletons, then any U-sequence in A i s good by ch.4 5 3.2. (2) . I n the following, we remove from U a l l singletons and suppose o< a t leas t equal t o a* . For each natural number i , l e t U i ments of U which begin with i . These elements, when ordered lexicographically,

be the inf in i te s e t of those ele-

form a sequence w i t h l e n g t h s t r i c t l y l e s s than o( ; indeed they a r e l e s s than any element of U which begins w i t h i+l . L e t f be a U-sequence i n A , and l e t

fi denote t h e r e s t r i c t i o n of f t o t h e domain Ui . Consider fi as an o r d i n a l sequence w i t h va lues i n l o Ui i n t h e l e x i c o g r a p h i c o r d e r i n g . Thus fi becomes a sequence i n A w i t h l e n g t h s t r i c t l y l e s s than o( . L e t t i n g i vary , i t f o l l o w s f rom t h e p reced ing p r o p o s i t i o n 5 .3 . (1 ) (dependent c h o i c e ) , t h a t t h e r e e x i s t two n a t u r a l

i, j > i such t h a t fi i s embeddable i n f . I n o t h e r words, t h e r e e x i s t s a f u n c t i o n h f rom Ui i n t o U , which i s s t r i c t l y i n c r e a s i n g w i t h r e s p e c t t o t h e

l e x i c o g r a p h i c o r d e r i n g , such t h a t f o r each e lement s o f Ui , we have f s Q f h s

modulo A . Suppose f i r s t t h a t s = {i,j) i s an element o f U . Then i t s image t = hs begins w i t h j , hence s 4 t and f s Q ft (mod A) , so f i s good. I n t h e o t h e r case, we have elements i n U beg inn ing w i t h i , j , u ( u j ) , whose images under h a r e elements beg inn ing w i t h j , hence w i t h j , v ( v 7 j ) . As u v a r i e s , t h e r e a r e

a t most f i n i t e l y many va lues v used. Indeed we can never reach t h e images under h o f elements beg inn ing w i t h i,j+l . Hence t h e r e e x i s t s a maximum v . Moreover,

t h e r e e x i s t s a maximum k f o r which a l l those elements beg inn ing w i t h i , j , k have as images, elements beg inn ing w i t h j f o l l o y e d by an i n t e g e r >/ k . Indeed

k = j+l

many k which v e r i f y t h i s c o n d i t i o n , s i n c e those k which do so, a r e l e s s than o r equal t o t h e maximum p o s s i b l e v . The elements o f U beg inn ing w i t h i , j , k have images which beg in w i t h j , k . Indeed, i f an element o f U beg inn ing w i t h i , j , k had an image beg inn ing w i t h

j , k ' & k + l , then t h e elements beg inn ing w i t h i , j , k + l would have images be- g i n n i n g w i t h

Suppose now t h a t s = { i , j , k ) i s an element o f U . Then i t s image t = hs

begins w i t h j , k , hence s -=It and f s Qft, (mod A ) , so f i s good. I n t h e oppos i te case, we have elements o f U b e g i n n i n g w i t h i , j ,k ,u > k , whose images

under h beg in w i t h j , k , v 7 k . I t e r a t i n g t h i s , we o b t a i n a s t r i c t l y i n c r e a s i n g w -sequence i ( 0 ) = i , i ( 1 ) = j , i ( 2 ) = k , ... such t h a t , f o r each n a t u r a l number r , t h e elements o f U which beg in w i t h i ( O ) , i(l), ... , i(r), .. have images under h which b e g i n w i t h i(l), ... , i ( r ) . By t h e d e f i n i t i o n ' o f b a r r i e r , t h e r e e x i s t s an r f o r which

s = i ( O ) , ..., i ( r ) i s an element o f U ; i t s image t = hs s a t i s f i e s s 4 t and f s ,<ft (mod A) , hence f i s good. 0

From t h e p reced ing p r o p o s i t i o n , i t f o l l o w s t h a t i f A i s an o( - b e t t e r p a r t i a l

o r d e r i n g f o r e v e r y denumerable o r d i n a l cg , then e v e r y b a r r i e r sequence w i t h va lues i n A i s good. I n o t h e r words, A i s a b e t t e r p a r t i a l o r d e r i n g w i t h r e s p e c t t o b a r r i e r s , o r a - b e t t e r p a r t i a l o r d e r i n g i n t h e sense o f 5 6 below.

A , by r e p l a c i n g each e lement o f Ui by i t s h e i g h t modu-

numbers

j

j

s a t i s f i e s t h i s c o n d i t i o n . And on t h e o t h e r hand, t h e r e a r e o n l y f i n i t e l y

j , k ' a k + l , c o n t r a d i c t i n g t h e m a x i m a l i t y o f k .

Chapter 8 225

Thus A i s a bet ter partial ordering, in view of the equivalence of b o t h notions, which will be proved in 9 7 below (modulo the axiom of choice).

5 .5 . Let us denote the barr ier of a l l pairs of natural numbers by i t s lexicographic rank &J . Given a par t ia l ordering A , a necessary and suff ic ient condition for the partial ordering of inclusion among i n i t i a l intervals of A t .be a well partial orderinq, i s that every barrier sequence of dent choice). Recall t h a t the par t ia l orderinq of inclusion among i n i t i a l intervals of A i s well-founded i f f A i s a well par t ia l ordering, hence i f f every a - s e - quence in A i s good: see ch.4 5 3.2.(2) and 4.1.

0 Suppose f i r s t t h a t the i n i t i a l intervals of A form a well oar t ia l ordering under inclusion. Consider a barr ier sequence from U 2 of natural numbers i , j > i , l e t ai . denote the value taken by th i s barrier

,J sequence a . Since A i s a well par t ia l ordering, extract an increasing sequence from the sequence of the a

(mod A ) : see ch.4 5 3.2.(3). Simultaneously, preserve only those values a for which bo th indices occur in the preceding extracted sequence. By successive i terat ions, extract from the double sequence of the a new inf i - nite double sequence, which upon renumbering, sa t i s f ies for each natural number i , the inequalities a i , i + l & ai , i+2 4 ... ,< a i , j ,<... (mod A ) for a l l j > i .

For each natural number i , l e t Ai be the i n i t i a l interval of A generated by the ai ( i fixed and j varies and > i ) . By hypothesis, the Ai form a well par t ia l ordering under inclusion: the i r sequence i s good (see ch.4 5 3.2 . (2) ) . I n other words, there ex is t two integers i and j > i with Ai included in A . Since the element ai belongs t o Ai thus t o A , then there exis ts an integer k > j with ai ,j. < a j , k (mod A ) . B u t we have the con-

secutivity { i , j } a { j , k \ , so that the barrier sequence a i s good. Conversely, suppose t h a t the i n i t i a l intervals of A do n o t form a well partial ordering under inclusion. Then ei ther th i s par t ia l ordering o f inclusion i s n o t well-founded: then A i s n o t a well par t ia l ordering (ch.4 5 4.1, dependent choice), and there exis ts a bad a-sequence u i ( i integer) in A (ch.4 3 3.2 . (2) , dependent choice). I n th i s case, the barrier sequence of W 2 into A , defined by ai - u i fo r a l l j > i , i s bad. Or the par t ia l ordering of inclusion among i n i t i a l intervals o f A i s well- founded b u t not a well partial ordering: hence there ex is t in f in i te ly many mutually non-inclusive i n i t i a l intervals . Choose an w-sequence of such intervals A D , A1, . . . , Ai , . . ( i natural number ; uses denumerable subset axiom).

2

W L into A i s good (sufficiency uses depen-

into A ; for each pair

, which we renumber by aD,l ,< aD,? 4 .... 0 , j ,<. a o , j d...

ai ,j

, j

j , j j

, j -

To each p a i r o f n a t u r a l numbers which belongs t o

Ai y e t n o t t o A ( coun tab le axiom o f c h o i c e ) . It s u f f i c e s t o p rove t h a t t h e

b a r r i e r sequence a thus d e f i n e d , i s bad (mod A) . Indeed, g i v e n any t h r e e i n t e - gers i < j < k we have {i,jj Q i j , k } and y e t aiij$ aj,k (mod A) . F o r i n

t h e oppos i te case t h a t ai,j< aj,k , s i n c e a . be longs t o A . by d e f i n i t i o n

o f t h e i n i t i a l i n t e r v a l A j , we would have a. e lement o f A . : c o n t r a d i c - t i o n . 0

I n e x e r c i s e 2, we g i v e an example o f a w e l l p a r t i a l o r d e r i n g , a re f i nemen t o f

Rado's p a r t i a l o r d e r i n g , which i s n o t a b e t t e r p a r t i a l o r d e r i n g , b u t f o r which

every b a r r i e r sequence d e f i n e d on w 2 i n t e r v a l s fo rm a w e l l p a r t i a l o r d e r i n g .

i, j 7 i , a s s o c i a t e an ai ,j

j

J ,k J

1 ,j J

i s good, thus f o r which t h e i n i t i a l

5.6. Given a p a r t i a l o r d e r i n g A , l e t JO(A) = A ; l e t 'J l(A) be t h e p a r t i a l o r d e r i n g o f i n c l u s i o n among t h e i n i t i a l i n t e r v a l s o f A ; f o r e v e r y n a t u r a l

number p , l e t 'J p+l(A) be t h e p a r t i a l o r d e r i n g o f i n c l u s i o n among t h e i n i t i a l

i n t e r v a l s o f 3 (A) . We then o b t a i n t h e f o l l o w i n g g e n e r a l i z a t i o n o f t h e p reced ing p r o p o s i t i o n 5.5; t h e

reader can e a s i l y adapt t h e p reced ing p r o o f t o o b t a i n t h e i n d u c t i o n s t e p f rom

p t o p + l . L e t us denote by I-,)

n a t u r a l numbers . Given a p a r t i a l o r d e r i n g A , a necessary and s u f f i c i e n t cond i -

t i o n f o r t h e p a r t i a l o r d e r i n g 3 (A) t o be a w e l l p a r t i a l O r d e r i n g i s t h a t

eve ry b a r r i e r sequence f rom

den t c h o i c e ) .

I n t h e a l ready mentioned e x e r c i s e 2, we g i v e , f o r each p o s i t i v e i n t e g e r p , an example o f a w e l l p a r t i a l o r d e r i n g which i s n o t a b e t t e r p a r t i a l o r d e r i n g , f o r

which eve ry b a r r i e r sequence w i t h domain wp i s good, y e t f o r which t h e r e e x i s t s a bad b a r r i e r sequence w i t h domain 3 p-l(A) i s a w e l l p a r t i a l o r d e r i n g y e t n o t 3 (A) , f o r t h e g i v e n p a r t i a l o r d e r i n g A .

P

t h e b a r r i e r formed o f a l l p-element subsets of t h e s e t o f

P ptl into A i s good ( s u f f i c i e n c y uses depen-

c3 ptl ; thus

P

5 6 - BETTER PARTIAL ORDERING WITH RESPECT TO BARRIERS, OR -BETTER

P A R T I A L ORDERING

We say t h a t A i s a b e t t e r p a r t i a l o r d e r i n g w i t h r e s p e c t t o b a r r i e r s , o r more

s imp ly a i s good (see t h e d e f i n i t i o n i n 5 2 ) . T h i s n o t i o n i s h i s t o r i c a l l y a n t e r i o r t o t h e b e t t e r p a r t i a l o r d e r i n g , and goes back t o NASH-WILLIAMS 1965' . It i s proved i n NASH-WILLIAMS 1968 t h a t eve ry - b e t t e r p a r t i a l o r d e r i n g i s a b e t t e r p a r t i a l o r d e r i n g ,

b e t t e r p a r t i a l o r d e r i n g , i f eve ry b a r r i e r sequence w i t h va lues i n A

Chapter 8 221

which we obtain in 5 7 tioned a t the end of 5 5 .4 above. A quasi-ordering i s said t o be a of i t s equivalence classes, each formed by elements simultaneously greater and lesser, i s a -better partial ordering.

below. Thus b o t h notions are equivalent, as already men-

better quasi-ordering, i f the partial ordering

- 6.1. Every restriction, and every partially ordered reinforcement of a partial ordering, i s a better partial ordering.

Every -better partial ordering i s a well partial ordering (uses dependent choice; ZF suffices for a countable partial ordering).

0 Since every barrier sequence i s good, in particular every W-sequence (which can be identified with a barrier on singletons) i s good. Then our statement f o l - lows by ch .4 g 3 . 2 . ( 2 ) , using dependent choice. 0

better

6 . 2 . The partial ordering formed by two incomparable elements, i s a -better par- t ial ordering.

0 Let U be a barrier and s an element of U , By 1.6 above, there exists a sequence of f in i te length h of elements ti of U ( i = l , . . . , h ) ; and another sequence of length h + l of elements ti ( i = 1, . . . , h + l ) , with s a tl 4 t2 4 . . . Q th

th = tA+l . Then there necessarily exist two successive elements with the same image in the partial ordering, since there are only two elements in i t . 0

and s U t i 4 t i 4 . . . at,!+, , b o t h having the same last term

6.3 . Every well-ordering, and in particular every finite chain, i s a -better partial ordering.

0 Let U be a barrier. Take an w-sequence of elements si of U ( i natural number) , such t h a t s i for each i . Then for each function f from U into the well-ordering A , there exists an i satisfying f ( s i ) s f ( s i+ l j (mod A ) ; so t h a t f i s good. 0

6 . 4 . Let A be a partial ordering. Part'tion i t s base into two disjoint subsets, thus obtaining the two restrictions B and C . B a n d C are -better partial orderings, then A i s a -better partial ordering.

0 Suppose,that our conclusion i s false. Let U be a barrier and f be a bad U-sequence in A . Partition the elements o f U into two classes, according t o whether the image under f belongs t o the base l B l or t o I C l . By the barrier partition theorem 1.4, there exists a sub-barrier V of U with f/V a bad barrier sequence from V into B or into C . Hence either B or C i s n o t a -better partial ordering. 0

Every f i n i t e par t ia l ordering i s a bet ter par t ia l ordering.

If A i s a -bet ter partial ordering, then so i s every par t ia l ly ordered extension of A t o i t s base augmented by a f i n i t e number of elements.

0 These statements follow from the preceding proposition, since the ordering on a singleton i s i t s e l f a -bet ter par t ia l ordering, and even a bet ter par t ia l ordering, as noticed in 5 5.1 above.

-

6.5. .If A B are both -bet ter par t ia l orderings, then the direct product A % B i s a -better par t ia l ordering (d i rec t product i s defined in ch.4 5 7 ) ; see NASH-WILLIAMS 1965' p . 706.

0 Let U be a barr ier and f a function from U into A % B . Call p the projection which, t o each ordered pair (a ,b) tes the f i r s t term a , and cal l q the projection which associates the second term A i s a -bet ter par t ia l ordering, by 2 . 1 there exis ts a barrier V included in U , with (p,f)/V perfect. Since B i s a -better partial ordering, there ex is t two elements s , t in V with sa t and qfs ,< q f t (mod B ) . Moreover p f s s pf t (mod A ) , since p,f i s perfect. Thus f s 4 f t modulo the product A x B : hence f i s good.

in the product of the bases, associa-

b . Since

6 .6 . To each natural number i , associate a f i n i t e s e t Fi of elements; so t h a t , fo r fixed i , the elements in Fi are mutually incomparable, and so that each element in Fi+l obtain a s t r a t i f i e d par t ia l ordering, in the sense of ch.2 5 5.2; moreover th i s ordering i s well-founded and Fi i s the s e t of elements of height i (ch.2 5 3.2) .

This s t r a t i f i e d partial ordering i s a bet ter par t ia l ordering. Particular case of the proposition 6.7 below. Assuming that the base i s denumerable (which in general requires the countable axiom of choice on f i n i t e s e t s ) , we have the following easy proof.

0 Star t with the se t of ordered pairs o f natural numbers, hence the direct pro duct of w with i t s e l f , which by the previous propositions 6.3 and 6 . 5 , i s a

take a s e t of ordered pairs of natural numbers of the form (x,y) with x + y = a. = Card Fo . Then for F1 take a s e t of ordered pairs (x,y) with x 2 a. and y > a o and x + y = a. + al where al =

Card F1 ; and so for th . The Given s t r a t i f i e d par t ia l ordering i s isomorphic with a res t r ic t ion of the direct Droduct W x W ; by 6.1, th i s i s a -bet ter par t ia l ordering. 0

i s s t r i c t l y greater than a l l the elements of Fi . We thus

- better partial ordering. For Fo

6.7. Let A be a par t ia l ordering. If every proper i n i t i a l interval of A i 2 f i n i t e , then A i s a -bet ter par t ia l ordering (POUZET 1977, unpublished).

Chapter 8 229

As in the preceding statement, th i s proposition gives well partial o rder ing whose elements have f i n i t e heights; yet they are more varied: i t i s no longer required t h a t each element of height i + l be greater t h a n every element of height i . 0 Note f i r s t t h a t every non-empty subset of the base has a minimal element; for otherwise, th i s would yield an in f in i te proper i n i t i a l interval . Moreover every free s e t , or every antichain, i s f i n i t e ; thus A i s a well partial ordering. By hypothesis, each element of the base has a height which i s a natural number. Moreover, i f A i s in f in i te , then by our hypothesis A i s directed, thus A i s an ideal . For otherwise, i f a and b are two elements without any common upper bound, then ei ther the s e t of non-upper bounds of a , o r the s e t of non-upper bounds of b , i s an in f in i te i n i t i a l interval which i s d i s t inc t from A : contradiction. Let U be a barrier and g a function from U into A . We shall prove that g i s good. Let h be the function which, t o each element in the base \ A 1 , associates i t s height (mod A ) , which i s a natural number. Let f be the compo- s i t ion hog . Since the chain (A) of the heights i s a bet ter par t ia l ordering, there exis ts a barrier V included in U , such that the rest r ic t ion f/V i s perfect (see 2 . 1 above). Let V ' be the subset of the squared barrier V , formed of the unions s u t of elements s , t in V , such t h a t s 4 t and f s = f t . By the barr ier par t i t ion theorem 1.4, there exis ts an in f in i te s e t H

of natural numbers which e i ther only contains elements of V ' , or only contains elements of V 2 - V ' . Let W be the barrier V rest r ic ted t o H ; then W i s e i ther included in V ' or included in V 2 - V ' . I n the f i r s t case, there ex is t s a natural number p such t h a t f s = p for every element s in W . Indeed, given two elements s , t of W , by 1.5.(2) there ex is t s a third element u of W with two f i n i t e sequences of successive elements of W , say s Q S ~ . . . Q U and t q t l 4.. .a u . Then we have

f s = f ( s ) = ... = fu and f t = f ( t l ) = ... = fu . For each element x of the base I A I whose height (mod A) i s p , l e t W x be the subset of W formed of those s such t h a t gs = x . The elements x are mutually incomparable (mod A ) , hence there are only f i n i t e l y many such. Thus there ex is t s an x with a barrier X included in W x . For two elements s , t in X , which we can take as succes- s ive, we have gs = g t ; hence g i s good. In the second case, recall t h a t f i s perfect. T h u s for every s , t in W , the condition s d t implies f s , < f t . B u t here f s # f t and so f s < f t . Let k be the function from the se t w of the natural numbers, into (13,

which t o each natural number i , associates the least j for which each element of height 6 i i s less (mod A ) t h a n every element of height j , thus also less than every element of height >, j . This value j = ki ex is t s , since

-

2

2

1

for each element a with height i , the s e t of non-upper bounds of a i s a proper i n i t i a l interval of A ; hence i t i s f i n i t e and there are only f i n i t e l y many heights of i t s elements. By 1 . 7 above, there e x i s t two elements s , t in W satisfying s a t and k fs< f t . Thus each element of the same height as g t i s greater (mod A) t h a n every element of the same height as gs 6 g t (mod A) : hence g i s good. 0

gs ; in par t icular

6.8. Let A be a well partial ordering which has f in i te ly many inf in i te ideals. Then A i s a -bet ter partial ordering. Let A be a par t ia l ordering which has only f in i te ly many inf in i te i n i t i a l intervals. Then A i s a -bet ter partial ordering (POUZET 1977, unpublished; uses dependent choice; ZF suffices i f A i s countable).

0 The second assertion follows from the f i r s t , since the partial ordering under consideration i s necessarily well-founded and f in i te ly f ree . Suppose f i r s t t h a t A i s a directed well partial ordering, with no other in f in i te ideal t h a n i t s e l f . Then A has no i n f i n i t e , proper i n i t i a l interval . Indeed, every inf in i te well partial ordering has as a res t r ic t ion , a t l eas t one inf in i te ideal (see ch.4 5 5 . 2 , dependent choice); so by the preceding proposition 6 . 7 , A i s a -better partial ordering. In the general case, we argue by induction. Given a positive integer p , suppose the proposition holds for any well partial ordering with a t most p in f in i te ideals, and l e t A be a well par t ia l ordering with p + l in f in i te ideals. Let I be an ideal of A , which i s maximal with respect t o inclusion. Partition the base I A I into the union C of those ideals d i s t inc t from I , and the complement D of C . The restr ic t ion A/C i s a well par t ia l ordering having only p in f in i te ideals, and the rest r ic t ion A/D i s s t i l l a directed well par t ia l ordering, hence an ideal , having no other in f in i te ideal than i t s e l f (provided D iS i n f i n i t e ) . Each i s thus a -better partial ordering; so by the previous 6.4, A i s a -better partial ordering. 0

A

6 .9 . Let A be a -better partial ordering. Then the se t of a l l words ( i . e . f i n i t e sequences) A forms a -bet ter par t ia l ordering under embeddability- (uses dependent choice).

Suppose on the contrary t h a t there exis ts a barr ier U and a bad U-sequence f taking as values words. The par t ia l ordering of words i s a well-founded par t ia l ordering: see ch.4 5 2 . So we can assume that f i s minimal bad: see theorem 2 . 2 above, dependent choice. Since f i s bad, for every s in U , the word f s i s non-empty.

-BETTER PARTIAL O R D E R I N G OF WORDS

Chapter 8 23 1

L e t g and h be b a r r i e r sequences w i t h domain U , d e f i n e d as f o l l o w s . Fo r each s i n U , t h e va lue gs i s t h e word composed o f t h e f i r s t t e rm o f f s ; t h e

va lue hs i s f s w i t h i t s f i r s t t e r m removed. By 2.1 above, t h e r e e x i s t s a sub- b a r r i e r V o f U such t h a t g/V i s p e r f e c t ( s i n c e A i s a b e t t e r p a r t i a l o r -

d e r i n g ) . On t h e o t h e r hand, t h e r e s t r i c t i o n h/V i s good; indeed s i n c e f i s minimal bad, i n go ing f rom f t o h , f o r each s i n V , t h e word f s i s r e -

p laced by hs which i s s t r i c t l y l e s s than f s w i t h r e s p e c t t o embeddab i l i t y . Thus t h e r e e x i s t two e lements s , t i n V w i t h s Qt and hs,< h t w i t h r e s -

p e c t t o embeddab i l i t y . Now, as gs,< g t (mod A) , we have f s , ( f t w i t h respec t

t o embeddab i l i t y ; so f i s good: c o n t r a d i c t i o n . 0

-

6.10. -BETTER QUASI-ORDERING OF ORDINAL-INDEXED SEQUENCES

Genera l i ze as f o l l o w s t h e p r e v i o u s p r o p o s i t i o n . L e t A be a - b e t t e r p a r t i a l o r d e r i n g . Then any s e t o f o rd ina l - i ndexed sequences w i t h va lues i n A

1968; t h e f o l l o w i n g p r o o f i s due t o MILNER 1984, unpubl ished; uses dep. cho ice ) . Cl L e t B be a s e t o f o r d i n a l - i n d e x e d sequences i n A : t h i s B i s quasi -ordered

under embeddab i l i t y (ch.4 5 2) . We can assume t h a t B reduces t o a p a r t i a l orde-

r i n g , by r e p l a c i n g sequences by t h e i r equ iva lence c lasses under embeddabi l i ty . Take t h e r a n k i n g f u n c t i o n s which t o each sequence

Suppose t h a t B i s n o t a - b e t t e r p a r t i a l o r d e r i n g : t h e r e e x i s t s a b a r r i e r U and a bad U-sequence w i t h va lues i n B . By theorem 3.5 (dependent cho ice ) , t h e r e e x i s t s a min imal bad b a r r i e r sequence t h e b a r r i e r Dom f . P a r t i t i o n t h e elements s o f U i n t o t h r e e d i s j o i n t c lasses , acco rd ing t o t h e t h r e e f o l l o w i n g p o s s i b i l i t i e s : e i t h e r t h e sequence f ( s ) has l e n g t h 1 , o r i t s

l e n g t h i s a l i m i t o r d i n a l , o r i t s l e n g t h i s a successor o r d i n a l s t r i c t l y g r e a t e r t han 1. Using t h e b a r r i e r p a r t i t i o n theorem 1.4, we can assume t h a t t h e e n t i r e

b a r r i e r U reduces t o one o f t h e t h r e e cons ide red c l a s s e s . I n t h e f i r s t case, f o r each s i n U t h e sequence f ( s ) reduces t o an element o f A . Since A i s a - b e t t e r p a r t i a l o r d e r i n g , t h e U-sequence f i s necessa- r i l y good: c o n t r a d i c t i o n .

Examine t h e second case where a l l l e n g t h s a r e l i m i t o r d i n a l s . Take any two

success ive e lements s 4 t i n U , so t h a t f ( s ) $. f(t) under embeddab i l i t y . S ince t h e l e n g t h o f f ( s ) i n t e r v a l of f ( s ) which i s non-embeddable i n f ( t ) : see ch.4 g 2.1. Consider

t h e square b a r r i e r V = U and t o each eiement v o f V assoc ia te i t s i n i t i a l i n t e r v a l s which belongs t o U and t h e f i n a l i n t e r v a l t = v minus i t s m i n i - mum i n t e g e r , so t h a t sa t and v = s u t . Then t o t h i s v a s s o c i a t e g ( v ) , t h e minimum p roper i n i t i a l i n t e r v a l o f f ( s ) which i s non-embeddable i n f ( t ) .

forms a - b e t t e r q u a s i - o r d e r i n g under embeddab i l i t y (NASH-WILLIAMS

u assoc ia tes t h e l e n g t h o f u.

(mod B, J' ) , say f ; we c a l l again U

i s a l i m i t o r d i n a l , t h e r e e x i s t s a p roper i n i t i a l

2

Note t h a t $ ( g ( v ) ) G s ( f ( s ) ) s ince g (v ) has length s t r i c t l y smaller than f ( s ) . Therefore f foreruns r~ (mod B , $ ) y e t g does not reduce t o a r e s t r i c t i o n of f . Fina l ly g i s bad: take any two successive elements v c l w i n V and the corresponding i n i t i a l i n t e rva l s S Q t i n U ; then g ( v ) i s non-embeddable in f ( t ) thus non-embeddable in g(w) which i s an i n i t i a l in te rva l of f ( t ) . This cont rad ic t s our hypothesis t h a t f i s minimal bad. Examine the t h i r d case where a l l lengths a r e successor ord ina ls d i f f e r e n t from 1. To each element s of U a s soc ia t e the l a s t term l ( s ) of the sequence f ( s ) and the sequence g ( s ) = f ( s ) minus i t s l a s t term. By 2 . 1 we can replace U by a sub-barrier again ca l led U , such t h a t t he U-sequence 1 i s per fec t . Therefore the U-sequence g must be bad, s ince f i s bad. Take the square b a r r i e r V = U 2 , and t o each element v of V a s soc ia t e the i n i t i a l in te rva l s of v which belongs t o U . Then p u t h ( v ) = g ( s ) . Note t h a t $ ( h ( v ) ) = $ ( g ( s ) ) < S ( f ( s ) ) , therefore f foreruns h (mod B , I? ) y e t h i s not a r e s t r i c t i o n of f . Fina l ly h is bad; indeed with the same notations than i n t he second case , h ( v ) = g ( s ) i s non-embeddable i n h ( w ) =

g ( t ) . This cont rad ic t s our hypothesis t h a t f i s minimal bad. 0

5 7 - EQUIVALENCE OF BOTH NOTIONS OF BETTER P A R T I A L O R D E R I N G : CHAIN SEQUENCE I N A PARTIAL ORDERING

Every -be t t e r pa r t i a l ordering i s a be t t e r pa r t i a l ordering (uses dependent chotce) ; therefore both notions coincide, by 5.4. 0 Let A be a - b e t t e r pa r t i a l ordering. By the preceding 6.10, every s e t of ordinal-indexed sequences i n A forms a -be t t e r quasi-ordering under embeddabi- l i t y , thus a well quasi-ordering by 6 .1 . Using 5 .3 . (2 ) , we see t h a t A is an * -be t t e r ordering f o r each o( : i n o ther words a be t t e r pa r t i a l ordering. 0

Consequently i n 6.10 we can replace be t t e r pa r t i a l ( o r quas i ) ordering.

b e t t e r pa r t i a l ( o r quas i ) ordering by

7 .1 . CHAIN SEQUENCE, DOMAIN CHAIN Let C be a chain and A be a pa r t i a l ordering. A chain sequence i n A , or C-sequence i n A i s a couple (C, f ) where f i s a function whose domain i s the base I C I and whose range i s included i n 1 A I . The chain C i s ca l l ed the domain chain of (C, f ) . I f C is an o rd ina l , we f ind again an ordinal-indexed sequence w i t h length C , as defined in ch.1 5 2 . 2 . In t h i s case i t i s unnecessary t o d is t inguish between (C, f ) and f , s ince Dom f i s well-ordered by the membership r e l a t ion . RESTRICTION, ISOMORPHIC SEQUENCE, EXTRACTED SEQUENCE Return t o the general case of a chain sequence (C, f ) . I f U i s a subset of the base I C l , then the sequence (C/U,f/U) obtained by r e s t r i c t i n g both the chain

Chapter 8 233

and the function t o U i s called the rest r ic t ion of (C,f) t o U . If h i s a n isomorphism taking C on to the chain D , then the chain sequence

(D,f,h-l) image of (C,f) under the isomorphism h . A chain sequence ( D , g ) i s said to be extracted from (C,f) i f f there exis ts a res t r ic t ion of (C,f) which i s isomorphic with ( D , g ) . I f C and D are bo th ordinals, we find again the definition of an extracted sequence (see ch.1 0 2 . 2 ) .

Given the chain sequence (C,f) in A , a chain sequence (C,g) with the same domain chain C i s less t h a n (C,f) i f f gx 4 fx (mod A) for each x in the domain I C l . Analogously for "greater than" . For ordinal-indexed sequences, we find again the notion as defined in ch.4 0 2. A chain sequence (D,g) i s said to be an inf-restr ic t ion of (C,f) i f f the chain D i s a res t r ic t ion of C and i f ( D , g ) i s less t h a n the rest r ic t ion (D,f/ D ) . Note the analogy with barrier sequences, see 2 . 2 above. We say that a chain sequence (D,g) is embeddable in (C,f) or that (C,f) admits an embedding of (D,g) i f f there exis ts an inf-restr ic t ion o f (C,f) which i s isomorphic with less than (C,f) from which (D,g) i s extracted. I n th i s case D i s embeddable in C . However D can be embeddable in C without (D,g) being embeddable in (C,f) : consider two w -sequences, one having the constant value a and the other having the value b incomparable with a . Every chain sequence extracted from a chain sequence (C,f) . However the converse i s fa lse: assume t h a t a < b and consider the sequences which reduce t o the singleton of a (resp. the singleton of b ). The notions of res t r ic t ion , isomorphism, extracted from, less than, inf-restriction and embeddability are reflexive and t ransi t ive. The only ones which are antisymmetric are the notions of res t r ic t ion, less than and inf-restr ic t ion.

i s said t o be isomorphic with (C,f) , and more precisely t o be the

LESSER SEQUENCE, INF-RESTRICTION, EMBEDDABILITY BETWEEN CHAIN SEQUENCES

(D,g) . Equivalently, i f f there ex is t s a chain sequence

(C,f) i s embeddable in

7.2. RIGHT AND LEFT INDECOMPOSABLE CHAIN SEQUENCE Let C be a chain and f a function from I C I into a partial ordering A . We say that the chain sequence i s right indecomposable i f f , for each non- empty final interval D of C , the chain sequence (C,f) i s embeddable in i t s

res t r ic t ion (D,f/ D ) . Analogously define a l e f t indecomposable chain sequence. I f a chain sequence (C,f) i s right indecomposable, then the domain chain C i s i t s e l f r ight indecomposable. However the converse i s fa lse: see 5 5 above. SUM OF CHAIN SEQUENCES Analogous t o the definition of the sum of chains along a chain called a homomorphic image of the sum: ch.2 0 3.6), we define the %of chain sequences f i , where the index i runs th rough the base of a chain I which shall again be called the image chain.

(C,f)

I (which is

For each fi , let Ci be its domain chain, and let C be the sum of the Ci along the image chain I , the bases lCil being taken to be mutually disjoint. Then the sum shall be the couple formed of C and the union of the functions in other words, the common extension f of the fi with domain ICl .

A chain sequence in a partial ordering A is said to be hereditarily indecompo- - sable, or h-indecomposable, iff it is obtained by induction from the following procedure. The chain sequence which reduces to the singleton of an element of the base is h-indecomposable. If a fi (i < a ) set of indices j (i < j < d ) for which fi is embeddable in f is cofinal in ~, then the sum of the fi along or along its converse are h-indecomposable. Moreover, the only h-indecomposable chain sequences in A are those which can be so constructed. Every h-indecomposable chain sequence is right or left indecomposable, according to whether it is a sum along an infinite regular aleph or along its converse. Moreover, the considered infinite regular aleph is the cofinality (in the case of a right indecomposable chain sequence) or the co-initiality (in the left case). The empty chain and the chain sequences reduced to a singleton are the only h-indecomposable chain sequences which are both right and left indecomposable.

fi ;

HEREDITARILY INDECOMPDSABLE OR H-INDECOMPOSABLE CHAIN SEQUENCE

I A I is either 0 or 1 or a regular infinite aleph, and

are h-indecomposable chain sequences such that for each i , the

j

7.3. We immediately extend LAVER'S theorems (ch.6 5 5.4 and 5.5, using axiom of choice) to the case of chain sequences. Let A be a partial ordering, and f be a chain sequence in A whose domain chain i s scattered. Suppose that the h-indecomposable restrictions of f form a well quasi-ordering under embeddability. Then f is a finite sum of h-indecomposable chain sequences. Let f be a chain sequence in A whose domain chain is scattered. Suppose that f is indecomposable and that the h-indecomposable restrictions of f form a well quasi-ordering under embeddability. Then f is h-indecomposable.

7.4. GOOD, BAD BARRIER SEQUENCE AND FORERUNNING REVISITED Consider a partial ordering or quasi-ordering A . Let U be a barrier and f be a barrier sequence which to each element of U associates a chain sequence with values in A . Then such a barrier sequence f is said to be iff there exist two elements r, s of U with r q s and fr,( fs under embeddability; f is said to be bad otherwise, following 5 2 above. To each chain sequence in A whose domain chain is scattered, associate the neighborhood rank of its domain chain: let $ be the ranking function thus defined.

-

Chapter 8 235

Then the notion of a barrier sequence which foreruns another, or which i s a successor of another (mod A, 6 ) with respect t o forerunning i s defined as in 5 3 . 3 above. However our present notation (mod A, $ ) replaces the notation (mod .-$, take the i r values in A and are quasi-ordered under embeddability (mod A) . This forerunning remains reflexive, antisymmetric and t rans i t ive , so that i t defines a partial ordering among barrier sequences which themselves take as values chain sequences in A . As in 3 .5 , we define the notion of a minimal bad barr ier sequence under our new forerunning (mod A, s ) . As before, given an arbi t rary bad barrier sequence f , there ex is t s a minimal bad barrier sequence (mod A, s ) which i s a successor of f

under forerunning. Now we generalize the proposition 4.1 by replacing the s e t of h-indecomposable chains by a s e t , s t i l l denoted by taking the i r values in a given bet ter partial ordering denoted by A . Our se t d i s quasi-ordered under embeddability. Moreover, we assume that (R i s closed under taking any interval of one of the considered chain domains, provided th i s interval yields an h-indecomposable chain sequence. NOW our proofs 4 . 1 and 4 . 2 extend t o the present case of chain sequences in except the f i r s t paragraph in 4.1. Indeed given two successive elements r Us in the barr ier , i f the chain sequence f r reduces to the singleton of an element in A , and i f the chain sequence f s does n o t admit an embedding of f r (since f i s assumed t o be b a d ) , then f s i s not necessarily empty nor necessarily a singleton: i t i s possibly inf in i te , and formed of terms none of which i s greater (mod A) than the unique element of the singleton f r . Moreover, our proof 4.1 corresponds t o the case where A reduces to a singleton. Now i t becomes necessary t o use our hypothesis t h a t A i s a bet ter partial ordering, with possibly inf ini- tely many elements. So that we must replace the f i r s t paragraph in 4.1 by the following argument. 0 Let U be a barrier, the domain of f . Partition U into two dis joint subsets U ' and U " , where U' i s formed of those elements whose image under f i s a singleton, and U " h-indecomposable chain sequence. By the barr ier par t i t ion theorem 1.4, a t l eas t one of these two subsets includes a barrier. I f U ' includes a barr ier , then th i s would yield a res t r ic t ion of f which i s a bad barrier sequence with values in the bet ter par t ia l ordering A : contradiction. Hence there exis ts a barr ier included i n U" . S t i l l denote by U t h i s barr ier , and take u p again the proof 4.1 beginning in the second paragraph, now knowing that a l l the values taken by f are inf in i te , h-indecomposable chain sequences.

) where J.7 designates the s e t of the considered chain sequences which

, of h-indecomposable chain sequences

A ,

i s formed of those elements whose image i s an inf ini te


Now we can take up again the proof 4.2 with "chain sequence in "chain" . Moreover in this proof, instead of assuming the existence of a bad w -sequence, we can assume only the existence of a bad barrier sequence. So that we obtain the following generalized statement. Let A be a better partial ordering. Then every set of h-indecomposable chain sequences in A forms a better qug-orderjny under-embeddability (uses axiom of choice).

A " instead of

7.5. In view of the preceding 7.3, we obtain the following generalization of 4.3. Let A be a better partial ordering. Then every chain sequence in A with a scattered domain chain is a finite sum of h-indecomposable chain sequences. In particular, every indecomposable chain sequence- a scattered domain chain is h-indecomposable (uses axiom of choice).

7.6. Let A be a better partial ordering. Then each set o f chain sequences with values in A embeddability (LAVER 1968, uses axiom of choice). In particular, each set of scattered chains is better quasi-ordered under embeddability: take the preceding statement where A reduces to a singleton. 0 We know by 7.5 that each set of h-indecomposable chain sequences forms a better quasi-ordering under embeddability. By 6.9 each set of words, or finite sequences composed o f such h-indecomposable chain sequences forms a better quasi-ordering. This subsists for each set of finite sums of h-indecomposable chain sequences, which yields a reinforcement of the previous quasi-ordering: see 5.1. Now by 7.5, every chain sequence with a scattered domain chain is such a finite sum of h-indecomposable chain sequences. 0

and with scattered domain chains is better quasi-ordered under _l__l ____

7.7. If. A i s a better partial ordering, so i s the partial ordering 3 ( A ) of initial intervals of A (with respect to inclusion); uses axiom of choice. 0 To each initial interval X associate any ordinal-indexed sequence of elements of X , such that every element of X

corresponding sequence. Then by 6.10 our sequences constitute a better quasi- ordering under embeddability. Therefore the corresponding initial intervals constitute, under inclusion, a partial ordering which reinforces the quasi- ordering o f sequences, hence a better partial ordering by 5.1. 0

has an upper bound in the

Chapter 8 2 3 1

EXERCISE 1 - Every f i n i t e l y f r e e p a r t i a l ordering has a cof ina l r e s t r i c t i o n which i s a b e t t e r pa r t i a l ordering (POUZET 1979, unpublished, answering a conjecture due t o GALVIN; uses axiom of choice) .

1 - Let A be a f i n i t e l y f r e e p a r t i a l ordering. Pa r t i t i on A i n t o a f i n i t e union of i dea l s : ch.4 5 5.3 , axiom of choice. For each i d e a l , take a cofinal r e s t r i c t i o n which i s a well-founded p a r t i a l ordering, hence a d i rec ted well p a r t i a l ordering: ch.4 5 5.4. Then in view of 6.4 above, i t would su f f i ce t o prove our asser t ion f o r each d i rec ted well pa r t i a l ordering.

2 - Let A be a d i rec ted well pa r t i a l ordering. By ch.7 5 3.11, there e x i s t s a cofinal r e s t r i c t i o n of A which i s isomorohic with the d i r e c t product of a f i n i t e number o f regular alephs. T h i s cof ina l r e s t r i c t i o n is a b e t t e r p a r t i a l ordering by 6 .3 and 6 .5 above.

EXERCISE 2 - A well pa r t i a l ordering f o r which every b a r r i e r sequence on W i s good, b u t which i s not a b e t t e r p a r t i a l ordering.

Let A be the pa r t i a l ordering defined on a l l ordered t r i p l e s of natural nunbers x , y , z by the following condition: ( x , y , z ) 4 ( x ' . y ' , z ' ) i f xb x ' and y,( y ' and z 4 z ' and addi t iona l ly e i t h e r x = x ' , or x K x ' and y < x ' , or f i n a l l y x < x ' and z C y ' .

1 - Prove the t r a n s i t i v i t y of the above. Note t h a t A i s well-founded, there being only f i n i t e l y many predecessors of (x ,y ,z ) . Prove tha t A i s f i n i t e l y f r e e , hence a well p a r t i a l ordering. For th i s , note t h a t t he d i r e c t nroduct ( x,< x ' and y,< y ' and z ,< z ' ) i s a well pa r t i a l ordering. Suppose t h a t there e x i s t s an w-sequence of t r i p l e s (x i ,y i , z i ) where i i s a natural number , which a re mutually incomparable (mod A) . Then e x t r a c t an a - sequence w i t h x i increas ing , yi increas ing , z i increasing. Since incomparabili ty requi res t h a t x i be s t r i c t l y increasing i n i , there e x i s t s an i f o r which xi > yo and xi > xo : cont rad ic t ion .

2 - Prove t h a t A i s not a b e t t e r p a r t i a l ordering; indeed we have the bad b a r r i e r sequence with domain i,s3

the s e t of natural numbers), ( w i t h x < y < z ) i s taken i n t o the ordered t r i p l e (x ,y , z ) . To see th i s , l e t s = {x ,y , z} and take an in t ege r r > z , and l e t t = { y ,z , r ) , this being the only possible manner t o ge t sa t . Then (x ,y ,z ) and ( y , z , r ) a re incomparable (mod A) . 3 - Now l e t f be a b a r r i e r sequence w i t h domain w ( the s e t of unordered pa i r s of i n t ege r s ) . For each p a i r s of i n t ege r s , l e t x ( s ) , y ( s ) , z ( s ) be the coordinates of f ( s ) . We sha l l prove t h a t f i s good.

( i . e . the set of a l l 3-element subsets of defined as follows: each 3-element subset {x ,y , z )

Denote by B the d i r e c t product ( x x ' and y , c y ' and z sz' ) , which i s a be t t e r pa r t i a l ordering by 6.3 and 6.5 above. Since A and B a re both based on the same s e t of ordered t r i p l e s , consider f as a b a r r i e r sequence i n B and then replace f by a r e s t r i c t i o n which be per fec t (mod B ) : see 2 . 1 above. Af te r renumbering, this r e s t r i c t i o n of f s t i l l has domain u2 , and now we again consider f as taking values in A . T h u s f o r any two pa i r s of i n t ege r s , say s and t w i t h s Q t , we have x(s),( x ( t ) , and s imi l a r ly w i t h y and with z . Then e i t h e r there e x i s t s , t w i t h s d t and x(s ) = x ( t ) , in which case f i s good and we a re f in i shed .

Or s 4 t implies necessar i ly t h a t x ( s ) < x ( t ) . Then by RAMSEY's theorem, we can require e i t h e r t h a t f o r a l l i n t ege r s t h a t we have the s t r i c t inequal i ty y ( j i , j j ) (y({ i ,k ] ) . In the f i r s t ca se , take a s t r i c t l y increasing Cr)-sequence of in tegers pa i r s give s t r i c t l y increas ing values f o r x . Then f o r h s u f f i c i e n t l y la rge

Y ( \ i O ' i h \ ) < x ( ( ih ' ih+l ) ) and obviously x ( { i O , i h ) ) < x ( i i h , i h + l j ) : so t h a t f i s good. In the second case , take again a s t r i c t l y increasing d - sequence of in te - gers g i v i n g , f o r h s u f f i c i e n t l y l a rge , z({io,il}) < y({ i l , i h ) ) and obviously x({i,-,,il})< x ( { i l , i h ) ) , and so f i s good.

4 - Generalize the preceding f o r the pa r t i a l ordering on the s e t of k-tuples of in tegers , defined by the d i r e c t product and ... and t i o n a l l y e i t h e r x1 = x i , o r x l < x i with e i t h e r x2 < x i o r x 3 < x i o r ...

i < j -= k we have y(J,i,j)) = y({i,k}) , or

i o < i l < ... , so t h a t successive

x l $ x i xke xk , and addi-

o r X k < x i - l .

EXERCISE 3 - Taking again BONNET'S chain show t h a t every non-empty f ina l in te rva l of R i s decomposable. Now using B O N N E T ' S r i g i d chain R such t h a t each loca l automorphism moves only countably many e l e - ments ( see ch.7 exerc. 4 ) , obtain t h a t every i n f i n i t e in te rva l of R i s decompo- sab le (communicated by POUZET).

EXERCISE 4 - Show t h a t each countable pa r t i a l ordering i s an d l - b e t t e r pa r t i a l ordering. More genera l ly take a pa r t i a l ordering A such t h a t every bad sequence in A i s countable.

R = AO+U+ ...+ U+Ai+U+ ... in 4.5 above,

EXERCISE 5 - Given two b a r r i e r s U, V having the same union s e t , ca l l U+V the s e t of unions u u v where u belongs t o U and v belongs t o V and Max u

s t r i c t l y l e s s than Min v . Show t h a t U+V i s a ba r r i e r . Examine the case where V i s the s e t of s ing le tons i n the union of U . Given a b a r r i e r U which does not contain any s ingle ton , ca l l U - t he s e t of elements s - 2 Max s) where s belongs t o U : then U- i s a ba r r i e r .

239

CHAPTER 9

ISOMORPHISM AND EMBEDDABILITY BETWEEN RELATIONS, LOCAL ISOMORPHISM, FREE INTERPRETABILITY, CONSTANT RELATION, CHAINABLE AND MONOMORPHIC RELAT I ON

§ 1 '- PERMUTATION, TRANSPOS ITI ON, LOCAL ISOMORPH ISM AND AUTOMORPHISM

LOCAL

The notions of isomorphism, automorphism and embeddability a re def ned in ch.2 5 3 and ch.5 5 1, and previously used f o r pa r t i a l ordering5 and chains ( o r t o t a l orde- r i n g s ) . We now apply these notions t o a r b i t r a r y r e l a t ions and mult r e l a t ions . For convenience, the de f in i t i ons and statements wi l l be given f o r the case of r e l a - t i ons . Unless otherwise ind ica ted , they can be extended t o the case of mul t i re la - t i ons : t he ro l e played by the a r i t y of the r e l a t i o n , t he re being played by the maximum a r i t y of the component r e l a t ions i n the mul t i r e l a t ion .

1.1. Let E be a s e t , f a permutation of E , and F a f i n i t e subset of E . There e x i s t s a sequence a l , ..., ah of elements of F , without r epe t i t i on , such t h a t f o r each x ,ii F , t he image fx i s equal t o the image of x obtained by composition of the successive t ranspos i t ions ( a l , f a l ) , ... , ( a h , f a h ) . 0 Par t i t i on the elements of F i n t o maximal pa r t i a l o r b i t s of the form (u1,u2 ,..., uk) ( k = pos i t i ve in t ege r ) where u2 = ful , ... , uk = fuk-l . Associate t o each maximal p a r t i a l o r b i t the sequence of t ranspos i t ions

(ukm1,uk) , ... , ( u l , u 2 )

( U ~ - ~ , U ~ ) , ... , (u1,u2) Then each element x of our maximal p a r t i a l o r b i t ( o r t o t a l o r b i t ) has image fx . I t s u f f i c e s t o order i n an a r b i t r a r y manner the s e t of our o r b i t s , hence the s e t of corresponding sequences of t ranspos i t ions .

( u k , f u k ) , i f fuk # u1 ; and the sequence of t ranspos i t ions

i f fuk = u1 ( the case of a cyc le , o r t o t a l o r b i t ) .

1 . 2 . Let R be a r e l a t ion w i t h base E and f a permutation of E . I f f o r each element x iE E , the t ranspos i t ion (x , fx ) i s an automorphism of R , .- then f is an automorphism of R ( L O P E Z 1969).

4 Let n be the a r i t y of R . I f f modifies R , then the re e x i s t s a sequence of n terms xi ( i = 1 ,... , n ) w i t h R ( x l , . .. , xn ) # R(fxl ,... , fxn) . By the preceding proposit ion, there e x i s t s a sequence y l , . . . ,yh formed of elements x i such t h a t the composition of the t ranspos i t ions takes (y l , fy l ) , ... , (yh , fyh )

each x . i n t o fx i and hence modifies R . Then a t l e a s t one of the considered t ranspos i t ions modifies R . 0

1.3. Let R be an n-ary r e l a t ion . I f a t ranspos i t ion ( a ,b ) modifies R , then there e x i s t s a subset F of the base I R I w i t h c a rd ina l i t y & n + l , which con- t a ins the elements a , b and such t h a t ( a ,b ) modifies R / F . 0 Let a l ,..., an be such t h a t R(al ,..., a n ) # R(fal ,..., fa,) , where f denotes the t ranspos i t ion ( a ,b ) . Then necessar i ly a and b occur among the ai and the F t o be the set of t he ai

and the ai = f ( a i ) except where one of these values i s a and the o ther i s b . 0

f ( a i ) ( i = 1, ..., n ) . T h u s i t su f f i ces t o take f ( a i ) , which has ca rd ina l i t y l e s s than o r equal t o n + l : indeed we have

1.4. LOCAL ISOMORPHISM Let R, R ' be two r e l a t ions of the same a r i t y . A local isomorphism from R i n to R ' i s an isomorphism from a r e s t r i c t i o n of R onto a r e s t r i c t i o n of R' . For example, i f R , R ' a r e two p a r t i a l o rder ings , then a local isomorphism from R i n to R' i s a b i j ec t ive mapping f from a subset of the base I R I onto a subset of I R ' l , w i t h f x 4 f y (mod R ' ) i f f x < y (mod R ) , f o r every x , y i n Dom f . In o ther words, f i s order preserving, a s well a s i t s converse f - l . The de f in i t i on of local isomorphism extends immediately t o the case of two multi- r e l a t ions of the same a r i t y . I t follows from ch.2 5 3 , t h a t given two mul t i re la - t ions R , R ' of the same a r i t y , where R = ( R 1 , . . . ,Rh) : the concatenation of the component r e l a t ions R ' = ( R i , . . . ,R,!,) : the concatenation of

R i , . . . , RA ; then a b i j ec t ive mapping f from a subset of I R ( onto a subset of I R ' I , i s a local isomorphism from R i n t o R ' i f f f i s simultaneously a local isomorphism from R1 i n t o R i and ... and from R h i n t o R,!, . For example, i f R , R' a r e chains and S , S ' a r e groups (thus te rnary r e l a t i o n s ) , then the concatenations RS and R ' S ' a re ordered groups provided t h a t t he known axioms f o r ordered groups a m s a t i s f i e d ; then a b i j ec t ive mapping f from a subset o f the base of RS onto a subset of the base of R'S' i s a local isomorphism i f , i n addi t ion t o being order preserving between R and R ' , we have ( f x ) . ( f y ) = f z modulo the group S ' i f f x.y = z modulo S , f o r a l l x , y , z i n Dom f . Consider the empty function introduced in ch.2 5 3 . Then extending the conventions of this paragraph, we say t h a t , f o r every in t ege r n 1 , the empty function i s a local isomorphism from every n-ary r e l a t ion i n t o every o the r n-ary r e l a t ion . Moreover f o r a l l s e t s E, E ' the empty function i s a local isomorphism from the 0-ary r e l a t ion (E,+) i n t o (El,+) and from (E,-) i n t o (El,-) , but no t from ( E , + ) into (El,-) or conversely by exchanging (+) and ( - ) .

R1, .. . , R h ; and

Chapter 9 241

Finally given two multirelations a local isomorphism from R into R ' e i ther when a l l the component relations have positive a r i t i e s , or when, for each index i corresponding t o the 0-ary components R i , R; , these l a t t e r have the same value (+) or ( - ) .

R, R ' of the same a r i t y , the empty function i s

1.5. I f f i s a local isomorphism from R in to R ' , then f res t r ic ted t o an arbi t rary subset of i t s domain i s s t i l l a local isomorphism from R into R ' . If f is a local isomorphism from R into R ' , then the inverse function f - i s a local isomorphism from R ' in to R . If additionally g i s a local isomorphism from R ' into R" , then the composition g,f i s a local isomorphism from R i n t o R " . In par t icular , i f f i s an isomorphism from R o n t o R ' , then every restr ic t ion of f i s a local isomorphism. More par t icular ly , the identity function w i t h domain a subset of the base o f R i s a local isomorphism from R into R i t s e l f . However, a local isomorphism from R R ' i s not i n qeneral ex.tendihl-Mto. an isomorphism from R onto R ' , even i f R and R ' are isomorphic or even ident ical . 0 I f R i s the chain of natural numbers and R ' the chain of negative integers, and f the function taking 0,1, ..., p into f0 = - p - l f l = -p , ... , fp = -1; then f i s not extendible since R and R' are not isomorphic. Another example. I f R = R ' = the chain of natural numbers , then the function taking 0 into 1 i s a local isomorphism; yet i t i s n o t even extendible t o a range which, apart from 1 , contains 0 .

1

1.6. Let R, R ' be two n-ary relations. A suff ic ient condition for a bijection f with domain F , t o be a local isomorphism from R into R ' , i s that for every subset X of F with cardinality 4 n , the function f res t r ic ted t o X . i s a local isomorphism. If R , R ' t ion s t i l l holds, by set t ing n t o be the maximum of the a r i ty . 0 Let x1 ,... ,xn be a sequence o f n elements in F . The s e t X = {xl ,... ,xn ) i s a subset of F and has cardinality 4 n . Hence by hypothesis we have R ' ( f x l ,..., fx,) = R ( x l ,..., x n ) and thus f i s a local isomorphism. Taking R ' = R and taking for f the ident i ty on the base, we find the statement in ch.2 $ 1.1.

are two multirelations w i t h the same a r i t y , then the preceding proposi-

1.7. LOCAL AUTOMORPHISM A jocal automorphism of a relation R i s a local isomorphism from R into R . An automorphism f of R , hence also the rest r ic t ion of f t o an arbitrary subse t of the base, i s a local automorphism of R .

However i n general, a l o c a l automorphism i s n o t ex tend ib le t o an automorphism: see our example i n 1.5 with the chain o f na tu ra l numbers and the l oca l automorphism which takes 0 i n t o 1 . The fo l l ow ing statement i s a p a r t i c u l a r case o f 1.6: Le t R be an n-ary r e l a t i o n and l e t f be a b i j e c t i o n whose domain and range are subsets o f the base I R I . I f every r e s t r i c t i o n o f f t o 6 n elements i s a l oca l automorphism o f R , then f i t s e l f i s a l o c a l automorphism o f R . Moreover i f f i s a permutation o f the base, then i t i s an automorphism of R .

1.8. (1) L e t R be o f a r i t y m and S o f a r i t y n , both having the same base. I f every l o c a l automorphism o f R , def ined on 6 n elements, i s a l o c a l automorphism o f S , then every l o c a l automorphism o f R i s a l o c a l automorphism o f S . I n p a r t i c u l a r , every automorphism o f R i s an automorphism o f S . (2) With the same nota t ions , i f R and S have the same l o c a l automorphisms on

cu la r they have the same automorphisms.

However, two d i s t i n c t chains isomorphic w i t h W , both having the same base, have the same automorphism ( the i d e n t i t y being t h e i r on l y automorphism), y e t do no t have the same l o c a l automorphisms.

4 Max(m,n) elements, then they have the same l o c a l automorphisms; and i n p a r t i -

§ 2 - FREE INTERPRETABILITY, DIMENSIONAL ARITY

Le t R, S be two m u l t i r e l a t i o n s w i t h the same base. We say t h a t S i s f r e e l y i n te rp re tab le i n R i f every l o c a l automorphism o f R i s a l o c a l automorphism o f s . For example, l e t R be a p a r t i a l order ing. Then the r e l a t i o n o f s t r i c t p a r t i a l o rder ing S(x,y) = + i f x t y (mod R) ( i n o ther words i f R(x,y) = t and x # y) i s f r e e l y i n te rp re tab le i n R ; and R i s f r e e l y i n te rp re tab le i n S . The r e l a t i o n o f intermediacy o r betweenness T(x,y,z) = + i f z i s between x and y (mod R) , is f r e e l y i n te rp re tab le i n R . I n general, R i s no t f r e e l y i n te rp re tab le i n T : f o r example take the usual o rder ing o f the in tegers 1, 2, 3; then the permutation which reverses t h i s o rder ing i n t o 3, 2, 1 i s a l o c a l automorphism f o r the betweenness r e l a t i o n T , bu t no t so f o r the order ing R . Another example. L e t R be a chain; the c y c l i c r e l a t i o n S associated with R (see ch.2 5 8.6) s a t i s f i e s S(x,y,z) = + i f x & y & z o r y < z g x o r z I ( x ,<y (mod R) . We see t h a t S i s f r e e l y i n te rp re tab le i n R , bu t i n general R i s no t f r e e l y i n te rp re tab le i n S : take the chain 1 < 2 < 3 and the c i r cu - l a r permutation.

Free i n t e r p r e t a b i l i t y i s r e f l e x i v e and t r a n s i t i v e , so t h a t i t def ines a pre- o rder ing on every se t of r e l a t i o n s w i t h common base. It i s no t antisymmetric;

Chapter 9 243

for example, take the case of an ordering 6 and the s t r i c t ordering < which are freely interpretable each by the other; or take the case of an arbi t rary relation R and i t s negation i R (see ch.2 5 1 . 7 ) . A multirelation freely interpretable in R i s s t i l l freely interpretable in any concatenation RS where S has same base as R . A necessary and suff ic ient condition f o r S t o be freely intrepretable in every component relation of S i s f reely intrepretable in R.

Let R be a relation with base E . If any bijection whose domain and range are subsets of E , i s a local automorphism of R , then R i s freely interpretable in every relation with base E . Take for example the identity relation on E , or i t s negation, or the n-ary re la t ion taking always the value (+) , e tc .

R i s t h a t

2 .1 . Let R, S be two relations with common base, and l e t n be the a r i ty of S (the maximum of th i s a r i t y in the case of a multirelation). Then a suff ic ient ( and obviously necessary) condition for s t o be freely interpretable in R i s that every local automorphism o f R whose domain has cardinality 4 n i s a local automorphism of S . This i s 1.8.(1) in other words.

2 . 2 . DIMENSIONAL ARITY Let R be a relation. The dimensional a r i ty of R shall be the least natural number n f o r which there exis ts a multirelation M whose a r i ty has n for maximum value, and in which R i s freely interpretable. For example, take an equivalence relation classes, say three classes so as t o f i x the ideas. Then in a unary birelat ion, formed of the relation U(x) = + i f f x belongs t o the f i r s t equivalence class , and of V ( x ) = + i f f x belongs t o the second equivalence class ( the third equivalence class being defined by U(x) = V(x) = - ) . Then the dimensional a r i t y o f R i s 1 . For each positive integer n , there ex is t s a denumerable rsl.atioj R whose a r i ty and dimensional a r i t y are b o t h equal t o n . In other words, our n-ary relation R

i s freely interpretable in no multirelation of a r i t y s t r i c t l y less than n (PABION 1970, unpublished).

0 Take a denumerable base E and an n-ary relation R on E , in which every f i n i t e n-ary relation be embeddable. This i s easy, since there are only f in i te ly

many n-ary relations on a s e t of p elements ( p integer): in fac t , power pn such relations. Now suppose that R i s freely interpretable i n a multirelation M formed of h components of a r i ty n-1 . Let (A denote the a r i t y of M , i .e. the sequence of h values equal t o n-1 . The number of e - a r y multirelations on a base of p elements, i s equal t o :

R having a f i n i t e number of equivalence R i s freely interpretable

2 t o the

2 to the power (h.pn-’) , T h u s when taking a base of

ca rd ina l i t y p > h , the number of n-ary r e l a t ions on this base i s s t r i c t l y g rea t e r than the number of p -ary mul t i re la t ions . I n our denumerable base of ca rd ina l i t y : 2 t o the power pn ; so tha t , the b i j ec t ion taking a l , ..., a 1,. . . , p , transforming the r e s t r i c t i o n

P base $1,. .. , p ) , these l a t t e r r e l a t ionsa re mutually d i s t i n c t . Then there necessa- r i l y exist two of our p-tuples, say and

r e s t r i c t i o n s mul t i re la t ion w i t h base l, 1 ,..., p ) . I t follows t h a t the function taking a of R . In o the r words R i s not f r ee ly in t e rp re t ab le i n M : cont rad ic t ion . 0

E , take a set of p-tuples ( a l , ..., a ) ( p i n t ege r > h ) i n t o

R/ial ,. . . ,a 1 i n t o an n-ary r e l a t ion w i t h

P P

( a l ,..., ap) (bl ,..., b p ) , f o r which the

M/{al, ..., ap\ and M/{bl, .... b 1 a re transformed i n t o the same P

a l ,..., i s a loca l automorphism of M , y e t not a local automorphism i n t o b l , ..., bp P

2.3. INTERPRETABILITY ARITY To obtain a common genera l iza t ion of the notions of dimensional a r i t y of a r e l a t i o n , as defined i n t he above 2 . 2 , and t h a t of the dimension of a pa r t i a l o rder ing , as defined i n ch.4 5 7.3 and going back t o DUSHNIK, MILLER 1941, we say t h a t an ord i - nal sequence I of length a , taking natural number values, i s an in t e rp re t ab i - l i t y a r i t y f o r the r e l a t ion R , i f there e x i s t s an o( -sequence obtained from I by replacing each term u i i n I ( thus i < d ) by a r e l a t ion Ri o f a r i t y u i and having the same base as R , so t h a t R i s f r ee ly in t e rp re t ab le i n the sequence of the R i , in the sense t h a t every local automorphism common t o the i s a local automorphism of R . Then f o r an a r b i t r a r y r e l a t ion R , i f we requi re t h a t I have f i n i t e length , then the dimensional a r i t y i s the l e a s t possible maximum of the sequences I which a re i n t e r p r e t a b i l i t y a r i t i e s of R . I f R i s a pa r t i a l ordering and i f we requi re t h a t the r e l a t ions Ri be cha ins , then the l e a s t poss ib le length of t he i n t e r p r e t a b i l i t y a r i t i e s l e s s than o r equal t o the dimension of R i n the sense of ch.4 5 7.3. Problem. I f R i s the i d e n t i t y r e l a t i o n , whose dimension i s 2 , and i f we again require t h a t the r e l a t ions Ri be cha ins , t h e n the l e a s t possible length under consideration i s 0 (the iden t i ty r e l a t ion being f r e e l y in t e rp re t ab le i n the multire- l a t ion reduced t o i t s base) . Apart t h i s case , t he dimension of a p a r t i a l ordering i s of chains.

Ri

I i s obviously

equal t o the l e a s t poss ib le length of i t s i n t e r p r e t a b i l i t y a r i t i e s by means

2.4. Let R , R' be two m-ary r e l a t ions and S , S ' be two n-ary r e l a t ions ; l e t E be the common base of R and S , and E ' the common base of R ' and S ' . I f every r e s t r i c t i o n of the concatenation R'S' 4 n elements i s embeddable i n RS , and i f S i s f r e e l y in t e rp re t ab le i n R , then S ' i s f r e e l y i n - t e rp re t ab le in R' .

Chapter 9 245

Let f ' be a l oca l automorphism o f R ' , such t h a t F ' = Dom f ' has c a r d i n a l i t y

less than o r equal t o n . By hypothesis, there e x i s t s an isomorphism g from the r e s t r i c t i o n (R 'S ' ) /F ' onto a r e s t r i c t i o n o f RS , and there e x i s t s an isomorphism from ( R ' S ' ) / ( f ' ) ' ( F ' ) onto a r e s t r i c t i o n o f RS . Let f = h,f',g-l : then f i s a l oca l automorphism o f R and hence o f S , the l a t t e r being f r e e l y i n te rp re tab le i n R . Thus f ' i s a l oca l automorphism o f S ' . Hence every l oca l automorphism

o f R ' def ined on a t most n elements i s a l oca l automorphism o f S ' . By 2.1

above, S ' i s f r e e l y i n te rp re tab le i n R ' (p roo f communicated by PAILLET). 0

For example, l e t R be a chain and S the associated te rnary c y c l i c r e l a t i o n (ch.2 8.6). Le t R ' and S ' be such t h a t every r e s t r i c t i o n o f R ' S ' w i th car-

d i n a l i t y < 3 i s embeddable i n RS . Then R ' i s a chain and S ' i s the c y c l i c r e l a t i o n associated w i t h R ' .

2.5. I f S i s f r e e l y i n te rp re tab le i n R , then f o r every subset X o f the base,

the r e s t r i c t i o n S/X i s f r e e l y i n te rp re tab le i n R / X . I f S i s f r e e l y i n te rp re tab le i n R and i f R ' i s an extension o f R , then

there e x i s t s an extension o f S which i s f r e e l y i n te rp re tab le i n R ' . 0 Let n be the a r i t y o f S , and (xl,. . . ,xn) be an n-tuple i n the base I R ' I . I f there e x i s t s a l oca l isomorphism f from R ' i n t o R having domain

{xl ,..., xn \ , then p u t S'(x l ,..., xn) = S(fxl ,..., fx,) . This l a s t value does no t depend on the chosen isomorphism, because S i s f r e e l y i n te rp re tab le i n R . Now i f there does not e x i s t such a l oca l isomorphism from R ' i n t o R having

domain {x l ,.... x n \ , then pu t S'(xl ,..., xn) = + . Then S ' i s an extension o f

S and i s f r e e l y i n te rp re tab le i n R ' . 0

2.6. Le t R, S have a common base, and n be the a r i t y o f S . If, f o r each subset X o f the base w i t h c a r d i n a l i t y less than o r equal t o 2n, the r e s t r i c t i o n

S/X i s f r e e l y i n te rp re tab le i n R/X ,& S i s f r e e l y i n te rp re tab le i n R . 0 Suppose t h a t S

automorphism f o f R which modif ies S . Thus there e x i s t s an n-tuple

(a l,...,an)

the se t X = { al,. . . .an,fal,. . . ,fan } has c a r d i n a l i t y a t most equal t o 2n ; and S/X i s no t f r e e l y i n te rp re tab le i n R/X . 0 I f R i s a chain, then the i n tege r 2n can be replaced by n+ l : see 5 5.1 below.

However i n general, for n = 2 , the value 4 cannot be replaced by a smal ler in teger . 0 Take a p a r t i a l o rder ing R constructed from two chains R1, R2 (each w i t h car-

d i n a l i t y >/ 2) , by s e t t i n g each element o f

P

-

i s not f r e e l y i n te rp re tab le i n

o f elements i n the base, w i t h

R . Then there ex i s t s a l oca l

S(al, ..., an) # S(fal, ..., fa,) . Then

R1 t o be incomparable w i t h each


element o f R2 . Take S t o be t h e p a r t i a l o r d e r i n g s i m i l a r l y o b t a i n e d f rom t h e cha ins Sl = R1 and S2 = R i = converse o f R2 . Then S i s n o t f r e e l y i n t e r p r e -

t a b l e i n R , y e t S / X i s f r e e l y i n t e r p r e t a b l e i n R/X f o r each subset X w i t h c a r d i n a l 1, 2 o r 3 . 0

§ 3 - FREE OPERATOR, CONNECTION WITH FREE INTERPRETABILITY

Given two f i n i t e sequences o f n a t u r a l numbers, say m and n , a f r e e o p e r a t o r

9 assoc ia tes t o each m-ary m u l t i r e l a t i o n R an n-ary m u l t i r e l a t i o n (R) hav ing t h e same base. We have t h e a d d i t i o n a l c o n d i t i o n t h a t f o r any two

r e l a t i o n s R, R ' each l o c a l isomorphism f rom R i n t o R ' i s a l s o a l o c a l i s o - morphism f rom T(R) i n t o T(R') . I n o t h e r words, 9 prese rves l o c a l isomorphism. The sequences m, n a r e c a l l e d t h e a r i t i e s o f 9 , and we say t h a t 7 i s an (m,n)-ary ope ra to r . We say t h a t any m-ary r e l a t i o n i s ass ignab le t o 9 . Note t h a t each f r e e o p e r a t o r i s comp le te l y determined by i t s va lues on m u l t i - r e l a t i o n s whose base i s a f i n i t e subset o f t h e s e t ~3 o f n a t u r a l numbers: t h e va lue i n t h e genera l case f o l l o w s immediate ly by u s i n g l o c a l isomorphisms. As

a l l t h e m u l t i r e l a t i o n s based on subsets o f w c o n s t i t u t e a s e t , we can d e f i n e an (m,n)-ary o p e r a t o r as a f u n c t i o n which, t o each m-ary r e l a t i o n based on a

f i n i t e subset o f CJ , assoc ia tes an n-ary r e l a t i o n on t h e same base, w i t h t h e

preceding c o n d i t i o n about l o c a l isomorphisms. A f t e r d e f i n i n g as i n d i c a t e d , we complete by d e f i n i n g t h e va lue taken by 9 f o r each m-ary r e l a t i o n : a l l t h i s w i t h i n t h e framework o f t h e axioms o f ZF. Example. F o r a g i v e n n a t u r a l number n , n e g a t i o n i s a f r e e o p e r a t o r which takes

each n-ary r e l a t i o n i n t o t h e r e l a t i o n hav ing t h e same base and always t a k i n g t h e oppos i te va lue. Another example. The symmetr iz ing o p e r a t o r which takes each b i n a r y r e l a t i o n R

i n t o t h e b i n a r y r e l a t i o n S s a t i s f y i n g S(x,y) = R(y,x) f o r a l l x, y i n t h e

base. A l s o t h e o p e r a t o r which takes each b i n a r y r e l a t i o n R i n t o t h e unary r e l a t i o n S (x ) = R(x,x) . An (m,n)-ary f r e e o p e r a t o r 9 i s determined by t h e ordered p a i r s (R, T(R)) f o r those m-ary m u l t i r e l a t i o n s R hav ing a base o f c a r d i n a l a t most equal t o Max n . Hence t h e r e a r e o n l y f i n i t e l y many f r e e opera to rs o f g i v e n a r i t i e s .

m-ary

3.1. The m u l t i r e l a t i o n S i s f r e e l y i n t e r p r e t a b l e i n R i f f t h e r e e x i s t s a f r e e o p e r a t o r t a k i n g R into S .

Consider t h e case o f two r e l a t i o n s : R i s m-ary and S i s n-ary and S i s

f r e e l y i n t e r p r e t a b l e i n R . F o r each m-ary r e l a t i o n X and each n - t u p l e

(xl,. . . ,xn) o f elements i n t h e base I X I , e i t h e r t h e r e e x i s t s an isomorphism

Chapter 9 2 41

f from X/\x l ,..., x n ) onto a res t r ic t ion R / { fxl ,... ,fxn) : then we define

(3 X ) ( x l , ..., x n ) = S(fx l.... , fxn) ; th i s l a s t value does n o t depend on the chosen isomorphism. Or no such isomorphism exis ts : then we s e t ((?X)(x l , . . . , x n ) = t . 0

3.2. INJECTIVE OPERATOR A free operator .j-> i s said t o be injective i f R # R ' for a l l R , R ' assignable to 9 ; or equivalently i f for a l l R , R ' every local isomorphism from q ( R ) into T(R') i s a local isomorphism from R into R ' . Every injective operator has an inverse. More precisely, i f i s inject ive, then there exis ts a f ree operator 2 such tha t $, T(R) = R for each R assignable

implies that Y(R) # ? ( R ' )

t o 9 . 0 Consider the case of re la t ions, and l e t m , n given an n-ary relation Y and an m-tuple ( x l , ..., x,) of elements in the base I Y I , we define

Y / { x l , ..., xm] i s the image under 9 of an m-ary relation X having the same base; o r ( a Y ) ( x l , ..., xm) = +

Hence, if 9 i s inject ive, then R and F(R) the other. A converse of th i s resul t shall be proved in 3.5 below.

COMPARISON BETWEEN ARITIES We say tha t the a r i ty n (of a multirelation) i s greater t h a n the a r i ty m , i f each term mi of m can be associated with a term n o f n , with m i $ n j , in an injective manner: i . e . two d is t inc t indices i , i ' in m correspond t o two d is t inc t indices j , j ' in n . If the a r i ty n i s greater t h a n the a r i ty m , then for any natural number P, there are more n-ary multirelations with base having cardinality p , than there are of m-ary multirelations with the same base. However, even i f the above condition i s t rue for every natural number p , th i s does n o t necessarily imply that the a r i t y n i s greater than m . 0 For example, take m = (1,l) and n = (0,2) . Then for a base of cardinality p , there are ( 2 t o the power ( l + p ) ) many n-ary birelat ions, with l + p >/ 2p ; yet n i s not greater than m. 0

Problem. I f the a r i t y n i s greater t h a n m , then there obviously exis ts an (m,n)-ary injective operator. Indeed t o each mi-ary component Ri of the multirelation R , i t suffices to associate the n.-ary component S j ( m i Q n j ) whose value only depends on the o f Ri . Conversely i f n i s n o t greater t h a n m , then we conjecture t h a t there exis ts no injective free operator with a r i t i e s ( m , n ) . For example i f m = (1,l) and

be the a r i t i e s of 9. Then

( d , Y ) ( x l ,..., xm) = X(xl ,..., xm) i f the rest r ic t ion

i f there i s no such X . 0

are each freely interpretable in

j

-

( 2 t o the power 2 p ) many m-ary birelat ions, and 2 2

J mi f i r s t terms, that value being equal to that


R = (0,2) , then there e x i s t s no i n j e c t i v e f ree (m,n)-ary operator. Indeed s t a r t w i t h a base o f three elements a, b, c and the b i r e l a t i o n (R1R2) where R1 takes

the value (+) on ly f o r a , and R2 on ly f o r b . Then the on ly l o c a l automorphisms o f t h i s b i r e l a t i o n are the i d e n t i t y on each subset o f the base. On the o ther hand,

any a r b i t r a r y b inary r e l a t i o n must take the same value, f o r instance f o r

and (c,c) , and thus admits a l o c a l automorphism o ther than the i d e n t i t y on a subset o f the base. Obviously the concatenation o f our b inary r e l a t i o n w i t h a

0-ary r e l a t i o n , changes nothing i n the previous discussion. F i n a l l y , the reader who i s tempted by the pseudo-solution which associates t o

(R1R2) the b inary r e l a t i o n S(x,y) = R1(x) A R2(y) , w i l l note t h a t i n j e c t i v i t y i s no longer s a t i s f i e d when R1

(a,a)

takes always the value ( - ) , f o r example.

3.3. PARTIAL OPERATOR Let m t i ons w i t h f i n i t e bases, which i s c losed under r e s t r i c t i o n and isomorphism ( t o be

r igorous i n the frame o f the axioms o f ZF, we assume t h a t the bases o f our mu l t i r e - l a t i o n s are f i n i t e subsets o f the se t

A (m,n)-ary p a r t i a l operator w i t h domain 4 i s a func t ion which t o each

m-ary m u l t i r e l a t i o n R belonging t o (R , associates an n-ary m u l t i r e l a t i o n T ( R ) w i t h the same base, such t h a t 9 preserves l oca l isomorphisms. A necessary and s u f f i c i e n t cond i t i on f o r a p a r t i a l operator 9 t o be i n j e c t i v e , i s again t h a t every l oca l isomorphism from ?(R) i n t o $(R') be a l o c a l isomorphism from R i n t o R ' , f o r every R, R ' belonging t o Dom 9 . CANONICAL EXTENSION

Le t 9 be a (m,n)-ary p a r t i a l operator, where the a r i t y n i s g rea ter than the

a r i t y m , i n the sense o f 3.2 above. We sha l l de f ine as fo l lows an operator 9 =

extending 9, and whose domain contains a l l

F i r s t , t o each term mi o f m , associate i n an i n j e c t i v e manner a term n . o f n , such t h a t mi$ n ; i n the fo l l ow ing we denote mi by m and n by n . Given an m-ary r e l a t i o n R and elements xl,.. ., x i n the base I R I ,

e i t h e r the r e s t r i c t i o n R ' = R/{ xl,. . . ,xn) i s an element o f Dom y, and then

we se t ( 9 =R)(xl,. . . ,xn) = ( T R')(xl,. . . ,xn) ; o r not, and then we se t

be a f i n i t e sequence o f na tura l numbers , and 18 a se t o f m-ary mu l t i r e la -

o f na tura l numbers ) .

m-ary m u l t i r e l a t i o n s .

J j j

R

The operator 9 = thus def ined i s c a l l e d the canonical extension o f . I f 9 i s a l ready def ined f o r a l l m-ary r e l a t i o n s o f c a r d i n a l i t y 6 n (replace by Max n f o r a m u l t i r e l a t i o n ) , then the canonical extension i s the unique f ree operator extending 9 . I n general, i n j e c t i v e .

can be i n j e c t i v e w i thout i t s canonical extension 9 = being

Chapter 9 2 49

Let m = n = 1 and l e t 9 associate t o each unary r e l a t i o n tak ing always the value (+),theunary r e l a t i o n always ( - ) ; y e t 9 i s undefined f o r unary re la t i ons taking a t l e a s t once, the value (-) . Then 9 = takes every unary r e l a t i o n i n t o

the unary r e l a t i o n w i t h same base, always ( - ) ; so t h a t p= i s no t i n j e c t i v e . 0

3.4. Le t k be a na tura l number . I f a p a r t i a l operator 7 i s def ined f o r every

m-ary r e l a t i o n o f c a r d i n a l i t y 4 k and on ly f o r such, and i f i s i n j e c t i v e ,

then the canonical extension 9 = i s i n j e c t i v e .

Moreover i n the case where 9 i s an (m,m)-ary p a r t i a l operator, then we have

( 9 =)- I - - (9 - I )=

3.5. Le t m, n be two a r i t i e s w i t h n greater than m . L e t 9 be an (m,n)-ary

i n j e c t i v e p a r t i a l operator w i t h domain and range @ . Le t k be the l a rges t in teger f o r which & contains every m-ary m u l t i r e l a t i o n o f c a r d i n a l i t y & k . Then there e x i s t s a p a r t i a l operator + extending 7, which i s i n j e c t i v e and def ined f o r every m-ary m u l t i r e l a t i o n o f c a r d i n a l i t y 6 k + l

Consequently every i n j e c t i v e p a r t i a l (m,n)-ary operator, where n i s g rea ter than m , i s ex tend ib le t o an i n j e c t i v e f r e e operator: go from k t o k + l , e tc . u n t i l

reaching Max n . Another consequence i s the fo l l ow ing converse o f 3.2: Le t R be an m-ary re la t i on , S an n-ary r e l a t i o n , w i t h n g rea ter than m . If R and S are each f r e e l y i n te rp re tab le i n the other, then there e x i s t s an i n j e c t i v e f ree (m,n)-ary operator which takes R iw S : s t a r t w i t h the i n jec -

t i v e p a r t i a l operator which takes every r e s t r i c t i o n o f R i n t o the r e s t r i c t i o n

o f S having the same base.

0 Proof o f the f i r s t assert ion. It su f f i ces t o consider the case of an m-ary

r e l a t i o n o f c a r d i n a l i t y k + l , which does no t belong t o the domain td , say R . Le t pk be the r e s t r i c t i o n o f our p a r t i a l operator 9 t o the se t o f a l l r e la - t i ons w i t h c a r d i n a l i t i e s ,< k . Consider the sequence o f r e l a t i o n s

So = ( 9 i ) ( R ) ;

t o the range @I ; then S2 = (9 ;)( 7 -l)(S,) provided t h a t S1 belongs t o

the range 63 ; and so on, u n t i l the f i r s t index h f o r which Sh = (9 E)(T -l)

(Sh-l) does no t belong t o the range @ . Then we def ine ( T+)(R) t o be Sh . To see t h a t t h i s procedure always terminates by g i v i n g a r e l a t i o n which does no t belong t o the range a, note t h a t the Si ( i = O , l , ...) phic, provided t h a t they belong t o @ . Indeed, i f

(POUZET 1973).

So, S1, ... , Sh ( h i n tege r ) , where

then S1 = ( 9 ;)( -')(So) provided t h a t So belongs

are mutual ly non-isomor- f were an isomorphism from

Si onto S . ( 1 6 i < j,< h) , then f would be an isomorphism from Si-l onto

Sj-l (because the i n j e c t i v i t y of 57 and 7 ; ) , and so fo r th ; thus f would be

an isomorphism from So onto Sj-i , hence from R onto 519 -l(Sj-i-l) . But

Sj-i-l belongs t o

Note t h a t 9 +

r e s t r i c t i o n of R which belongs t o the domain 4 , then y+ acts on t h i s r e s t r i c -

t i o n as does . F i n a l l y i t remains t o see t h a t

R , R ' w i t h c a r d i n a l i t y k + l and suppose t h a t 7 ' (R) = F+(R') : we must show t h a t R = R ' . F i r s t no t i ce t h a t R belongs t o J? i f f P+(R) belongs t o a, by the preceding cons t ruc t ion . E i the r R and R ' belong t o 8, and then we have

o f . O r ne i the r R nor R ' belongs t o & . Then we c la im tha t , i f we denote by h the number o f successive t ransformat ions Si associated w i t h R , and by h ' the number s i m i l a r l y associated w i t h R ' , then h = h ' . Indeed suppose t h a t

h < h ' . Then we obta in t h a t R = T-l(S,l,,-h-l) ; and since t h i s S,l,,-h-l belongs

t o @i , i t fo l lows t h a t R belongs t o '4 : con t rad i c t i on . F i n a l l y we ob ta in

Si = S; f o r each i and hence R = R ' . 0

Problem due t o POUZET 1973. Can the above be general ized t o the case o f i n te rp re - t a b i l i t y by l o g i c a l formulas ( f i r s t order p red ica te ca lcu lus w i t h i d e n t i t y ) . That i s , i f R , S are r e l a t i o n s o f the same a r i t y , each o f which i s i n te rp re tab le i n the o ther v i a a l o g i c a l formula, then does there e x i s t a l o g i c a l formula which

operates i n j e c t i v e l y and takes R i n t o S .

J

hence R belongs t o ,& : cont rad ic t ion .

obviously preserves l o c a l isomorphisms. And i f we take a proper

+ i s i n j e c t i v e . Indeed, take two m-ary r e l a t i o n s

y + ( R ) = T(R) and s i m i l a r l y w i t h R ' , so t h a t R = R ' because the i n j e c t i v i t y

§ 4 - CONSTANT RELATION

A r e l a t i o n R w i t h base E i s sa id t o be constant i f every permutation o f E i s

an automorphism o f R . For example the n-ary r e l a t i o n tak ing always the value (+) , the n-ary r e l a t i o n always (-) , the b inary r e l a t i o n o f i d e n t i t y , t ak ing the value (+) when x = y and ( - ) when x # y . The d e f i n i t i o n extends t o m u l t i r e l a t i o n s ; we see t h a t a m u l t i r e l a t i o n i s constant

i f f i t s component r e l a t i o n s are a l l constant. I f R i s constant, then every r e s t r i c t i o n o f R i s constant.

4.1. An n-ary r e l a t i o n R i s constant i f f R(xl ,..., xn) = R(yl ,..., yn) f o r a l l sequences x1 ,. . . ,xn and yl,. . . ,y, such t h a t the t ransformat ion which takes x1

i n t o y1 , ... , and xn i n t o yn , i s an i n j e c t i v e func t i on . I n o ther words, i f f f o r each p a i r o f ind ices (1 5 i < j 6 n) , we have t h a t i, j

Chapter 9 25 1

xi = x . i f f yi = y j . J It fo l lows t h a t R i s constant i f f : (1) a l l r e s t r i c t i o n s o f R .having a same c a r d i n a l i t y , less than o r equal t o the

a r i t y , are isomorphic, and ( 2 ) these r e s t r i c t i o n s are constant.

Nei ther o f the above cond i t ions (1) and (2 ) i s alone s u f f i c i e n t . 0 For every unary r e l a t i o n , any r e s t r i c t i o n o f c a r d i n a l i t y 1 i s constant.

For every r e f l e x i v e and symnetric b inary r e l a t i o n , any r e s t r i c t i o n o f c a r d i n a l i t y 1 o r 2 i s constant. For every tournament (see ch.5 5 2 .7 ) , a l l r e s t r i c t i o n s w i t h

c a r d i n a l i t y 1 are isomorphic; s i m i l a r l y a l l r e s t r i c t i o n s w i t h c a r d i n a l i t y 2. 0

Given an n-ary constant r e l a t i o n R and an n-ary r e l a t i o n S : i f every r e s t r i c - t i o n o f S w i t h c a r d i n a l i t y 6 n i s embeddable i n R , then S i s constant.

Moreover i f S has the same base as R , then S = R . Given an n-ary constant r e l a t i o n R w i t h base E , then f o r every superset E '

o f E , there e x i s t s a constant extension o f R t o E ' . If Card E n , then t h i s extension i s unique.

4.2. A r e l a t i o n R ,w i th - base E i s constant i f f , f o r a l l elements a, b iL E , the t ranspos i t i on (a,b) R . 0 I f R i s constant, then our conclusion i s obvious. Conversely i f R i s not constant, then there e x i s t s a permutation f o f E which modif ies R . Using 1.2 above, there e x i s t s an element a i n E such t h a t the t ranspos i t i on (a,fa) modif ies R . 0

4.3. A r e l a t i o n R i s constant i f f every b i j e c t i o n between two subsets o f the

- base i s a l oca l automorphism o f R . 0 I f our cond i t ion holds, then i n p a r t i c u l a r every permutation o f the base i s an automorphism, hence R i s constant. Conversely i f R i s constant, then an a rb i -

t r a r y b i j e c t i o n f between two f i n i t e subsets o f the base i s extendible t o a permutation o f the base, which by hypothesis i s an automorphism o f R . F i n a l l y we ob ta in the case f o r a b i j e c t i o n w i t h i n f i n i t e domain ( inc luded i n the base), by apply ing 1.7 above. 0

4.4. A necessary and s u f f i c i e n t cond i t i on f o r an n-ary r e l a t i o n R t o be cons- tan t , i s t h a t f o r each sequence xl, ..., xn i n R and each index i ( 1 6 i 6 n) , and f o r each element u i n the base, d i s t i n c t from x1 , ... , xn , the b i j e c t i - ve func t i on f w i t h domain 4 xl,...,xn) which takes xi 1% u and which preserves each x . # xi (1 6 j 6 n) , s a t i s f y R(xl,. . . ,xn) = R(fxl,. . . ,fxn) (communicated by HODGES).

J

If R i s constant, then our conclusion fo l lows from the preceding statement 4.3. Conversely i f our cond i t ion holds, then by t r a n s i t i v i t y , f o r each m,< n , every

b i j e c t i o n o f an m-element se t onto another i s a l oca l automorphism o f R , Using 1 .7 above, every b i j e c t i o n between any two subsets o f the base i s a l o c a l automor-

phism o f R , and so by the preceding 4.3, the r e l a t i o n R i s constant. 0

4.5. An n-ary r e l a t i o n R i s constant i f f every r e s t r i c t i o n o f R having card i -

n a l i t y n+ l i s constant.

I f R i s constant, we already no t iced t h a t each r e s t r i c t i o n i s constant. Conver-

sely, i f each r e s t r i c t i o n t o a t most n+ l elements i s constant, then the condi-

t i o n i n our preceding statement 4.4 i s s a t i s f i e d , since the se t { X l , . . . , ~ n , ~ ) has c a r d i n a l i t y 6 n t l . 0

The propos i t ion i s f a l s e i f we replace ples o f any unary r e l a t i o n , o r any r e f l e x i v e and symmetric b inary r e l a t i o n .

n+ l by n : see i n 4.1 above, the exam-

4.6. For each na tura l number n , l e t H(n) be the number o f p a r t i t i o n s of the

se t {1,2 ,..., n ) ; i n o ther words, the number o f b inary equivalence r e l a t i o n s on t h i s se t . For n = 0 , we adopt the convention t h a t H(0) = 1 , by consider ing t h a t the empty b inary r e l a t i o n i s an equivalence r e l a t i o n .

Then H(0) = H ( l ) = 1 and we have the fo l l ow ing recurrence r e l a t i o n :

H(n+l) = C: H(0) + C; H ( l ) + ... + C: H(n) , where the Cf

usual b inomial coe f f i c i en ts .

Given an equivalence r e l a t i o n on J, l , Z , . . . ,n t l ) , l e t A be the equivalence class o f n + l , and l e t k be the c a r d i n a l i t y o f the complementary se t 21 ,... ,n+l} - A : so 0 4 k 6 n . For each k , the number o f poss ib le choices

fo r A , o r equ iva len t ly f o r the complement o f A , i s C L . Hence the number o f

poss ib le p a r t i t i o n s o f t h i s complement, i s the product

L e t E be a se t w i t h c a r d i n a l i t y g rea ter than o r equal t o n . Then there are ( 2 t o the power H(n) ) many constant n-ary r e l a t i o n s w i t h base E , where H(n) i s the prev ious ly def ined func t i on g i v i n g the number o f p a r t i t i o n s o f the se t

{I ,..., n ) .

Le t U be an equivalence r e l a t i o n w i t h base 41 ...., n} . We say t h a t an

n-tuple (xl, ..., xn) o f elements i n E i s a (U,n)-tuple i f f o r a l l i, j ( 1 6 i < j 6 n) we have xi = x

4.1 above, an n-ary r e l a t i o n R w i t h base E i s constant i f f t o each equivalence r e l a t i o n U based on { 1,. . . ,n} i s associated a value v(U) (equal t o (+) o r ( - ) ) , such t h a t R takes the value v(U) on a l l (U,n)-tuples o f

-

( i s n) are the

C:.H(k) .O

i f f i and j are equivalent (mod U) . By j

Chapter 9 25 3

elements o f E ; the propos i t ion fo l lows. 0

Some i n i t i a l values o f H : we have H(2) = 2 ; t h i s y i e l d s the 4 constant b inary re la t ions : always + , always - , i d e n t i t y and i t s negation. We have H(3) = 5 : t h i s y i e l d s 32 constant te rnary re la t i ons ; then H(4) = 15 ; H(5) = 52 ; and H(6) = 203 .

5 5 - CHAINABLE RELATION

Let A be a chain w i t h base E . Then a r e l a t i o n R w i t h base E i s sa id t o be

A-chainable i f i t i s f r e e l y i n te rp re tab le i n A . I n o ther words, i f every l oca l automorphism o f A i s a l oca l automorphism o f R . A r e l a t i o n R i s sa id t o be chainable i f there e x i s t s a chain A f o r which R

i s A-chainable. For example the s t r i c t t o t a l o rder ing < , obtained from the chain o r t o t a l orde-

r i n g < (mod A) by changing the value (t) i n t o ( - ) along the diagonal, i s A-chainabl e.

The te rnary c y c l i c r e l a t i o n def ined i n ch.2 5 8.6, by g i v i n g (+) t o the t r i p l e (x,y,z) i f f x s y , < z o r y 6 z ,< x o r z & x , c y (mod A) , i s A-chainable.

S i m i l a r l y f o r the r e l a t i o n o f intermediacy o r betweenness (mod A) .

5.1. Le t R and R ' be two n-ary, A-chainable r e l a t i o n s (on the same base I A I ) . I f there e x i s t s an n-element s e t F f o r which R/F = R'/F , then R = R ' . I n o ther words, an n-ary , t i o n t o an n-element set . This fo l lows from ch.2 5 1.1, and the uniqueness o f the isomorphism from a f i n i t e chain onto another equipotent chain.

If R i s A-chainable, then f o r every subset X o f the base, the r e s t r i c t i o n R / X i s (A/X)-chainable.

I f f o r each subset X o f the base o f R w i t h c a r d i n a l i t y 6 n + l ( n = a r i t y o f R), the r e s t r i c t i o n R / X i s (A/X)-chainable, then R & A-chainable.

0 Using 1.8.(1) above, i t su f f i ces t o see t h a t f o r any two sequences o f n elements

x1 < x2 < .. . < xn and y1 < y2 < ... < yn (mod A) , the func t ion tak ing

i n t o yi f o r i = 1, ..., n i s a l oca l automorphism o f R . For t h i s , f i r s t def ine

zi = Min(xi,yi) (mod A)

Now s t a r t w i t h the sequence x1,x2 ,..., xn , then go t o the sequence z1,x2 ,..., xn , then t o the sequence z1,z2,x3 ,..., xn

t o Z1, . . . Jn-1 ,Yn and so f o r t h u n t i l yl,. . . ,yn . 0

Given two m u l t i r e l a t i o n s R, S , then the concatenation RS i s A-chainable i f f R and S are both A-chainable.

A-chainable r e l a t i o n i s determined by i t s r e s t r i c -

-

xi

f o r i = 1 ,..., n : thus zl< z2 < ... < zn (mod A) .

, and so f o r t h u n t i l z1 ,..., zn , and then

An n-ary r e l a t i o n R i s A-chainable i f f , f o r each na tura l number i,< n , the

r e s t r i c t i o n s o f the concatenation AR w i t h c a r d i n a l i t y i are a l l isomorphic. These two observations fo l l ow from the uniqueness o f the isomorphism between two f i n i t e chains o f a same c a r d i n a l i t y .

5.2. A constant r e l a t i o n w i t h base E i i A-chainable f o r every chain A having base E . 0 Indeed every b i j e c t i o n between two subsets o f E , and i n p a r t i c u l a r every l oca l

automorphism o f an a r b i t r a r y chain on E , i s a l oca l automorphism o f the given constant re la t i on : see 4.3 above. 0

However, i f a r e l a t i o n R w i t h base E i s A-chainable f o r every chain A on E , then R i s no t necessar i l y constant. 0 For example, a chain A on two elements i s both A-chainable and (A-)-chainable (where A- i s the converse o f A ) . Another example: the te rnary c y c l i c r e l a t i o n on three elements (see ch.2 9 8.6) i s

A-chainable f o r any one o f the 6 chains A based on these- 3 elements. 0

-

Let R be an n-ary r e l a t i o n w i t h base E . Assume t h a t Card E >/ n+ l and t h a t e i t h e r E i s f i n i t e , o r t h a t E i s i n f i n i t e and orderable: see ch.2 § 2.3.

I f R i s A-chainable f o r every chain A on E , then R i s constant.

It even su f f i ces tha t , f o r every (n+l)-element subset F o f E and every chain A ')" F , the r e s t r i c t i o n R/F is- A-chainable. Compare w i t h exercise 2 below.

- - - - .-

0 It su f f i ces t o see t h a t R s a t i s f i e s the cond i t i on o f 4.4 above. Indeed, take a sequence xl, ..., xn i n E , and l e t u be an element o f E , d i s t i n c t from the

xi ( i = 1, ... ,n) . Now f i x i and l e t A be a chain on E f o r which xi and u

are consecutive, and such t h a t those x . # xi (14 j 4 n) less than xi o r s t r i c t l y g rea ter than u (mod A) . Then the b i j e c t i o n w i t h

domain { xl,. .. ,xn) , which takes xi i n t o u and preserves each x . # xi , i s a l oca l automorphism o f the chain A , hence o f R : thus the cond i t i on i n 4.4

i s s a t i s f i e d (p roo f communicated by HODGES). 0

are e i t h e r s t r i c t l y J

J

5.3. If R i s a chainable r e l a t i o n w i t h base E , then f o r every superset E' - o f E , there e x i s t s a chainable extension o f R w i t h base E ' ; f o r E l - E i n f i - n i t e , t h i s uses the order ing axiom, ch.2 5 2.3. 0 Let A be a chain w i t h base E , and R be an A-chainable r e l a t i o n . It s u f f i -

ces t o take a t o t a l l y ordered extension A ' o f A t o E ' , and then t o apply 2.5 above, t o ob ta in an extension o f R which be A'-chainable. 0

5.4. A necessary and s u f f i c i e n t cond i t ion f o r a r e l a t i o n R t o be chainable i s t h a t every f i n i t e r e s t r i c t i o n o f R be chainable ( s u f f i c i e n c y uses the u l t r a f i l t e r axiom; ZF su f f i ces i f R i s countable).

Chapter 9 255

0 If R i s A-chainable, then we sa id i n 5.1 t h a t f o r every subset F o f the

base, the r e s t r i c t i o n R/F i s (A/F)-chainable. Conversely suppose t h a t every f i n i t e r e s t r i c t i o n o f i s chainable. Then associ-

ate, t o each f i n i t e subset F o f the base, the se t UF o f chains X on F , f o r which R/F i s X-chainable. By hypothesis UF i s non-empty f o r every f i n i t e subse t F o f the base. Given F f i n i t e and a subset G o f F , then each chain

which i s an element o f UF , when r e s t r i c t e d t o

By the coherence lemma i n ch.2 0 1.3 (equ iva len t w i t h the u l t r a f i l t e r axiom, y e t ZF s u f f i c i e n t f o r a countable base), there e x i s t s a r e l a t i o n A based on I R ( , such t h a t f o r each F , the r e s t r i c t i o n A/F belongs t o UF ( so t h a t A/F i s a chain). This r e l a t i o n A i s thus a chain based on I R I . Each loca l automor-

phism o f A having f i n i t e domain, i s a l oca l automorphism o f R : hence R i s A-chainable by 2.1. 0

The preceding p ropos i t i on w i l l be strengthened i n ch.12 5 3.4 and 3.5: f o r each

a r i t y n , there e x i s t s a na tu ra l number p(n) such t h a t every n-ary r e l a t i o n

i s chainable, provided t h a t a l l i t s r e s t r i c t i o n s o f c a r d i n a l i t i e s 4 p(n) .are chainable.

Given a se t o f chainable r e l a t i o n s R , which i s d i rec ted under extension, then

the common extension o f these R , o n the union o f t h e i r bases, i s chainable. 0 This fo l lows from the preceding propos i t ion . Indeed l e t S be the common exten-

s ion of r e l a t i o n s R . Then f o r each f i n i t e subset F o f the base I S 1 , the

r e s t r i c t i o n S/F has an R as an extension: thus S/F i s chainable. 0

R

G , gives an element o f UG .

-

5.5. (1 ) Given an a r b i t r a r y denumerable r e l a t i o n R , there ex i s t s a denumerable subset D o f the base and a chain A w i t h base D , which i s isomorphic w i t h W , and such t h a t the r e s t r i c t i o n R/D ix A-c-. (2) Given an a r i t y n and a na tura l number p , there e x i s t s an i n tege r q>/ p

such t h a t every n-ary >/ q has a chainable r e s t r i c t i o n

o f c a r d i n a l i t y p . 0 (1) Le t n be the a r i t y o f R , which we can assume i s >/ 2 . Le t C be a

chain w i t h base I R I , which i s isomorphic w i t h W ; consider the concatenated m u l t i r e l a t i o n RC . We say t h a t two n-element subsets X, X ' o f the base, are

equivalent iff the r e s t r i c t i o n s (RC)/X and (RC)/X' are isomorphic; s i m i l a r l y f o r i-element subsets ( i Q n) . There are on ly f i n i t e l y many equivalence classes

f o r the equivalence r e l a t i o n thus def ined. Using RAMSEY's theorem (ch.3 5 l.l), there e x i s t s a denumerable subset D of the base, i n which a l l n-element subsets are equivalent, as we l l as a l l i-element

subsets, f o r each i 6 n . L e t A = C/D , a chain isomorphic w i t h W . Then every l o c a l automorphism o f A having domain o f c a r d i n a l i t y \< n i s a l oca l automorphism o f R/D . By 2.1, the r e s t r i c t i o n R/D i s f r e e l y i n te rp re tab le i n A . 0

_ _ ~

0 (2 ) Analogous proof, using the f i n i t a r y version o f RAMSEY's theorem: ch.3 fj 1.3.0

5.6. Given the ord ina l w , the chain o f na tura l numbers , each G)-chainable re la - t i o n i s minimal w i t h respect t o embeddabil ity, among the denumerable re la t i ons ; i .e., every denumerable r e l a t i o n which i s embeddable i n an W-chainable r e l a t i o n R i s isomorphic w i t h R . Le t a be an aleph, i . e . an o rd ina l which i s equipotent w i t h no s t r i c t l y smal ler

o rd ina l . Then each a-chainable r e l a t i o n i s minimal w i t h respect t o embeddabil ity, among the r e l a t i o n s o f c a r d i n a l i t y a . The proposition 5 . 5 . ( 1 ) above asserts tha t , f o r every denumerable r e l a t i o n R , there e x i s t s a denumerable minimal r e l a t i o n which i s embeddable-i!

This r e s u l t does no t extend t o the c a r d i n a l i t y o f the continuum. Indeed by DUSHNIK, MILLER, ch.5 fj 5.2.(1), every chain w i t h continuum c a r d i n a l i t y and which i s embed-

dable i n the rea l s lesser , w i t h respect t o embeddabil ity.

Problem. Does there e x i s t a r e l a t i o n w i t h continuum c a r d i n a l i t y , which i s minimal

among r e l a t i o n s w i t h continuum c a r d i n a l i t y , y e t which i s no t chainable by mean o f the continuum aleph ( i . e . the smal lest o rd ina l w i t h continuum c a r d i n a l i t y ; the

axiom o f choice being used).

R .

has a r e s t r i c t i o n o f continuum c a r d i n a l i t y which i s s t r i c t l y

5.7. For each ordered p a i r of na tu ra l n and r < n , l e t S: denote

the S t i r l i n g number, o r number o f p a r t i t i o n s o f an n-element se t i n t o r non- empty classes. Zn o ther words, the number o f equivalence re la t i ons w i t h cardina- l i t y n and having exac t l y r non-empty equivalence classes. We have So = 1 : the equivalence r e l a t i o n w i t h empty base i s supposed t o e x i s t ;

obviously i t has exac t l y 0 non-empty equivalence classes. We have $ = 0 f o r each n 31 , since an equivalence r e l a t i o n on n elements has a t l e a s t one

non-empty equivalence class.

We have S; = S i = 1

the reCUrSiOn e q u a l i t y S: = Sn-' + r.S:--l f o r every r ( 1 6 r < n ) . Suppose t h a t the se t { 1,2, ... ,n} i s p a r t i t i o n e d i n t o r non-empty classes,

and pu t aside the element n . Then an equivalence r e l a t i o n having r non-empty classes can be obtained, e i t h e r by s t a r t i n g w i t h an equivalence r e l a t i o n on

{1,2 ,..., n-1) having r-1 classes, t o which we add the s ing le ton o f n as an

rth class: t h i s y i e l d s S::: p o s s i b i l i t i e s ; o r by s t a r t i n g w i t h an equivalence

r e l a t i o n on {1,2 ,..., n-1) having r classes, and then adding the i n tege r n

t o one o f these r classes: t h i s y i e l d s r.S;-' p o s s i b i l i t i e s . 0

For example S2 = 3 , corresponding t o the 3 poss ib le p a r t i t i o n s o f { 1,2,3) i n t o

numbers

0

f o r each s t r i c t l y p o s i t i v e i n tege r n ; moreover we have

r- 1

3

Chapter 9 25 7

4 a s ing le ton and i t s complement. Another example,

4 p a r t i t i o n s o f 21,2,3,4} i n t o a s ing le ton and i t s complement, p lus the 3 p a r t i -

t ions i n t o 2 classes each having 2 elements.

Let n be a na tura l number and A be a chain w i t h a base o f c a r d i n a l i t y 3 n . Then there are ( 2 t o the power V(n) ) many A-chainable n-ary re la t i ons , where

S2 = 7 , corresponding t o the

V(n) = O ! S: + l! S! + 2! S; + ... + n! Sl . For p o s i t i v e n , we can

remove the f i r s t term, since S: = 0 . 0 Let F be a subset o f the base, w i t h c a r d i n a l i t y n . We have already seen t h a t

an A-chainable n-ary r e l a t i o n i s determined by i t s r e s t r i c t i o n t o F : see 5.1

above. Thus i t suff ices t o ca l cu la te the number o f n-ary re la t i ons based on F and (A/F)-chainable. Le t us say t h a t two n-tuples xl, ..., xn and yl, ...,yn i n F are equ iva len t if the func t ion mapping xi i n t o yi ( i = 1, ..., n) i s a l oca l automorphism o f the chain v ing (mod A) . Then an n-ary r e l a t i o n R i s (A/F)-chainable i f f we have

R(xl ,..., xn) = R(yl ,..., yn)

Now there are exac t l y V(n) equivalence classes. Indeed, each c lass i s def ined by p a r t i t i o n i n g the se t o f ind ices 1,2, ..., n i n t o a f i n i t e number r o f non-

empty classes, by l e t t i n g xi = x ( 1 G i < j b n ) i f f i and j are i n the

same equivalence c lass : t h i s y i e l d i n g SF p o s s i b i l i t i e s ; and f i n a l l y by t o t a l l y order ing the se t o f these r classes and l e t t i n g xi < x . iff the class o f i i s less than the class o f j , modulo t h i s t o t a l o rder ing o f the classes: t h i s y i e l d s r! = 1.2. ... .r p o s s i b i l i t i e s .

F i n a l l y each A-chainable r e l a t i o n i s def ined by g i v i n g the value (+) o r t he value ( - ) t o each o f the equivalence classes o f the n-tuples i n F : t h i s y i e l d s

(2 t o the power V(n)) p o s s i b i l i t i e s . Note t h a t f o r n = 0 , we have V(0) = 1 , since O! = 1 and So = 1 ; and we obta in the two 0-ary r e l a t i o n s (F,+), (F,-).

Ca lcu la t ion o f the f i r s t values o f V . We have V ( l ) = 1 , g iv ing the two unary re la t i ons , always (+) and always ( - ) . We have V(2) = 3 , g iv ing 8 b inary re la - t ions : always (+) , the chain A , the converse chain, the i d e n t i t y r e l a t i o n , p lus

the 4 o ther r e l a t i o n s obtained by interchanging (+) and ( - ) ( o r equ iva len t l y by rep lac ing (+) by ( - ) on the diagonal) . Then we have V(3) = 13, V(4) = 75 .

A , thus if t h i s func t i on i s b i j e c t i v e and order preser-

f o r a l l equ iva len t n-tuples.

j

J

0

5.8. CONJUNCTION OF A SET OF RELATIONS

I n ch.2 5 1.7, we def ined the con junc t ion o f two r e l a t i o n s having the same base; i n ch.2 5 4.1, we def ined the reinforcement o f a r e l a t i o n , and i n ch.4 5 7.3, the

general ized conjunct ion o f p a r t i a l order ings. Now we extend t h i s no t ion t o a rb i -

t r a r y re la t i ons . An n-ary r e l a t i o n R i s sa id t o be the conjunct ion o f the r e l a t i o n s Ri which are

re in forcements o f R , i f R takes t h e va lue (t) f o r e x a c t l y t hose n - t u p l e s f o r which each Ri t akes t h e va lue (+) . R e c a l l t h a t by d e f i n i t i o n , a r e i n f o r c e m e n t

o f R has t h e same base and a r i t y as those o f R . Reca l l t h a t eve ry p a r t i a l o r d e r i n g A i s t h e c o n j u n c t i o n o f t h e t o t a l l y o rde red

re in fo rcemen ts o f A : t h i s i s an immediate consequence o f t h e re in fo rcemen t axiom i n ch.2 5 4.2 (ZF be ing s u f f i c i e n t i f A I n o t h e r words, eve ry p a r t i a l o r d e r i n g i s t h e c o n j u n c t i o n o f i t s cha inab le r e i n f o r -

cements, which a r e t h e chains, o r t o t a l l y ordered re in fo rcemen ts , t o g e t h e r w i t h

t h e r e l a t i o n always (+) . However t h e r e e x i s t s a b i n a r y r e l a t i o n which i s n o t t h e c o n j u n c t i o n o f i t s chaina-

b l e re in fo rcemen ts . 0 Take t h e b i n a r y c y c l e , o r c y c l e o f c o n s e c u t i v i t y assoc ia ted w i t h t h e c h a i n o f i n t e g e r s f rom 1 t o p ; so t h a t C ( i , i + l ) = + f o r each i = 1, ...,p- 1 and f i n a l l y

C(p, l ) = t C o n l y has t h e

r e l a t i o n always (+) , and t h e r e l a t i o n # ( n e g a t i o n o f t h e i d e n t i t y ) , as i t s

cha inab le re in fo rcemen ts . Hence C i s n o t t h e c o n j u n c t i o n o f i t s cha inab le r e i n - forcements. 0

I f a r e l a t i o n R i s f r e e l y i n t e r p r e t a b l e i n a p a r t i a l o r d e r i n g , t hen R i s n o t

n e c e s s a r i l y t h e c o n j u n c t i o n o f i t s cha inab le re in fo rcemen ts (communicated by HODGES).

0 L e t a,b,c be t h r e e e lements and A t h e p a r t i a l o r d e r i n g d e f i n e d by a < c , and b < c and a I b . L e t R be c o m p a r a b i l i t y modulo A , thus R(a,b) =

R(b,a) = - , va lue (+) o the rw ise . Then a cha inab le re in fo rcemen t o f R i s neces-

s a r i l y a b i n a r y r e l a t i o n always (t): t h e c o n j u n c t i o n o f a l l t hese cha inab le r e i n - forcements cannot be R . 0

Problem. I n t h e oppos i te d i r e c t i o n , does t h e r e e x i s t a r e l a t i o n which i s n o t f r e e l y i n t e r p r e t a b l e i n any p a r t i a l o r d e r i n g , y e t which i s t h e c o n j u n c t i o n o f i t s cha inab le re in fo rcemen ts .

i s coun tab le ) .

; va lue ( - ) o the rw ise (see ch.2 5 8.6). T h i s c y c l e

§ 6 - MONOMORPHIC RELATION

L e t p be a n a t u r a l number; a r e l a t i o n R i s s a i d t o be p-monomorphic i f t h e

r e s t r i c t i o n s o f R t o any two p-element subsets o f t h e base, a r e isomorphic . Every r e l a t i o n i s 0-monomorphic.

Every r e l a t i o n based on a t most p elements, i s p-monomorphic. Any r e f l e x i v e b i n a r y r e l a t i o n i s 1-monomorphic. Any b i n a r y tournament i s 2-mono-

morphic; r e c a l l t h a t a tournament i s a r e f l e x i v e , an t i symmet r i c and comparable

b i n a r y r e l a t i o n : see ch.5 5 2.7. A r e l a t i o n R i s s a i d t o be (6 p)-monomorphic i f R i s q-monomorphic f o r each n a t u r a l number q,<p .

Chapter 9 259

A r e l a t i o n number. p

For example

s s a i d t o be monomorphic i f i t i s p-monomorphic f o r e v e r y n a t u r a l

a chain, and more g e n e r a l l y a cha inab le r e l a t i o n , i s monomorphic. There e x i s t monomorphic non-chainable r e l a t i o n s : f o r i n s t a n c e t h e b i n a r y c y c l e , o r c y c l e o f c o n s e c u t i v i t y , on 3 e lements. I f R i s p-monomorphic, t hen eve ry r e s t r i c t i o n o f R i s p-monomorphic.

S i m i l a r l y f o r Every denumerable r e l a t i o n has a denumerable, monomorphic r e s t r i c t i o n ; s ince , by 5.5. (1) above, i t has a denumerable, cha inab le r e s t r i c t i o n . Fo r each n a t u r a l number p , t h e r e e x i s t s a unary r e l a t i o n which i s p-monomor-

. p h i c y e t n o t (p-1)-monomorphic.

0 Take a base o f p elements, and a unary r e l a t i o n t a k i n g t h e va lue (+) on a s i n g l e e lement , and t h e va lue ( - ) on t h e o t h e r s . 0

For each p , t h e w e x i s t s a r e l a t i o n which i s p-ponomorphic y e t n o t (p+l)-=- morphi c . 0 Take a ( p t 1 ) - a r y r e l a t i o n R s a t i s f y i n g R(xl,. . . , x ~ + ~ ) = + whenever a t l e a s t

two o f t h e e lements x1 , ... , xp+l a r e equa l . Thus a l l r e s t r i c t i o n s o f ca rd ina - l i t y p a r e isomorphic . To a v o i d t h a t R be (p+l)-monomorphic, t ake two d i s - t i n c t (p+ l ) -e lemen t s e t s F and G ; and d e f i n e R t o take t h e va lue (+) f o r every (p+l)-sequence o f elements i n F w i t h o u t r e p e t i t i o n , and ( - ) f o r eve ry (p+l)-sequence i n G w i t h o u t r e p e t i t i o n . 0

(6 p)-monomorphic and f o r monomorphic.

6.1. A r e l a t i o n R i s p-monomorphic i f f each r e s t r i c t i o n o f R w i t h c a r d i n a l i t y p + l & p-monomorphic.

0 Indeed we can go f rom one p-element subset o f t h e base, t o another , by adding an e lement and removing ano the r element; o r by a f i n i t e sequence o f these two opera t i ons . 0

More g e n e r a l l y , g i v e n r & p , a r e l a t i o n R i s r-monomorphic i f f each r e s t r i c - t i o n o f R w i t h c a r d i n a l i t y p + l i s r-monomorphic ( f o r r / p + l ) . A r e l a t i o n i s

i s (6 p)-monomorphic.

-

( & p)-monomorphic i f f each r e s t r i c t i o n w i t h c a r d i n a l i t y 6 p + l

6.2. Every monomorphic r e l a t i o n hav ing i n f i n i t e base i s cha inab le . I n o t h e r words,for

an i n f i n i t e r e l a t i o n , mononorphism f i l t e r axiom and t h e denumerable subset axiom; ZF s u f f i c e s f o r a denumerable r e l a - t i o n .

0 L e t R be o u r monomorphic r e l a t i o n w i t h i n f i n i t e base E . Take a denumerable subset D o f E (denumerable subset axiom); t hen u s i n g 5.5.(1), t h e r e e x i s t s a c h a i n A on D such t h a t R/D i s A-chainable. S ince R i s monomorphic,

c o i n c i d e s w i t h c h a i n a b i l i t y ; uses t h e u l t r a -


t o each f i n i t e subset F o f E , associate an isomorphism f tak ing R/F onto a r e s t r i c t i o n R/f"(F) such t h a t f " (F ) i s a f i n i t e subset o f D . To t h i s isomor-

phism f , associate the image o f the chain A / f " (F) under f-' ; then R/F i s chainable by t h i s image. For each f i n i t e subset F o f E , l e t UF be the se t o f chains w i t h base F and by which R/F i s chainable: by the preceding, UF i s non-empty f o r each F . Moreover, f o r every f i n i t e subset F o f E and every G

included i n F , each chain belonging t o UF , when r e s t r i c t e d t o G , gives an

element o f UG . By the coherence lemma i n ch.2 5 1.3 (equ iva len t t o the u l t r a f i l - t e r axiom), there e x i s t s a chain C based on E , such t h a t f o r every f i n i t e sub-

se t F , the r e s t r i c t i o n C/F belongs t o UF . Then f o r every F , the r e s t r i c t i o n

R/F i s (C/F)-chainable, and hence R i s C-chainable by 5.1. 0

6.3. Le t R be a r e l a t i o n w i t h base E , and p a na tura l number 4 Card E . - If R & p-monomorphic, then R i s a lso r-monornorphic f o r each na tura l number

I n p a r t i c u l a r i f E i s i n f i n i t e , o r even w i t h c a r d i n a l i t y >/ 2p-1 , then R i s

p-monomorphic i f f R i s (6 p)-monomorphic.

Le t F be an a r b i t r a r y r-element subset o f E . Le t (+) denote the isomorphism

type of the r e s t r i c t i o n R/F . We say t h a t an r-element subset o f E has the co lo r (+) i f f the r e s t r i c t i o n of R t o t h i s se t i s isomorphic w i t h R/F , Since R i s p-monomorphic, every p-element subset includes the same number o f r -e le -

ment subsets having the c o l o r (+) . On the o ther hand, we have r+p 6 Card E ;

using ch.3 5 4.3.(1), we see t h a t a l l the r-element subsets o f E have the co lo r (t) , and hence t h a t R i s r-monomorphic. 0

r 6 Min(p,(Card E)-p) (POUZET 1976).

6.4. I f a unary r e l a t i o n R i s 1-monomorphic, then R i s constant and hence

chainable. I f R i s a b inary r e l a t i o n , then R can be (4 2)-monomorphic w i t h i n f i n i t e

base, y e t no t be chainable. Take R t o be r e f l e x i v e , antisymmetric and comparable, i . e . a tournament

w i t h a t l e a s t a cyc le R(a,b) = R(b,c) = R(c,a) = - I f R i s a b inary , (6 3)-monomorphic r e l a t i o n and based on a t l e a s t 4 elements, then R i s chainable.

0 To f i x ideas, suppose t h a t R i s r e f l e x i v e . E i the r the r e s t r i c t i o n s o f R t o

two elements are s y n e t r i c ; then since, by hypothesis, these r e s t r i c t i o n s are a l l

isomorphic, i t fo l lows t h a t R i s a constant, thus a chainable r e l a t i o n .

O r the r e s t r i c t i o n s o f R t o two elements are chains, o r o r i en ted r e s t r i c t i o n s . Then the r e s t r i c t i o n s o f R t o 3 elements cannot be c y c l i c : s ince otherwise, i f we consider 4 elements, the f o u r c y c l i c r e s t r i c t i o n s t o 3 elements would be mutu-

a l l y incompatible. Thus a l l r e s t r i c t i o n s t o 3 elements are chains, and then R

. 0

-

Chapter 9 261

i s i t s e l f a chain. 0

The two p reced ing cases, t h a t o f unary and b i n a r y r e l a t i o n s , a r e b u t t h e f i r s t

terms i n a genera l process. As a l r e a d y s a i d abou t c h a i n a b i l i t y ( i n 5.4 above), we s h a l l see i n ch.12 5 3.4 t h a t f o r each i n t e g e r n , t h e r e e x i s t s an i n t e g e r p>, n

such t h a t , i f R i s an n-ary, (6 p)-monomorphic r e l a t i o n w i t h c a r d i n a l i t y a t l e a s t p , then R i s cha inab le . More s p e c i f i c a l l y , f o r each n t h e r e e x i s t s a l e a s t i n t e g e r p a n and f o r t h i s p , t h e r e e x i s t s a l e a s t q>/ p such t h a t i f R i s a (6 p)-monomorphic n -a ry r e l a t i o n w i t h c a r d i n a l i t y >, q , then R i s

chainable. By t h e above 6.3, we can suppose s imp ly t h a t R i s p-monomorphic,

i ns tead o f (L p)-monomorphic. On the o t h e r hand, f o r each i n t e g e r n >, 2 , t h e r e e x i s t s an n-ary r e l a t i o n R

w i t h i n f i n i t e base, such t h a t R & (4 n)-monomorphic .vet n o t cha inab le . 0 To each n-element s e t , a s s o c i a t e a un ique n - tup le , formed o f t h e elements o f

t h i s s e t w i t h o u t r e p e t i t i o n , i n such a way t h a t t hese p a r t i c u l a r n - tup les do n o t form a cha in , f o r which t h e y would be t h e r e s t r i c t i o n s o f c a r d i n a l i t y n . Then

l e t R t a k e t h e va lue (+) f o r o u r p a r t i c u l a r n - tup les , and t h e va lue ( - ) f o r

a l l o t h e r n - t u p l e s . 0

- Problem. Does t h e r e e x i s t a p o s i t i v e i n t e g e r no such t h a t f o r each n >,no and

each n a t u r a l number p , t h e r e e x i s t s an n-ary r e l a t i o n which i s p-monomorphic

y e t n o t (p+l)-monomorphic ( f r o m t h e p reced ing i t f o l l o w s t h a t such r e l a t i o n s would necessary be f i n i t e ) .

6.5. The n o t i o n of a tournament r e l a t i o n was d e f i n e d i n ch.5 5 2.7. Every r e s t r i c -

t i o n o f a tournament t o a 3-element se t , i s e i t h e r t h e c h a i n o f c a r d i n a l i t y 3 , o r the b i n a r y c y c l e o f c a r d i n a l i t y 3 : more e x a c t l y t h e r e f l e x i v e b i n a r y c y c l e o f car-

d i n a l i t y 3 : f o r s h o r t we s h a l l c a l l i t t h e 3-cycle. Given an element u i n t h e base o f a b i n a r y r e l a t i o n , we s h a l l say t h a t a 3-cyc le

which i s a r e s t r i c t i o n o f o u r r e l a t i o n , passes th rough u i f i t s base con ta ins the element u . Le t A be a tournament and E i t s base, o f f i n i t e c a r d i n a l i t y pa 5 . - If A & (p-2)-monomorphic, t hen

(1) f o r each e lement u of E , t h e number o f 3-cyc les pass inq th rouqh u 1 1

independent o f u ; f o r each p a i r o f elements u, v o f E , t h e number o f 3-cvc les pass inq th rouqh u and v i s independent o f u, v ; (2) l e t (x,y) and ( x ' , y ' ) be any two a r b i t r a r y o rde red p a i r s o f elements i m E

which s a t i s f y t h e c o n d i t i o n s x # y , x ' # y ' , A(x,y) = A ( x ' , y ' ) = + ; and l e t

f be an isomorphism f rom A/(E-{x,y)) o n t o A/ (E-{x ' ,y ' ) ) ; then f o r each element

z i n E-\x,y) and f o r i t s image z' = f z , t h e r e s t r i c t i o n s A/-(x,y,z)=

A/{T,y' , z ' ) a r e e i t h e r b o t h 3 -cyc les , o r b o t h cha ins (POUZET 1977, unpubl ished) .


0 (1) L e t u, v be two d i s t i n c t elements o f E . Since A i s (o-2)-monomorphic,

t h e number o f 3-cyc les which pass n e i t h e r th rough u n o r th rough v , i s indepen- den t o f u and v ; l e t k2 denote t h i s number. Given an e lement u o f E , t h e number o f 3 -cyc les which do n o t pass th rough u i s independent o f u : app ly ch.3

5 4.1. (1) w i t h p = 3 and p t q = (Card E)-2 ; l e t kl denote t h i s number. Now l e t k denote t h e number o f a l l t h e 3 -cyc les which a r e r e s t r i c t i o n s o f A . For each element u , t h e r e e x i s t hl = k-kl many 3 -cyc les which pass th rough u . Given u and v d i s t i n c t , t h e r e e x i s t kl many 3 -cyc les which do n o t pass th rough

v , and among these k2 many which do n o t pass th rough u e i t h e r , hence (kl-k2) many which pass th rough u w i t h o u t pass ing th rough v . F i n a l l y t h e r e e x i s t

h = k-2kl+k2 many 3 -cyc les which pass th rough u and th rough v . 0

( 2 ) F o r elements x,y,z i n E , l e t h(x,y,z) denote t h e number, e i t h e r 0 o r 1, o f t h e 3-cyc les pass ing th rough x,y,z . L e t h(-x,y,z) denote t h e number o f 3-cyc les which pass through y and z w i t h o u t pass ing th rough x , e t c .

We r e t u r n t o t h e two o rde red p a i r s (x,y) and ( x ' , y ' ) and t h e elements z and f z i n o u r s ta tement . The number o f 3 -cyc les which pass th rough z i s equal t o

hl

h(x,y,z) + h(-x,y,z) + h(x,-y,z) + h(-x,-y,z) . The number o f 3-cyc les which pass

through x and z i s h2 by t h e above ( l ) , and i s a l s o equal t o h(x,y,z) + h(x,-y,z) . S i m i l a r l y t h e number o f 3 -cyc les pass ing th rough y and z i s h2

and equals h(x,y,z) + h(-x,y,z) . Adding t h e second and t h i r d e q u a l i t i e s and then s u b t r a c t i n g t h e f i r s t , we o b t a i n

same r e s u l t when s u b s t i t u t i n g x ' , y ' , z ' f o r x,y,z . Now, s i n c e f i s an isomorphism f rom A/(E-{x ,y} ) o n t o A / (E - { x ' , y ' } ) , t h e numbers h(-x,-y,z) and

h ( - x ' , - y ' , z ' ) a r e equa l : so t h a t h(x,y,z) = h ( x ' , y ' , z ' ) . 0

by t h e above (1). On t h e o t h e r hand, t h i s number i s equal t o t h e f o l l o w i n g sum:

h(x,y,z) = 2h2 - hl + h(-x,-y,z) . We have t h e

6.6. L e t A be a tournament and E i t s base, which i s f i n i t e and o f c a r d i n a l

p 2 5 . Suppose t h a t A (p-2)-monomorohic. Then e i t h e r A i s a chain; o r e l s e , f o r any two o rde red p a i r s (x,y) and ( x SY') s a t i s f y i n g t h e c o n d i t i o n s x # y , x ' # y ' , A(x,y) = A ( x ' , y ' ) = + , e v e r y i s o - morphism f rom A/(E-{x,y)) onto A / ( E - { x ' , y ' j ) , when extended by t a k i n g x to x ' y & y ' , i s an automorphism o f . A (JEAN 1969; t h e f o l l o w i n g p r o o f i s due t o POUZET 1977) . 0 E i t h e r a l l r e s t r i c t i o n s of A w i t h c a r d i n a l i t y 3 a re chains: t hen A i s a

cha in . O r t h e r e e x i s t s a t l e a s t one 3 -cyc le . By t h e p reced ing 6.5, f o r each p a i r o f d i s t i n c t elements t h e r e passes a t l e a s t one 3-cyc le. Suppose f rom t h i s p o i n t

on, t h a t we a r e i n t h i s case. L e t (x,y) and ( x ' , y ' ) be t h e two o rde red p a i r s i n o u r s ta tement , and z an

e lement i n E-{x,y} , and z ' t h e image o f z under t h e isomorphism i n o u r

Chapter 9 263

statement. Suppose t h a t A(x,z) = + : we s h a l l prove t h a t A ( x ' , z ' ) = + by a rgu ing towards a c o n t r a d i c t i o n i n supposing t h e c o n t r a r y , t h a t A ( x ' , z ' ) = - and hence t h a t A ( z ' , x ' ) = t . The r e s t r i c t i o n A/jx,y,z) i s n o t a c y c l e : by t h e p reced ing 6.5. (2) , t h e r e s t r i c -

t i o n A / \ x ' , y ' , z ' ) i s n o t a c y c l e ; hence A ( z ' , y ' ) = + . L e t u be an element such

t h a t A/(x,y,u} i s a c y c l e : we know t h a t such an e lement e x i s t s . Then we have

A(y,u) = A(u,x) = + . We c l a i m t h a t A(u,z) = + . I f n o t , t hen A(z,u) = + , so A ( z ' , u ' ) = + as w e l l , where u' i s t h e image o f u under t h e isomorphism i n ou r

hypo thes i s . Then no 3 -cyc le pass ing th rough z ' and u' would pass through x '

no r th rough y ' ( r e c a l l t h a t A / { x ' , y ' , u ' J i s a c y c l e , by t h e p reced ing 6 .5 . (2 ) ) .

Yet A/{x,z,u) would be a c y c l e ; so t h a t t h e number o f 3-cyc les pass ing through

z and u would be s t r i c t l y g r e a t e r t han t h e number o f 3 -cyc les pass ing through z ' and u' , c o n t r a d i c t i n g 6.5. (1) . Thus A(u,z) = A ( u ' , z ' ) = + . It now f o l l o w s t h a t A(z,y) = + ; f o r i f n o t , t hen

no 3 - c y c l e pass ing th rough z and u would pass th rough x n o r through y , and y e t A / { y ' , z ' , u ' j would s t i l l be a c y c l e , c o n t r a d i c t i n g 6.5. (1) .

Thus o u r hypo thes i s a l l owed us t o determine t h e va lues o f

t he tournament A f o r a l l p a i r s i n c l u d e d i n 1 x,y,z,u] and i n { x ' , y ' , z ' , u ' l . We t e r m i n a t e t h e p r o o f by t a k i n g an element t f o r which A / { x , z , t j i s a c y c l e : we know t h a t such an e lement e x i s t s . Thus t i s d i s t i n c t f rom y and f rom u . Moreover, no 3 - c y c l e pass ing th rough z' and th rough t h e image t ' o f t (under the isomorphism i n o u r hypo thes i s ) , can be completed by x ' o r by y ' . Hence the number o f c y c l e s pass ing th rough z, t on t h e one hand, and th rough z ' , t '

on t h e o t h e r hand, i s d i f f e r e n t , c o n t r a d i c t i n g 6.5.(1). 0

Problem communicated by POUZET, 1978. L e t A and A ' be two tournaments hav ing

t h e same base E w i t h f i n i t e c a r d i n a l i t y p a 5 . I f f o r each subset X ob ta ined by removing two elements f rom E , t h e r e s t r i c t i o n s A / X and A ' / X a r e isomorphic ,

p h i c , t hen a r e A and A ' isomorphic .

A(x,z) = A ( z ' , x ' ) = +

6.7. L e t A be a b i n a r y r e l a t i o n w i t h f i n i t e c a r d i n a l i t y p>/ 5. If A 2 (p-2)-monomorphic, t hen A & (p-1)-monomorphic. Moreover, i f A i s a n o n - t o t a l l y o rde red tournament, t hen f o r any two elements x a& x ' i n t h e base, t h e r e e x i s t s an automorphism o f A which takes x - i n t o x ' (JEAN 1969).

0 Since t h e r e l a t i o n A i s (p-2)-monomorphic, A i s a l s o ( 4 2)-monomorphic

by 6.3 ( i ndeed p 5 , so t h a t 2 < p-2 ) . S ince A i s 1-monomorphic, t o f i x ou r

ideas, suppose t h a t A i s r e f l e x i v e . S ince A i s 2-monomorphic, e i t h e r i t s res - t r i c t i o n s t o 2-element s e t s a r e symmetric, i n which case A i s .a c o n s t a n t r e l a - t i o n , hence monomorphic. O r t h e r e s t r i c t i o n s t o 2-element s e t s a r e o r i e n t e d , i n which case A i s a tournament. By t h e p reced ing 6.6, e i t h e r A i s a chain, and

~

hence monomorphic, o r A i s a n o n - t o t a l l y o rde red tournament.

I n t h e l a t t e r case, l e t E be t h e base o f A , and l e t x, x ' be two elements

o f E . E i t h e r t h e r e e x i s t two o t h e r elements y, y ' w i t h A(x,y) = A ( x ' , y ' ) = + ; then by t h e p reced ing p r o p o s i t i o n , t h e r e e x i s t s an automorphism o f A t a k i n g x

i n t o x ' and y i n t o y ' . Thus we have an isomorphism o f t h e r e s t r i c t i o n s A / ( E - {XI) and A/(E - i x ' ) ) ; so t h a t A i s (p-1)-monomorphic.

O r t h e r e do n o t e x i s t two such elements y, y ' . Then A takes t h e va lue (+) f o r

eve ry ordered p a i r w i t h f i r s t t e r m x , w h i l e t a k i n g t h e va lue ( - ) f o r eve ry orde-

r e d p a i r w i t h f i r s t term x ' ( t o f i x i d e a s ) . S ince E has c a r d i n a l i t y s t r i c t l y

l a r g e r than 3 , l e t t be an e lement d i s t i n c t f rom x and x ' . Then A(x,x ' ) = A(x, t ) = + ; t h u s we have an automorphism o f A which preserves x and takes

t i n t o x ' . S i m i l a r l y A(x ' ,x) = A ( x ' , t ) = - ; so we have an automorphism o f A which takes x i n t o t . By composi t ion, we have an automorphism o f A which

takes x i n t o x ' : thus A i s (p-1)-monomorphic. 0

I n o p p o s i t i o n t o t h e preceding: f o r each i n t e g e r p a4 , t h e b i n a r y c y c l e based

on p elements i s (p-1) b u t n o t (p-2)-monomorphic.

Another example: f o r p = 2q , t h e p a r t i a l o r d e r i n g formed o f q cha ins o f c a r d i - n a l i t y 2 , taken t o be m u t u a l l y incomparable.

Note t h a t , f o r a b i n a r y r e l a t i o n w i t h c a r d i n a l i t y

morphic, t hen i t i s I n genera l , f o r r a 3 , e v e r y b i n a r y r e l a t i o n based on p + r+3 elements, which i s (p-r)-monomorphic, i s (4 3)-monomorphic, and thus cha inab le .

Problem (JEAN 1976, unpub l i shed) . L e t n 3 2 and k,< n . Is every n -a ry r e l a t i o n

based on p>, 2n+l elements, which i s (p-k)-monomorphic, n e c e s s a r i l y (p -k+ l ) - monomorphic. Note t h a t i t i s

F o r n > / 6 , i s eve ry n -a ry r e l a t i o n based on p>/ 2n+l elements, which i s

(p-6)-monomorphic, n e c e s s a r i l y cha inab le ( t h i s i s connected t o Jordan hypo thes i s about pe rmuta t i on groups ;

p a 6 , i f i t i s (p-3)-mono- (< 3)-monomorphic, hence cha inab le : see 6.3 and 6.4.

(,< k)-monomorphic by 6.3.

see ch.11 5 2.2) .

6.8. L e t A be a tournament w i t h c a r d i n a l i t y p and which i s (p-2)-monomorphic. - If A i s n o t a chain, t hen p = 3 , modulo 4 . 0 L e t E be t h e base of A , o f c a r d i n a l i t y p ; and l e t u be an e lement i n E . L e t q denote t h e number of successors o f u (mod A) ; i n o t h e r words, t h e num-

b e r o f x such t h a t A(u,x) = + . I f we r e p l a c e u by ano the r e lement u ' , then by t h e p reced ing d i s c u s s i o n t h e r e e x i s t s an automorphism o f A t a k i n g u i n t o u ' , hence t h e number q o f successors i s preserved. The p r o d u c t p.q i s

t hus equal t o t h e t o t a l number o f o rde red p a i r s o f elements g i v i n g t h e va lue (+) t o A ; hence equal t o t h e number p. (p-1) /2 o f unordered p a i r s o f elements. Thus

q = (p-1) /2 , which a l r e a d y shows t h a t p i s odd.

Chapter 9 265

Given an element u i n E , p a r t i t i o n the se t o f elements d i s t i n c t from u i n t o the se t F o f (p-1)/2 many successors o f u , and the se t G o f (p-1)/2 many predecessors o f u (elements x such t h a t A(x,u) = + ) . L e t v and v ' be two elements i n F . By the preceding discussion, there e x i s t s an automorphism o f A which preserves u and takes v i n t o v ' . Since i t preserves u , t h i s automorphism preserves F and G . Thus the number r o f elements i n F which are suc-

cessors o f v remains the same number when passing t o v ' . The product r . (p -1) /2 i s thus equal t o the t o t a l number o f ordered pa i r s i n

equal t o the number (p - l ) (p -3 ) /8 o f unordered p a i r s i n F . Hence r = (p-3)/4 , and thus p-3 i s a m u l t i p l e o f 4 . 0

F g i v i n g value (+); hence

6.9. An t i c ipa t i ng the not ions o f ch.11 5 2.2, we say t h a t a group o f permutation i s 2 -se t - t rans i t i ve iff f o r any two unordered p a i r s o f elements i n the base, there ex i s t s a permutation o f the group, which takes the f i r s t p a i r i n t o the second. The above propos i t ion 6.6 then takes the fo l l ow ing form: i f A i s a (p-2)-mono-

morphic, non - to ta l l y ordered tournament w i t h c a r d i n a l i t y p , then the group o f automorphisms o f A i s 2 -se t - t rans i t i ve (p i n tege r 3 5) . Groups which are 2 -se t - t rans i t i ve have been s tud ied i n p a r t i c u l a r by DEMBOWSKI 1968, p. 96 note 2, under the name o f 2-homogeneous groups. The term "se t - t rans i -

t i v e " i s equa l ly used and w i l l be employed i n t h i s book t o avoid confusion w i t h homogeneous re la t i ons (see ch.11 5 1 ) . From the above c i t e d work, i t fo l lows t h a t f o r every l y ordered tournament based on p a 5 elements, this p i s an odd power o f a prime congruent t o 3 (mod 4) . Thus, a f t e r the b inary cyc le on 3 elements, we have a tournament on 7 elements which i s 5-monomorphic, and hence 6, a z 2 , and 1-monomorphic, y e t ne i the r 3 nor 4-monomorphic; t h i s tournament has chains o f c a r d i n a l i t y 3 and cycles o f cardina- l i t y 3 as r e s t r i c t i o n s .

To cons t ruc t i t , s t a r t w i t h a heptagon, o r polygon w i t h 7 ve r t i ces

C y c l i c a l l y o r i e n t the edges ab, bc, ... , fg , ga . C y c l i c a l l y o r i e n t the s ta r red heptagon acegbdf i n the d i r e c t i o n ac, ce, ... , df , f a . C y c l i c a l l y o r i e n t the second s ta r red heptagon adgcfbe i n the "opposite" d i r e c t i o n da, gd, ... , eb,

ae . I n view o f the r o t a t i o n a l symmetry, i t su f f i ces t o v e r i f y the isomorphism between the th ree sub-tournaments o f c a r d i n a l i t y 5 : one which is obtained by removing two consecutive ver t i ces , a second obtained by removing two ver t i ces

which are separated by one intermediate vertex, and a t h i r d obtained by removing two ve r t i ces which are separated by two intermediate ve r t i ces .

(p-2)-monomorphic, non-total-

a,b,c,d,e,f,g .


§ 7 - PROFILE OF A RELATION, PROFILE INCREASE THEOREM (POUZET)

Le t R be a r e l a t i o n w i t h base E . To each na tura l number p , associate the f i n i t e number f ( p ) o f isomorphism types o f the p-element r e s t r i c t i o n s o f R . The numerical func t ion f thus def ined i s c a l l e d the p r o f i l e o f R (no t i on due t o POUZET 1972). Note t h a t f ( 0 ) = 1 and i f Card E = h ( f i n i t e ) , then f ( h ) = 1 and f ( p ) = 0 f o r a l l p > h . Examples. I f R i s a chain, o r i n general a monomorphic r e l a t i o n , then the p r o f i l e

func t ion has constant value equal t o 1 (when p & Card E ) . I f R i s a unary r e l a t i o n tak ing the value (+) f o r a f i n i t e number a o f elements

and ( - ) on a l l o ther elements, then the p r o f i l e increases from f ( 0 ) = 1

f ( a ) = a+l , and then remains s ta t i ona ry a t t h i s l a t t e r value. I f R

set , then the p r o f i l e i s f ( p ) = p+ l . By taking, f o r instance, the consecut iv i t y r e l a t i o n on the na tura l numbers , we

obta in a p r o f i l e func t ion w i t h a f a s t e r growth ra te . I n the case where every f i n i t e r e l a t i o n w i t h the same a r i t y i s embeddable i n R , the p r o f i l e o f R i s maximum, hence f o r each p i s equal t o the number o f isomor-

phism types o f re la t i ons o f the given a r i t y w i t h c a r d i n a l i t y p .

t o

i s unary and takes the value (+) and the value ( - ) , each on an i n f i n i t e

7.1. PROFILE INCREASE THEOREM

Le t p, q be two na tura l numbers and R a r e l a t i o n w i t h c a r d i n a l i t y a t l e a s t

equal t o 2p+q . Then the number o f isomorphism types o f the r e s t r i c t i o n s o f R

t o p+q elements i s a t l e a s t as g rea t as the number o f isomorphism types o f the r e s t r i c t i o n s o f R 2 p elements.

More prec ise ly , there e x i s t s an i n j e c t i v e func t ion which, t o each isomorphism type u o f a r e s t r i c t i o n o f R 5 p

t r i c t i o n t o p+q elements, which i s an extension o f u (POUZET 1976).

0 This fo l lows from the m u l t i c o l o r theorem, ch.3 5 5.3, where the isomorphism

types on p elements p lay the r o l e o f the co lo rs o f the p-element sets, and two isomorphic

Consequently, i f a r e l a t i o n has an i n f i n i t e base, then i t s p r o f i l e i s increasing. I f a r e l a t i o n has even c a r d i n a l i t y 2h , then i t s p r o f i l e i s inc reas ing f o r

integers less than o r equal t o h . I f a r e l a t i o n has odd c a r d i n a l i t y 2h+l , then i t s p r o f i l e i s inc reas ing f o r in tegers less than o r equal t o h+ l . For the case o f an i n f i n i t e base, an a l t e r n a t i v e proo f w i l l be given i n ch.10

5 9.9, about "almost chainable" re la t i ons . Note t h a t the f i r s t o f the preceding propos i t ions i s stronger than the second. For example, consider a r e l a t i o n on 7 elements. Not only does i t s p r o f i l e increase

-

-

(p+q)-element r e s t r i c t i o n s have the same mu l t i co lo r . 0

Chapter 9 267

f o r in tegers 0 t o 4 , but i t s value f o r 5 i s g rea ter than o r equal t o the value f o r 2 ; and i t s value f o r 6 i s g rea ter than o r equal t o the value f o r 1

7.2. Le t p, q be two na tura l numbers and R a r e l a t i o n w i t h base E , where Card E >/ 2p+q , and l e t f be a permutation o f E . I f the image o f every r e s t r i c -

t i o n o f R t o p+q elements (under f ) i s an isomorphic r e s t r i c t i o n o f R , then the image of every r e s t r i c t i o n o f R 5 p elements i s an isomorphic r e s t r i c - t i o n .

0 Take an a r b i t r a r y r e s t r i c t i o n o f R t o p elements, and l e t U denote i t s

isomorphism type. A l l p-element subsets o f E w i t h t h i s isomorphism type sha l l

have the co lo r U . Now a p-element subset o f E i s sa id t o have the co lo r V i f i t s image under f has the co lo r U . By hypothesis, f o r each (p+q)-element subset a o f E , the r e s t r i c t i o n s R/a and R/ f " (a ) are isomorphic, hence a and i t s image f " ( a ) have the same number o f p-element subsets o f co lo r U . Thus a includes the same number o f p-ele-

ment subsets o f co lo r U as we l l as o f co lo r V . It fo l lows t h a t the co lo rs U

and V are i d e n t i c a l , by ch.3 5 4 .3 . (2 ) . Hence f takes each p-element subset

i n t o another o f the same co lo r . 0

-

-

§ 8 - HOMOMORPHIC IMAGE OF A N ARBITRARY RELATION

I n t h i s paragraph, we attempt t o general ize t o a r b i t r a r y r e l a t i o n s the c lass i ca l no t ion o f homomorphic image

S t a r t i n g w i t h a r e l a t i o n R w i t h base E and w i t h a se t E" o f subsets o f E , l e t U1 , ... , Un be a f i n i t e sequence o f these subsets. Construct a l o g i c a l formula w i t h the connections 3 (no t ) , A (and), v ( o r ) , rJ ( i f ... then), etc. ; w i t h the var iab les x,y,z, ... represent ing elements o f E , w i t h the quan t i f i e rs

v x and 3 meaning " f o r every element x o f E " and " there e x i s t s an

element x o f E " ; and f i n a l l y w i t h predicates the r e l a t i o n R and the unary re la t i ons Ui ( i = 1, ..., n) where Ui(x) takes the value (+) i f x belongs t o Ui and ( - ) i f x belongs t o E-Ui . The usual semantic t r u t h value o f such a formula being obvious, we say t h a t an n-ary r e l a t i o n R" w i t h base E" i s a homomorphic image o f R i f f there ex i s t s

a l o g i c a l formula whose t r u t h value i s Ro(U1, ..., Un) when U1 , ... , Un run through ED .

among chains (see ch.2 5 3.6) o r among groups.

*

8.1. F i r s t example. Le t R be a p a r t i a l o rder ing w i t h base E , and l e t E" be

the se t o f a l l subsets o f E . Then the unary r e l a t i o n R" such t h a t R"(U) = + i f f U i s an i n i t i a l i n t e r v a l o f R i s a homomorphic image o f R , v i a the


formula vx,y (U(x) A R(y,x)) 3 U(Y) . S im i la r l y , we express the r e l a t i o n tak ing (+) i f f U i s a c o f i n a l subset (mod R)

v i a the formula vx 3y R(x,y) A U(y) . Now tak ing f o r E" the se t o f i n i t i a l i n t e r v a l s (mod R), the p a r t i a l o rder ing o f i nc lus ion w i t h base E" , whose arguments are denoted by U , V , i s a homomor-

ph ic image o f R v i a the formula vx U(x) =7 V(x)

Second example. Suppose t h a t R i s a chain w i t h base E , and t h a t E" i s a s e t

o f pairwise d i s j o i n t i n te rva l s . Then the corresponding homomorphic image o f R , i n the usual sense o f ch.2 5 3.6, i s expressed by the formula:

.

vx,y (u(X) A v(Y)) 3 R(x,Y) . Problem. Returning t o the p a r t i a l o rder ing R w i t h base E , and l e t t i n g E" be the se t o f a l l subsets o f

l o g i c a l formula o f the preceding kind, which would express t h a t the subset U o f E i s a co f i na l subset (mod R) w i t h minimum c a r d i n a l i t y ( s o de f in ing the c o f i n a l i -

t y o f R ).

E ; then we conjecture t h a t there does no t e x i s t any

8.2. Take R t o be a te rnary r e l a t i o n o f a group based on E , and f o r E" the

se t of a l l subsets o f E . We l e a w i t t o the reader t o see t h a t the unary r e l a t i o n s " the r e s t r i c t i o n R/U i s a subgroup " and " R/U i s a normal subgroup " , are homomorphic images o f R . Also the b inary r e l a t i o n " R/U i s a normal subgroup and V i s an equivalence class modulo U " . Now take a f i x e d normal subgroup 0 , and f o r E" take the s e t o f equivalence classes modulo 0 : the reader w i l l see t h a t the usual homomorphic image, o r

quot ien t group, def ined on the s e t o f these equivalence classes, i s represented by a l o g i c a l formula o f the preceding kind.

Problem o f t r a n s i t i v i t y o f c e r t a i n homomorphic images. S t a r t w i t h a r e l a t i o n R

w i t h base E and a se t E" of mutua l ly d i s j o i n t subsets o f E , and w i t h a l o g i c a l formula de f i n ing a homomorphic image R " o f R , w i t h base E" . I t e r a t e , by tak ing a se t E"" o f (no t necessar i l y mutual ly d i s j o i n t ) subsets o f E " , and

a homomorphic image R o o o f R" , w i t h base En" . Associate t o each element o f E"" i t s union, which i s a subset o f E . Note tha t , t o two d i s t i n c t elements o f E"" there correspond two d i s t i n c t unions, because o f the above d i s junc t i on of elements o f E" . Hence we have a b i j e c t i o n from E"" onto a se t o f subsets o f

E , and t h i s b i j e c t i o n transforms R o n i n t o a r e l a t i o n on subsets o f E . I s the r e l a t i o n thus transformed, necessar i l y a homomorphic image o f R .

--. -

Chapter 9 269

§ 9 - BIVALENT TABLE

A b iva len t t ab le i s the system formed by two d i s j o i n t sets: the s e t E

and the se t product E Y F , associates the value (+) o r the value ( - ) . Given the tab les T on E x F and T' on E'x F ' , we say t h a t T i s embeddable i n T ' , o r T,< T ' , i f there e x i s t s an i n j e c t i o n e from E i n t o E ' and an

i n j e c t i o n f from F i n t o F ' , preserving the values: T ' (ex , fy ) = T(x,y) f o r a l l x i n E and y i n F . The tab le T i s sa id t o be extensive by the tab le X , r e l a t i v e l y t o rows, i f e i t h e r T Q X , o r there e x i s t s an X+ obtained from X by adding a row, such

t h a t T+ Xt . Otherwise, i f T 4 X X by adding a row, then we say t h a t T i s inextensive by X ( r e l a t i v e l y t o rows).

We l e a v e i t t o the reader t o see t h a t the t a b l e T having two columns (below on the l e f t ) , which i s no t embeddable i n the t a b l e X having fou r columns (below on the r i g h t ) , i s inex tens ive by X , r e l a t i v e l y t o rows:

o f columns

F o f rows; and a func t ion which, t o each element i n the Cartesian

and y e t T,< X+ f o r every X+ obtained from

t + + - + + - - + - + - t - t -

+ - - - - + - t

Hint . F i r s t note t h a t you cannot add d i f f e r e n t values t o the t h i r d and fou r th columns i n X , wi thout embedding T ; i f you add + + t o these columns, then

you must add - and then T i s embedded i n the second and fou r th co1umns:contradiction. Now i f you add - - t o the t h i r d and fou r th column, then you must add + t o the f i r s t column (because o f the f i r s t and fou r th columns); and then T i s embedded i n the f i r s t and t h i r d columns: con t rad i c t i on .

We say t h a t a t a b l e T i s p-extensive ( p natura l number ) i f f T i s extensive

fo r every tab le having p rows. I n o ther words, f o r every tab le X having p rows and such t h a t T # X , there e x i s t s X+ (obtained from X by adding a row)

which respects the non-embeddability T $ Xt . F i n a l l y T i s sa id t o be extensive ( r e l a t i v e l y t o rows) i f T i s extensive by

every tab le ; i t i s sa id t o be inex tens ive otherwise. Ev&ry inextensive tab le must have a row o f (t), a row o f ( - ) , and a t l e a s t two

i d e n t i c a l rows. This i s the case f o r the above example T w i t h two columns and fou r rows. It i s proved by LOPEZ 1977, t h a t the above inex tens ive t a b l e T i s however p-ex-

tensive f o r every i n tege r p 3 5 . Problem. For each b i va len t t a b l e which i s f i n i t e , i . e . which has a f i n i t e se t o f rows and columns, do there e x i s t i n f i n i t e l y many in tegers p f o r which T i s p-extensive; and even does there e x i s t s an i n tege r p such t h a t T i s q-extensive fo r every i n tege r q g rea ter than p .

t o the second column (because o f the second and t h i r d columns);


We terminate t h i s paragraph by g i v ing a sketch o f the proof, by LOPEZ, o f the above

statement. Arguing towards a cont rad ic t ion , we suppose t h a t there e x i s t s a tab le

Cartesian product E x F , w i t h Card F 3 5 , w i t h T inextensive by X . We can requ i re t h a t the se t remove any column from E , then there e x i s t s an X+ , obtained from X by adding

a row, i n which T i s no more embeddable. Consequently a l l the columns are d i s t i n c t (s ince the two columns o f T

t a ins a t l e a s t one (+) and one ( - ) : indeed otherwise, i f f o r example one column

contains only (+), then by adding another (+) t o t h i s column, i t could no t c o n t r i - bute t o any embedding o f T . Given an ordered p a i r (a,b) o f columns and two values v(a) and v (b) , each equal t o (+) o r t o ( - ) , we say t h a t the ordered p a i r (v (a ) ,v (b ) ) (a,b) , i f T i s embeddable i n the t a b l e o f two columns a, b completed by v (a) and v (b) . Given two columns a, b, there are on ly f i v e p o s s i b i l i t i e s . E i t h e r (+,+) i s the

only bad p a i r f o r (a,b) : i n the case where (+.+) i t s e l f i s no t a row i n (a,b) , y e t (-,-) and tw ice (+,-) o r tw ice (-,+) arc rows i n (a,b) . O r (-,-) i s

the only bad p a i r ; o r (+,-) i s the only bad (ordered) pa i r ; o r (-,+) ; o r f i n a l - l y (+,-) and (-,+) are both bad pa i r s . As an immediate consequence, we see t h a t

a tab le X which reduces t o two columns cannot y i e l d t o the i n e x t e n s i v i t y o f T . Now given three columns a, b, c, we have the fo l l ow ing lemma: i t i s impossible t h a t (+,+) be bad f o r (a,b) and t h a t (-,-) be bad f o r (b,c) . Indeed by the preceding, assume t h a t the th ree columns are d i s t i n c t and t h a t every column contains a t l e a s t one (+) and one ( - ) . There necessar i l y e x i s t s a f i r s t row

w i t h values (-,-) f o r (a,b) (s ince (+,+) i s bad). Then the value f o r c i n

t h i s row i s necessar i l y (+); f o r i f i t were ( - ) , then (-,-) would already be pre-

sent i n (b,c) , and thus could no t be a bad p a i r f o r (b,c) ; so our f i r s t row i s (-,-,+) . S i m i l a r l y there e x i s t s a second row w i t h (+,+) f o r (b,c) , and then ( - ) f o r a : so our second row i s (-,+,+) . Since the column a must conta in a t l e a s t one (+) , we have a t h i r d row w i t h (+) f o r a , and then ( - ) f o r b , and then

(+) f o r c : so t h a t our t h i r d row i s (+,-,+) . S i m i l a r l y c must conta in a t l e a s t one ( - ) , so t h a t our fou r th row i s (-,+,-) . F i n a l l y T i s embedded i n the columns (a ,c) : cont rad ic t ion . From the preceding lemma, i t fo l lows tha t , i f we requ i re t h a t T

by X , then X cannot reduce t o th ree columns (assumed t o be d i s t i n c t and t o con-

t a i n (+) and ( - ) ) : t o see tha t , i t su f f i ces t o add t o X . the row (+,+,+) and then the row (-,-,-) ; so t h a t i f T i s inex tens ive by X , then there e x i s t a t l e a s t one bad p a i r (+,+) and one bad p a i r (-,-) , con t rad i c t i ng the lemma.

X on the

E o f columns be minimal, i n the sense tha t , i f we

are themselves d i s t i n c t ) ; moreover, every column con-

i s bad f o r

be inex tens ive

Chapter 9 27 1

EXERCISE 1 - EXTENDIBLE LOCAL AUTOMORPHISM

Given a r e l a t i o n R , we say t h a t a l o c a l automorphism f i s e x t e n d i b l e iff t h e r e e x i s t s an automorphism o f R ex tend ing f . Consider t h e b i n a r y r e l a t i o n R d e f i n e d as f o l l o w s on a base o f t e n elements a,b,c,d,r,s,t,u,i,j . L e t R(a,r) = R(a,u) = R(b,r) = R(b,s) = R(c,s) = R(c, t

R(d, t ) = R(d,u) = R ( i , r ) = R ( i , t ) = R( j , s ) = R( j ,u) = + , w h i l e R takes t h e va lue ( - ) f o r a l l o t h e r o rde red p a i r s . The i d e n t i t y mapping on { a,b) i s o b v i o u s l y a l o c a l automorphism o f R , and is o b v i o u s l y e x t e n d i b l e t o t h e i d e n t i t y automorphism on t h e e n t i r e base. The mapping

which preserves a and which takes i i n t o j , i s e x t e n d i b l e t o t h e automorphism which p rese rves a and preserves c , and in te rchanges b and d , r and u , s and t , i and j . S i m i l a r l y , t h e mapping which p rese rves b and which takes

i i n t o j , i s e x t e n d i b l e t o t h e automorphism which p rese rves b and d , and in te rchanges a and c , r and s , t and u , i and j . L e t f denote t h e mapping which preserves a , and p rese rves b , and takes i i n t o j . Then f i s a l o c a l automorphism, s i n c e t h e r e l a t i o n R i s b i n a r y and t h e t h r e e r e s t r i c t i o n s o f f t o {a,b'J , t o (a,') and t o { b , i } a r e l o c a l automorphisms. Prove t h a t f i s n o t e x t e n d i b l e t o t h e e n t i r e base. N o t i c e t h a t

r i s t h e o n l y element x such t h a t R(a,x) = R(b,x) = + ; hence any automorphism ex tend ing f must p rese rve r . Taking i i n t o j , we must have R ( i , r ) = R ( j , r ) , which i s f a l s e .

Thus a l though a l l t h r e e r e s t r i c t i o n s o f f t o 2 elements, a re e x t e n d i b l e t o t h e e n t i r e base, f i t s e l f i s n o t e x t e n d i b l e .

EXERCISE 2 - CONDITIONS FOR A RELATION TO BE CONSTANT

1 - L e t A be t h e usual c h a i n o f t h e n a t u r a l numbers , and B be a c h a i n on t h e

same base, b u t which i s isomorphic w i t h t h e c h a i n Q o f t h e r a t i o n a l s , hence dense and hav ing n e i t h e r a maximum n o r a minimum element . Prove t h a t , f o r each p o s i t i v e i n t e g e r n and each pe rmuta t i on f o f { 1,2, ... ,nJ , t h e r e e x i s t s a s e t F o f n i n t e g e r s ul, ..., un such t h a t , assuming t h a t these i n t e g e r s a r e

o rde red by u1 < u2 < . . . 4 un (mod A) , then t h e y a r e reordered, modulo B ,

U f ( l ) < ' f ( 2 )< * * . < 'f(n) *

2 - L e t n be a p o s i t i v e i n t e g e r , and R an n -a ry r e l a t i o n based on t h e n a t u r a l numbers. Prove t h a t i f R i s b o t h A-chainable and B-chainable, t hen f o r eve ry

pe rmuta t i on f o f t h e s e t {1 ,2, ..., n ) , t h e r e e x i s t s a s e t F o f n i n t e g e r s

such t h a t , assuming t h a t these e lements of F a r e u l < ... < un (mod A/F) , then t h e pe rmuta t i on f t ransposed by t h e b i j e c t i o n t a k i n g 1 i n t o u1 , ... , n i n t o un , i s an automorphism o f R/F . Hint: t a k e an n-element s e t G f o r which t h e r e s t r i c t i o n s A/G and B/G a r e i d e n t i c a l , and then use t h e isomorphisms


o f A/F onto A/G and o f B/F onto B/G . From t h i s , deduce t h a t f o r each

n-element se t F , the r e s t r i c t i o n R/F i s preserved by every permutation o f F , thus i s a constant r e l a t i o n . Using 4.1, deduce t h a t R i t s e l f i s a constant re1 a t ion .

3 - Suppose t h a t R i s a denumerable n-ary r e l a t i o n s a t i s f y i n g the fo l low ing :

f o r every denumerable r e l a t i o n X , i f every r e s t r i c t i o n t o a t most n elements i s embeddable i n R , then X i s isomorphic t o R . Then R i s a constant re la -

t i on . H in t : f i r s t prove t h a t R i s chainable; more s t rong ly t h a t , f o r every denumerable chain, there e x i s t s an isomorphic chain C w i t h base I R I , such t h a t

R i s C-chainable.

4 - Using the axiom o f choice, extend the above r e s u l t i n ( 3 ) t o the case o f an

a r b i t r a r y i n f i n i t e r e l a t i o n .

273

CHAPTER 10

AGE, RICH RELATION, INEXHAUSTIBLE RELAT EXISTENCE CRITERION FOR A RICH RELATION

ON, SATURA OF A GIVEN

ED RELATION, AGE

§ 1 - PROJECTION FILTER, OLDER RELATION, EXTENSION, (~,P)-MORPHISM AND (1, PI- ISOMORPHISM

Consider two sets E and E ' and a func t ion f from E i n t o E ' . Le t m be

a na tura l number ; consider two m-ary re la t i ons , R w i t h base E and R ' w i th base E ' . Then R i s sa id t o be the inverse p ro jec t i on o f R ' under f , i f f o r every x1 ,..., xm i n E , we have R ' ( fx l ,..., fx,) = R(xl ,..., xm) . I n t h i s case we w r i t e R = ( f - ) " ( R ' ) . I f f i s a b i j e c t i o n from E onto E ' , then we f i n d the usual d e f i n i t i o n of

an isomorphism from R onto R ' : see ch.2 5 3. Given sets E and E ' , consider the se t o f a l l func t ions from E i n t o E ' . A f i l t e r on t h i s se t s h a l l be c a l l e d a p r o j e c t i o n f i l t e r from E onto E ' . The r e l a t i o n R i s c a l l e d the inverse p ro jec t i on o f R ' under the f i l t e r F, which we w r i t e by R = - ( R ' ) , i f f o r every xl,. . . ,xm i n E , the se t o f

funct ions f s a t i s f y i n g R'(fxl, ..., fx,) = R(xl, ..., xm) i s an element o f the f i l t e r

t i o n (mod 3- ) . I f R ' and f are given, then the inverse p ro jec t i on ( f - ) ' ( R ' ) ex is t s and i s

unique. I f R ' pro jec t i on .

I f the inverse p ro jec t i on 3 -'(R')

' $ . - l ( R ' ) = T - ' ( R ' ) . - If 0 Indeed e i t h e r the se t o f func t ions analogous se t w i t h the value (-), i s ' a n element o f the u l t r a f i l t e r . 0

I f the u l t r a f i l t e r i s t r i v i a l , more p rec i se l y i f i t i s formed o f those sets o f

func t ions which contain as an element the func t i on f , then the inverse projec-

t i o n under the u l t r a f i l t e r coincides w i t h the inverse p ro jec t i on under f .

1

1

. I n other words, the preceding equa l i t y holds f o r almost every func-

1 -

and the f i l t e r 3 are given, then there ex i s t s a t most one inverse

e x i s t s and i f i s a f i l t e r f i n e r than 9 ';r , then

i s an u l t r a f i l t e r , then the inverse p ro jec t i on e x i s t s and i s unique.

f such t h a t R'(fxl, ... ,fx,) = t , o r the

1.1. INJECTIVE FILTER

Let be a p ro jec t i on f i l t e r from the se t E onto E ' . Two elements a, b o f

E are sa id t o be equ iva len t (mod 3 ) i f f a = f b f o r almost every f (mod 9). This cond i t i on i s r e f l e x i v e , symmetric and t r a n s i t i v e .


A p r o j e c t i o n f i l t e r from E o n t o E ' i s s a i d t o be i n j e c t i v e , i f f o r any two

d i s t i n c t elements a, b o f E , we have f a # f b f o r a lmos t eve ry f u n c t i o n f . To each f i n i t e subset F o f E , a s s o c i a t e t h e s e t UF o f those f u n c t i o n s f rom E i n t o E ' which a r e i n j e c t i v e on F . When F v a r i e s , t h e s e t s UF and t h e i r superse ts , c o n s t i t u t e an i n j e c t i v e f i l t e r , which i s minimum i n t h i s sense t h a t any g i v e n p r o j e c t i o n f i l t e r f rom E o n t o E '

i s i n j e c t i v e i f f i t i s f i n e r than o u r i n j e c t i v e f i l t e r .

Every f i l t e r which i s f i n e r t han an i n j e c t i v e f i l t e r , i s i n j e c t i v e .

I n t h e case o f a t r i v i a l u l t r a f i l t e r , formed o f t hose s e t s o f f u n c t i o n s which con- t a i n t h e f u n c t i o n f , t h e u l t r a f i l t e r i s i n j e c t i v e i f f t h e f u n c t i o n f i s i n j e c - t i v e . I n t h i s case, n e c e s s a r i l y Card E ' 3 Card E .

e c t i v e f i l t e r f rom E 0% E ' .

1.2. OLDER RELATION, YOUNGER RELATION

Given two r e l a t i o n s R , S o f t h e same a r i t y , we say t h a t R i s younger t h a n S , o r t h a t S i s older t han R , i f eve ry f i n i t e r e s t r i c t i o n o f R i s embeddable i n S . Th is comparison i s r e f l e x i v e and t r a n s i t i v e , b u t n o t an t i symmet r i c , even up t o

isomorphism: f o r example t h e c h a i n w o f t h e n a t u r a l numbers , and i t s converse, a r e each o l d e r than t h e o t h e r .

L e t R be a r e l a t i o n w i t h base E , and R ' be a r e l a t i o n o f t h e same a r i t y , w i t h base E ' . Then t h e r e e x i s t s an i n j e c t i v e p r o j e c t i o n f i l t e r s a t i s f y i n g

R = 7 -'(R') iff R '

0 L e t m denote t h e common a r i t y o f R and R ' . Suppose t h a t R = T - ' ( R ' )

and l e t F be a f i n i t e subset o f E . The s e t o f f u n c t i o n s f which a r e i n j e c t i v e

on F , i s qn element o f 'F . Fo r each m- tup le al,.. . ,am i n F , t h e s e t o f func-

t i o n s f s a t i s f y i n g R ' ( f a l ,..., fa,,,) = R(al ,..., a,,,) i s again an element o f . Since F i s f i n i t e , t h e s e t of a l l m- tup les i n F i s f i n i t e , so t h a t t h e i n t e r -

s e c t i o n o f t h e p reced ing s e t s o f f u n c t i o n s i s an element o f f which belongs t o t h i s s e t g i v e s an isomorphism f rom R/F o n t o R ' / f " ( F ) .

Conversely suppose t h a t R ' i s o l d e r than R . F o r each f i n i t e subset F o f E , t h e r e e x i s t s a f u n c t i o n f f rom F i n t o E ' , which i s an isomorphism f rom R/F on to R ' / f " ( F ) . L e t UF des igna te t h e s e t o f those f u n c t i o n s f rom E i n t o

E ' , whose r e s t r i c t i o n t o F i s such an isomorphism. Then these UF and t h e i r supersets c o n s t i t u t e a p r o j e c t i o n f i l t e r , o b v i o u s l y i n j e c t i v e , under which R i s t h e i n v e r s e image ( o r p r o j e c t i o n ) of R ' . 0

i s o l d e r than R .

. And eve ry f u n c t i o n

1.3. The p reced ing d e f i n i t i o n s and p r o p o s i t i o n s immediate ly ex tend t o t h e case of m u l t i r e l a t i o n s . Thus we have t h e f o l l o w i n g s tatement . L e t R , R ' be two m u l t i r e l a t i o n s w i t h r e s p e c t i v e bases E , E ' .

Chapter 10 215

.- I f R ' i s o lde r than R , then f o r every S ' w i t h base E ' , there e x i s t s an S

o f the same a r i t y and w i t h base E , such t h a t the concatenation R ' S ' i s o lde r

than RS (uses the u l t r a f i l t e r axiom; ZF su f f i ces i f E i s denumerable).

0 Take an i n j e c t i v e u l t r a f i l t e r 7 such t h a t R = 7 - '(R') , then se t S =

7 -1 9 ( S ' ) . Another proof: f o r each f i n i t e subset F o f E , there ex i s t s an SF

w i th base F , such t h a t (R/F,SF) i s embeddable i n R ' S ' . Le t U F designate the se t o f these SF F , and apply the coherence lemma o f ch.2 5 1.3. I n

the denumerable case, the u l t r a f i l t e r i s unnecessary. 0

f o r a given

1.4. FILTER IDENTICAL ON A SET

Le t E , E ' be two sets and D a subset of the i n te rsec t i on E n E ' . A projec- t i o n f i l t e r 7 from E onto E ' i s sa id t o be 0 - iden t i ca l , i f there ex i s t s an element o f F , formed o f functions from E i n t o E ' I each o f whose r e s t r i c t i o n t o D i s the i d e n t i t y . I n other'words, almost every func t ion (mod ) , when

r e s t r i c t e d t o D , i s the i d e n t i t y .

I f R = ' T - ' ( R ' ) , where i s a D- ident ica l f i l t e r , then R and R ' have the same r e s t r i c t i o n t o D . I f E ' i s i n f i n i t e and E i s a superset o f E ' , then there e x i s t s a f i l t e r from E onto E ' which i s i n j e c t i v e and E l - i den t i ca l .

0 For each f i n i t e subset F o f E , take the se t UF o f those funct ions from E

i n t o E ' whose r e s t r i c t i o n t o E ' i s the i d e n t i t y and which are i n j e c t i v e on F . Then the supersets o f these

Let R be a r e l a t i o n w i t h i n f i n i t e base E , and l e t E+ be a superset of E . An extension Rt o f R t o E+ i s sa id t o be a 1-extension o f R , i f f o r every f i n i t e subset F o f E+ , there e x i s t s an isomorphism from R+/F onto a r e s t r i c - t i o n o f R , which reduces t o the i d e n t i t y on F n E . For example, i f we add a l a s t element t o the chain w o f the na tura l numbers , then we have a 1-extension. However i f we add an element before the minimum ele-

ment 0 , then we ob ta in an extension isomorphic w i t h ~1 , y e t which i s no t a 1-extension.

Every 1-extension o f R has the same f i n i t e r e s t r i c t i o n s as R , up t o isomorphism. The converse i s f a l se : take again the chain o f na tura l numbers , and i t s

extension by adding an element before 0 . The no t i on o f 1-extension i s t r a n s i t i v e . Moreover, i f R+ w i t h base E+ i s a 1-extension o f R w i t h base E , then fpr every intermediate se t H ( E 5 H E+) , the r e s t r i c t i o n R+/H 1-extension o f R .

UF cons t i t u te the desired f i l t e r . 0

1-EXTENSION

-


L e t R be a r e l a t i o n w i t h base E , and R+ be a r e l a t i o n w i t h t h e same a r i t y , which i s basel on a superse t E+ o f E . Then R+ 1-extens ion o f R i f f t h e r e e x i s t s a p r o j e c t i o n f i l t e r T from E+ onto E , which i s i n j e c t i v e and E - i d e n t i c a l , and s a t i s f i e s R+ = f - l ( R ) . 0 Take up t h e p r o o f o f 1.2 w i t h t h e f u n c t i o n s f rom E+ i n t o E which reduce t o t h e i d e n t i t y when r e s t r i c t e d t o E . 0 I n connec t ion w i t h l o g i c , n o t e t h a t fi 1-extens ion o f R i f f t h e r e e x i s t s

an e x t e n s i o n o f R+ which i s a l o g i c a l e x t e n s i o n o f R ( c a l l e d , ve ry improper l y ,

an e lementary e x t e n s i o n o f R ) ; i . e . an e x t e n s i o n which s a t i s f i e s t h e same

l o g i c a l f o rmu las as R , where t h e f r e e v a r i a b l e s i n t h e fo rmu las a r e r e p l a c e d by elements o f t h e base I R I . See f o r example CHANG, KEISLER 1973, p rop . 5.2.2.

R+

1.5. L e t R be a m u l t i r e l a t i o n w i t h i n f i n i t e base E , and l e t R+ be a l -ex ten - s i o n o f R , based on a superse t E+ o f E . Then f o r e v e r y S w i t h base E , t h e r e e x i s t s an e x t e n s i o n S+ o f S t o Et , such t h a t t h e conca tena t ion RfSt

- i s a 1 -ex tens ion o f RS (uses t h e u l t r a f i l t e r axiom).

0 Take an u l t r a f i l t e r which i s i n j e c t i v e , E - i d e n t i c a l and which v e r i f i e s

R+ =

I n p a r t i c u l a r , f o r each i n f i n i t e s e t E , every superse t o f E , cons ide red as a m u l t i r e l a t i o n reduced t o i t s base, i s a 1 -ex tens ion o f E . Thus f o r eve ry r e l a t i o n R w i t h i n f i n i t e base E and f o r e v e r y superse t E+ of E , t h e r e e x i s t s a 1 -ex tens ion o f R which i s based on E+ . T h i s i s a weak fo rm of t h e upward Lowenheim-Skolem theorem.

- 1

- l ( R ) , and then s e t S+ = ;F -'(S) . 0 .

1.6. L e t R be a r e l a t i o n w i t h i n f i n i t e base E , and l e t D be an i n f i n i t e

subset o f E . Then t h e r e e x i s t s a s e t D+ e q u i p o t e n t w i t h D , s a t i s f y i n g

D 5 0's E , such t h a t R i s a 1 -ex tens ion o f R/D+ (uses axiom o f choice; weak fo rm o f t h e downward Lowenheim-Skolem theorem).

0 L e t us say t h a t two f i n i t e subsets F , F ' o f E+ a r e e q u i v a l e n t , i f t h e y

have t h e same i n t e r s e c t i o n G = F n D = F' n D , and i f a d d i t i o n a l l y t h e r e e x i s t s

an isomorphism f rom R/F o n t o R/F' , which reduces t o t h e i d e n t i t y on G . F o r a g i ven f i n i t e subset G o f D , t h e r e e x i s t coun tab ly many equ iva lence c lasses . Indeed i t s u f f i c e s t o choose a r e p r e s e n t a t i v e F f r om each c l a s s (ax iom o f cho ice ) , and then t o d e f i n e D+ as t h e u n i o n o f t h e F corresponding t o a l l t h e G . 0

1.7. Given t h r e e r e l a t i o n s R , R ' , S such t h a t R and R ' a r e b o t h younger - than S , then t h e r e e x i s t s an i somorph ic coey o f R '

t h i s copy and R , which i s younger than S

ces i f R and R ' a r e coun tab le ) .

and a common e x t e n s i o n o f (uses t h e u l t r a f i l t e r axiom; ZF s u f f i -

Chapter 10 271

0 Le t E , E ' be the bases o f R , R ' , which we assume t o be d i s j o i n t ; and l e t 0 be the base o f S . To each ordered p a i r (F,F') where F i s a f i n i t e subset o f E and F ' a f i n i t e subset o f E' , associate the se t LJF,F, o f a l l func t ions h from E u E ' i n t o D , whose r e s t r i c t i o n s t o F and t o F' are respec t ive ly an isomorphism from R / F onto S/h"(F) and an isomorphism from R'/F ' onto S/ho(F') . By hypothesis, f o r a l l F and F ' , the s e t UF,Fl i s non-empty. Moreover, f o r any f i n i t e supersets G o f F and 6 ' o f F ' , we have the i nc lu - s ion UG,Gl c_ UF,Fl . Thus the supersets o f the UF,Fl cons t i t u te a f i l t e r on the se t o f func t ions from E u E ' coherence lemma, ch.2 5 1.3). Match an element a i n E , w i t h an element a' i n E' , i f ha = ha' f o r almost every func t ion h , modulo the u l t r a f i l t e r . Note t h a t i t i s impossible t o match up an element i n E t o more than one element i n E ' , as w e l l as t o match up an element i n E t o another element i n E , and s i m i l a r l y when exchanging E and E ' . Indeed, the matching i s an equivalence r e l a t i o n , and i f a , b are d i s t i n c t e lements o f E , then almost every func t i on h takes R/{a,b) isomorphical ly onto S/{ha,hb) . Thus ha # hb f o r almost a l l h . Let n be the common a r i t y o f the th ree re la t i ons . Take a sequence o f n elements al, ..., a

according t o whether S(hal, ..., han) = + f o r almost every h , o r = - f o r almost every h , modulo the u l t r a f i l t e r . This r e l a t i o n Rt i s a common extension o f R and R ' . Take an isomorphic copy R" o f R ' , by keeping each element o f E ' which i s matched t o no element, and f o r each element o f E ' which i s matched, by tak ing t h a t element matched t o it; then R" i s isomorphic w i t h R ' . F i n a l l y , the desired common extension o f R and R" , i s the r e s t r i c t i o n o f Rt t o E augmen- t e d by those elements o f E ' which are matched t o no o the r element. 0

i n t o D . Take a f i n e r u l t r a f i l t e r (o r use the

t i n E u E ' and de f ine t h e r e l a t i o n Rt by R (al, ..., an) = + o r - n

1.8. (1,p)-MORPHISM AND (1,p)-ISOMORPHISM L e t p be a na tura l number . A l o c a l isomorphism f w i t h domain F , from a r e l a t i o n R i n t o a r e l a t i o n S , i s sa id t o be a (1,p)-morphism from R i n t o S , i f f o r every se t F ' = F augmented by a t most p elements from the base IRI , there e x i s t s a l o c a l isomorphism from R i n t o S which i s an extension o f f t o the domain F ' . The loca l isomorphism f i s sa id t o be a (1,p)-isomorphism from R i n t o S , if f i s a (1,p)-morphism and the inverse func t i on f-' i s a (1,p)-morphism from S i n t o R . Every l oca l isomorphism i s a (1,O)-isomor~hism. Every (1,p)-morphism i s a (1,q)-morphism f o r each qs p ; s i m i l a r l y f o r a (1,p)-isomorphism.


1-MORPHISM, 1- ISOMORPHISM

We say t h a t a l o c a l isomorphism f f rom R i n t o S i s a 1-morphism, i f i t i s a (1,p)-morphism f o r e v e r y n a t u r a l number p . I n o t h e r words i f f i s e x t e n d i - b l e t o eve ry f i n i t e superse t o f Dom f . We say t h a t f i s a 1-isomorphism, i f i t i s a (1,p)- isomorphism f o r e v e r y i n t e g e r p . I n o t h e r words i f f i s e x t e n d i b l e t o eve ry f i n i t e superse t o f Dom f and t o

eve ry f i n i t e superse t o f Rng f . O r aga in i f f and f - l a re b o t h 1-morphisms.

I f S i s an e x t e n s i o n o f R , t h e n t h e i d e n t i t y on any subset o f t h e base I R I i s a 1-morphism f rom R i n t o S . For example, l e t R be t h e c h a i n o f t h e n a t u r a l numbers , and S be t h e c h a i n o f

t h e p o s i t i v e and nega t i ve i n t e g e r s , so t h a t S i s an e x t e n s i o n o f R . Then t h e i d e n t i t y on t h e s i n g l e t o n o f 0 i s a 1-morphism f rom R i n t o S . Yet it i s n o t

even a (1, l ) - isomorphism, s i n c e t h e n e g a t i v e i n t e g e r (-1) i s s t r i c t l y l e s s t h a n 0

(mod S ) , and t h e r e e x i s t s no e lement i n

L e t S be an e x t e n s i o n o f R . Then S l y e x t e n s i o n i f f , f o r each f i n i t e subset F o f I R I , t h e i d e n t i t y on F 1-morphism f rom S into R , hence

a l s o a 1-isomorphism.

( R I which i s s t r i c t l y l e s s than 0 (mod R).

-

§ 2 - CLOSED UNDER EMBEDDABILITY, DIRECTED UNDER EMBEDDABILITY, AGE

A s e t & o f r e l a t i o n s o f t h e same a r i t y , i s s a i d t o be c l o s e d under embeddab i l i t y

(up t o isomorphism), i f f o r a l l r e l a t i o n s X, Y , i f X belongs t o & and Y 4 X then t h e r e e x i s t s a r e l a t i o n i somorph ic w i t h

A s e t 6;1 i s s a i d t o be d i r e c t e d under embeddab i l i t y , i f f o r a l l r e l a t i o n s X,Y i n (R , t h e r e e x i s t s a r e l a t i o n Z i n s a t i s f y i n g Z >,X and Z>/ Y . Given a r e l a t i o n R and an i n t e g e r p , t h e s e t o f r e s t r i c t i o n s o f R o f c a r d i - n a l i t y l e s s than o r equal t o p

Fo r c e r t a i n r e l a t i o n s R , t h i s s e t i s a l s o d i r e c t e d : f o r i n s t a n c e i f R i s a

cons tan t r e l a t i o n , o r a cha in , o r a monomorphic r e l a t i o n : see ch.9 5 4 and 6. However i f R i s t h e c o n s e c u t i v i t y r e l a t i o n on t h e i n t e g e r s , t hen excep t f o r p = 0 and p = 1 , t h i s s e t i s n o t d i r e c t e d .

Given a r e l a t i o n R and a p o s i t i v e i n t e g e r p , t h e s e t o f f i n i t e r e s t r i c t i o n s o f R o f c a r d i n a l i t y g r e a t e r t han o r equal t o p i s d i r e c t e d b u t n o t c l o s e d under embeddabi 1 i ty . AGE, REPRESENTATIVE RELATION

L e t R be a r e l a t i o n . Then t h e s e t o f f i n i t e r e s t r i c t i o n s o f R i s c l o s e d and

d i r e c t e d under embeddab i l i t y . Considered up t o isomorphism, t h i s s e t s h a l l be c a l l e d t h e o f R ; we s h a l l say t h a t R rep resen ts t h i s aqe, o r aga in t h a t R i s a r e p r e s e n t a t i v e o f t h e age.

Y which belongs t o (J&.

i s c l o s e d under embeddab i l i t y .

Chapter 10 279

For the reader des i r i ng a r igorous d e f i n i t i o n , the age o f R phic copies o f the f i n i t e r e s t r i c t i o n s o f o f the se t o o f the in tegers . For a l og i c ian , an age i s a un iversa l theory o f f i r s t order predicate calculus

w i th i d e n t i t y ; see f o r example KRAUSS 1971. Since ages are countable sets, they can be compared under inc lus ion . For example the age o f a l l f i n i t e chains, which i s represented by an i n f i n i t e chain, i s included i n the age o f a l l f i n i t e p a r t i a l order ings, which i t s e l f i s

represented by those i n f i n i t e p a r t i a l order ings, i n which every f i n i t e p a r t i a l

o rder ing i s embeddable.

Given two r e l a t i o n s R, S o f the same a r i t y , R i s younger than S ,or S & older than R (see 1.2 above) i f f t h e age o f R i s included i n the aqe o f S . The r e l a t i o n s R and S have the same age i f f each i s o lde r (and younger) than

the o ther . To each age @ there corresponds the negation age o f , obtained by rep lac ing

each element o f @, by i t s negat ion ( i . e . the r e l a t i o n tak ing always the opposite value: see ch.2 5 1.7); o r equ iva len t l y by rep lac ing a representat ive by i t s

negation. For example, the negation o f the age o f a l l f i n i t e chains, o r t o t a l orderings, i s

the age o f a l l s t r i c t f i n i t e t o t a l order ings ( w i t h < instead o f ,< ) . An age may be i d e n t i c a l t o i t s negation: f o r example the se t o f a l l f i n i t e binary re la t i ons , o r the se t o f a l l f i n i t e symmetric b inary r e l a t i o n s .

i s the se t o f isomor- R , which are based on f i n i t e subsets

2.1. Given a se t & o f f i n i t e r e l a t i o n s o f the same a r i t y , which i s closed and

d i rec ted under embeddabil ity, there e x i s t s a countable r e l a t i o n whose age i s & (up t o isomorphism). I n o ther words, an age i s an i d e a l f o r the p a r t i a l order ing

o f embeddabil ity among f i n i t e re la t i ons .

0 Since the se t If?, i s countable (up t o isomorphism), enumerate i t by using in teger

ind ices then B1 = the common extension, up t o isomorphism, o f Bo and A1 , which belongs

t o & , hence t o our sequence, w i t h the minimum poss ib le index; and so on. Then take the common extension o f the Bi , on the union o f t h e i r bases.

Given a se t o f ages, t o t a l l y ordered under inc lus ion , the union and the in te rsec- t i o n are both an age. Given a se t of ages f o r which i nc lus ion i s a d i rec ted par- t i a l order ing, then the union i s an age. However i n general, the union o r the i n te rsec t i on o f two ages i s no t an age. 0 For the case o f a union, take the age formed o f a l l f i n i t e unary re la t i ons always (+), and the age o f f i n i t e unary r e l a t i o n s always ( - ) . For the case o f an i n te rsec t i on , s t a r t w i t h two d i s j o i n t , non-empty sets Take the b inary r e l a t i o n A based on the union E v F w i t h A(x,y) = + f o r

i , thus ob ta in ing a sequence o f f i n i t e r e l a t i o n s Ai . Let Bo = A. ;

E, F .


x y belonging t o E , and A(x,y) = - otherwise. Now take the r e l a t i o n B w i t h B(x,y) = + f o r x y belonging t o E , and value (-) otherwise. Then the on ly common f i n i t e r e s t r i c t i o n s o f A and B are the r e l a t i o n s always (+) and the re la t i ons always (-) : they do no t cons t i t u te a d i rec ted se t . 0

2.2. A se t 6% o f f i n i t e r e l a t i o n s o f common a r i t y , which i s c losed under embed- d a b i l i t y , i s an age i f f &, i s no t t he union o f two subsets d i s t i n c t from &@ closed under - embeddabil ity. This i s a c lass i ca l p roper ty o f idea ls : ch.4 5 5.1 .

- -_-_-

2.3. Le t A be a r e l a t i o n o f a r i t y 3 2 , with f i n i t e base. Then the set, up t o isomorphism, o f those f i n i t e r e l a t i o n s which do no t admit an embedding o f A an i n f i n i t e age. This fo l lows from ch.5 5 1.3.(1) and (2 ) ( f a i t h f u l common extension)

2.4. Given a r e l a t i o n R and an age inc lud ing the age o f R , there e x i s t s an extension o f R which i s a representa t ive o f ,'f (uses the u l t r a f i l t e r axiom; ZF su f f i ces i f R i s countable).

0 Take a r e l a t i o n S represent ing the age s" , then a common extension o f R and o f an isomorphic copy o f S , t h i s extension being chosen t o be younger than S : see 1.7 above. 0

On the o ther hand, s t a r t i n g wi th a r e l a t i o n R younger than S , there does n o t necessar i ly e x i s t a r e s t r i c t i o n o f S having the same age as. R . 0 Take R t o be the binary, denumerable r e l a t i o n always (t). For each p o s i t i v e i n tege r i , def ine Si t o be the b ina ry r e l a t i o n o f c a r d i n a l i t y i , always tak ing the value (+) , where the bases Ei o f the Si are taken t o be d i s j o i n t . Le t S be the common extension t o the Si , based on the union of the Ei ; two elements x, y from d i s t i n c t Ei , g i v i n g the value (-) , 0

-.

2.5. L e t us give, w i thou t p roo f , the fo l l ow ing ref inement o f 1.7, which concerns the existence o f a common 1-extension, ins tead o f a common extension o f two re la - t i ons having the same age; see FRAISSE 1975.

(1) Le t R and R ' two r e l a t i o n s having the same age. There e x i s t s an isomorphic c m R" o f R ' and a common extension o f R and R" on the union o f t h e i r bases, which i s a 1-extension o f R . ( 2 ) Given R , R' having the same age. A necessary and s u f f i c i e n t cond i t i on f o r t he existence of an isomorphic copy R" o f R ' w i t h a common 1-extension o f R and R" i s t ha t , f o r every f i n i t e subsets F of I R I and F' Lf IR'I and every i n tege r p , there e x i s t s a r e l a t i o n T w i t h a (1,p)-isomorphism from R i n t o T w i t h domain F , and a (1,p)-isomorphism from R ' iK T w i t h domain F' ( f o r d e f i n i t i o n s o f 1-extension and (1,p)-isomorphism, see 1.4 and 1.8).

.. .

Chapter 10 28 1

2.6. L e t us app ly t h e above s tatement 2 . 5 . ( 2 ) t o t h e conc re te example o f chains.

Consider t h e age o f a l l f i n i t e chains, whose r e p r e s e n t a t i v e s a r e a l l i n f i n i t e

cha ins . Given two i n f i n i t e cha ins R and T , and a l o c a l isomorphism f f rom R i n t o T w i t h a f i n i t e domain F = Dom f , we see t h a t f i s a (1,p)- isomorphism

( p p o s i t i v e i n t e g e r ) i f f , f o r each o rde red p a i r o f consecu t i ve elements a and b modulo R/F , e i t h e r t h e two co r respond ing open i n t e r v a l s (a,b) (mod R) and

( f a , f b ) (mod T) have t h e same c a r d i n a l i t y < p , o r b o t h have c a r d i n a l i t i e s 3 p . S i m i l a r l y f o r t h e open i n i t i a l i n t e r v a l s b e f o r e a ( i f a i s t h e minimum o f R/F) and b e f o r e f a ; and f o r t h e open f i n a l i n t e r v a l s ( i f a i s t h e maximum o f R/F).

It f o l l o w s e a s i l y t h a t , g i v e n two i n f i n i t e cha ins R , R ' and f i n i t e subsets F

o f I R I and F ' o f I R ' I , t h e r e e x i s t s an i n f i n i t e c h a i n T and two l o c a l i s o - morphisms f f rom R i n t o T and f ' f rom R ' i n t o T , w i t h r e s p e c t i v e domains

F and F ' , which a r e (1,p)- isomorphisms: i t s u f f i c e s t o assoc ia te , t o each open i n t e r v a l i n R o r i n R ' , e i t h e r an i n t e r v a l w i t h t h e same f i n i t e c a r d i n a l i t y ,

o r ano the r i n f i n i t e i n t e r v a l . Consequently, g i v e n two i n f i n i t e cha ins R , R ' , t h e r e e x i s t s an isomorohic CODY

R" gJ R ' and a common 1 -ex tens ion o f R and R" .

2.7. On t h e o t h e r hand, t h e r e e x i s t ages where t h e common

always e x i s t .

Take f o r R t h e usual c h a i n o f n a t u r a l numbers , whose d iagona l i s changed by

s e t t i n g R(x,x) = + o r - acco rd ing t o whether x i s even o r odd. S i m i l a r d e f i -

n i t i o n f o r R ' by exchanging va lues (+) and ( - ) on t h e d iagona l . Then R and R ' have t h e same f i n i t e r e s t r i c t i o n s , up t o isomorphism. Yet a 1-extens ion o f R i s ob ta ined , as f o r t h e usual c h a i n o f i n t e g e r s , by adding new e lements a f t e r a l l t h e i n t e g e r s : so t h a t t h i s 1 -ex tens ion s t i l l has minimum 0 , w i t h va lue + f o r t h e o rde red p a i r (0,O) . S i m i l a r l y f o r a 1 -ex tens ion o f R ' : t h e y cannot be iden-

t i c a l . T h i s example i s due t o PAILLET. Another example. Take f o r R a p a r t i a l o r d e r i n g w i t h a minimum element, f o r R '

ano the r p a r t i a l o r d e r i n g w i t h two incomparable min imal e lements. It i s easy t o ensure t h e same f i n i t e r e s t r i c t i o n s , up t o isomorphism. Moreover t h e minimum i s

p rese rved when pass ing t o a e lements: we cannot o b t a i n a common 1-extens ion. Example due t o BONNET. 0

1-extens ion does n o t

1-extens ion; s i m i l a r l y f o r t h e incomparable min imal

5 3 - RELATIVE RESTRICTION, RELATIVE ISOMORPHISM, REL-AGE, MAXIMALIST RELATI ON

3.1. A-RELATION, A-RESTRICTION L e t A be a r e l a t i o n w i t h f i n i t e base. An A - r e l a t i o n i s a r e l a t i o n e x t e n s i o n

o f A . An A - r e s t r i c t i o n o f an A - r e l a t i o n R i s a r e s t r i c t i o n o f R t o a


superset o f the base I A J . I f A i s no t mentioned, then we sha l l say r e l a t i v e r e s t r i c t i o n .

A-ISOMORPHISM, A-EMBEDDABILITY

We sha l l c a l l an A-isomorphism, o r more exac t l y an IAl-jsomorphism from an A-rela- t i o n R onto another A - re la t i on S , any isomorphism from R onto S whose res-

t r i c t i o n t o the base o f A i s the i d e n t i t y . Two extensions o f A are sa id t o be A-isomorphic i f such an isomorphism ex i s t s . We say t h a t an extension R o f A i s A-embeddable i n another extension S o f

A , i f there e x i s t s an A-isomorphism from R onto a r e s t r i c t i o n o f S , again c a l l e d an A-embedding o f R i n t o S . Here again we sha l l speak o f r e l a t i v e i so -

morphism and o f r e l a t i v e embeddabi l i ty .

&AGE OR REL-AGE Given an extension R o f A , the A - s o f R w i l l be the se t o f a l l f i n i t e

A - res t r i c t i ons o f A-isomorphism. For the reader d e s i r i n g

a r igorous d e f i a i t i o n , the A-age o f R i s the se t o f those A-relat ions whose bases are f i n i t e sets i nc lud ing I A 1 and inc luded i n the union I A J uw , where

I A l i s d i s j o i n t from w . We say t h a t R .represents, o r i s a representa t ive , o f the considered A-age.

I f A i s no t mentioned, we s h a l l speak o f re l -age. Each A-age i s a se t o f f i n i t e A-relat ions, which i s closed and d i rec ted under A-embeddability.

Conversely given a se t ft o f f i n i t e under A-embeddability, there e x i s t s a countable A- re la t ion which i s a represen: t a t i v e o f 4 . Same proo f as i n 2.1.

R , considered up t o

W i s the se t o f in tegers . It may be convenient t o assume t h a t the base

A-relat ions, which i s - c l o s e d and d i rec ted

3.2. A-OLDER RELATION, A-YOUNGER RELATION We say t h a t an A- re la t ion S i s A-older than another A - re la t i on R , o r t h a t

R i s A-younqer than S , i f every f i n i t e A - r e s t r i c t i o n o f R i s A-embeddable i n S . O r equ iva len t ly , i f the A-age represented by R i s included i n the A-age represented by S . Given th ree A- re la t ions R , R ' and S where R and R ' are both A-younqer than - S , the re e x i s t s an A-isomorphic copy R" of R ' and a common extension 0-f R and R" , which i s A-younger than S (uses the u l t r a f i l t e r axiom; ZF suf-

f i ces i f R and R ' are countable; same p roo f as i n 1.7 above).

Consequently, given an A- re la t ion R and an A-age f i nc lud ing the A-age o f

R , there e x i s t s an extension o f R w i t h the same c a r d i n a l i t y , which i s a representa t ive o f . Same cond i t ions and same proo f as 2.4.

Chapter 10 283

3.3. INDUCED REL-AGE, SPECIFICATION OF A REL-AGE

L e t A be a f i n i t e r e l a t i o n and B a f i n i t e e x t e n s i o n o f A . S t a r t i n g w i t h a B-age @ , t h e A - r e s t r i c t i o n s o f t h e e lements o f (I?, , considered up t o A-isomorphism, c o n s t i t u t e an A-age which i s s a i d t o be induced by @ . I n p a r t i c u l a r i f A i s empty, t hen t h e A-age reduces t o t h e a l ready known n o t i o n o f an age: we s h a l l speak o f t h e age induced by a g i v e n B-age.

The n o t i o n o f induced r e l - a g e i s r e f l e x i v e , t r a n s i t i v e and an t i symmet r i c , and thus d e f i n e s a p a r t i a l o r d e r i n g among re l -ages .

Fo r eve ry A-age U$ and e v e r y r e l a t i o n B be long ing t o b4, t h e r e e x i s t s a t l e a s t

- one B-age which induces c& . We say t h a t such a B-age s p e c i f i e s 4 o r i s a s p e c i f i c a t i o n o f 4 . For A and B g iven, i n genera l t h e r e e x i s t many B - a g e r s p e c i f i c a t i o n s o f c,+ . 0 S t a r t w i t h t h e age o f f i n i t e chains, so t h a t A i s empty, and l e t B be t h e

c h a i n on two elements u < v . There e x i s t s a s p e c i f i c a t i o n , a l l o f whose elements a r e cha ins beg inn ing w i t h u , v . There e x i s t s ano the r s p e c i f i c a t i o n f o r which we have a t most p e lements between u and v ( p g i v e n i n t e g e r ) . F i n a l l y t h e r e

e x i s t s ano the r s p e c i f i c a t i o n w i t h as many e lements as one m igh t w ish between u and v ; o r w i t h as many e lements as one w ish b e f o r e u and between u and v

and a f t e r v , e t c . 0

Consider two f i n i t e r e l a t i o n s A and B e x t e n s i o n o f A , w i t h an A-age '4 and a B-age (13 . Then t o say t h a t 6 i s a s p e c i f i c a t i o n o f 64 i s e q u i v a l e n t t o say ing t h a t every r e p r e s e n t a t i v e o f @ i s a r e p r e s e n t a t i v e o f '4 . I f @ i s a s p e c i f i c a t i o n o f s i o n of B , i s n o t , i n genera l , a r e p r e s e n t a t i v e o f @ : cons ide r aga in t h e case o f cha ins w i t h A empty and B d e f i n e d on two elements.

It may happen t h a t be a r e p r e s e n t a t i v e o f & and t h a t no r e p r e s e n t a t i v e o f

0 Take f o r R t h e c o n s e c u t i v i t y r e l a t i o n on t h e p o s i t i v e a n d n e g a t i v e i n t e g e r s . Take A empty; l e t S be a c o n s e c u t i v i t y r e l a t i o n formed o f two components each isomorphic w i t h R ; and f o r B , t a k e t h e r e s t r i c t i o n o f S t o two elements,

each i n one o f t h e components. Then e v e r y B - r e l a t i o n r e p r e s e n t i n g t h e B-age o f S , has a t l e a s t two components, hence cannot be isomorphic w i t h R . 0

, then a r e p r e s e n t a t i v e o f &? which i s an exten-

R be isomorphic w i t h R .

3.4. L e t A be a f i n i t e r e l a t i o n and B be a f i n i t e e x t e n s i o n o f A . Consider two A-ages d and h?' i n c l u d i n g A . Then f o r e v e r y B-age s p e c i f i c a -

tx@ of & , t h e r e e x i s t s a B - a 2 i n c l u d i n g 63 , and which i s a s p e c i f i - c a t i o n o f A' . 0 Take a coun tab le r e p r e s e n t a t i v e R o f L+'and a coun tab le r e p r e s e n t a t i v e S o f

@) . This r e l a t i o n S

A' : see 3.2 above. This r e l a t i o n T i s a B- re la t ion , and i s a representa t ive

i s s t i l l a representa t ive o f '4 . There e x i s t s a common

extension T o f R and S , up t o A-isomorphism, which i s a representa t ive o f

o f a c e r t a i n B-age inc lud ing @ . 0

3.5. MAXIMAL REL-AGE

Given an age @, and a f i n i t e r e l a t i o n A belonging t o a;? , an A-age spec i f i ca- t i o n o f & i s sa id t o be maximal (mod &,), i f no o ther A-age s p e c i f i c a t i o n o f

s t r i c t l y includes i t . Example. Take the age o f a l l f i n i t e chains, and l e t A be the chain on two e le -

ments u < v . Then the on ly maximal

f i n i t e number o f elements before u , a f t e r v , and between u and v . Another example. S t a r t from the age represented by the consecut iv i t y r e l a t i o n on the na tura l numbers ; and l e t A be the consecu t i v i t y r e l a t i o n o f a f i x e d f i n i t e

chain. Then there e x i s t s on ly one maximal A-age s p e c i f i c a t i o n o f the given age:

namely the A-age o f a l l the consecut iv i t y r e l a t i o n s o f f i n i t e chains i nc lud ing A as an i n t e r v a l , and t h e i r A - res t r i c t i ons .

Now tak ing the preceding age and tak ing f o r A the r e l a t i o n on two elements u and v w i t h A(u,v) = A(v,u) = - , we have a maximal A-age by p u t t i n g a unique

element which i s the successor o f u and the predecessor o f v , and as many e le - ments as one wishes before u and a f t e r v . Given a p o s i t i v e i n tege r p , we

have another maximal A-age, by p u t t i n g exac t l y p elements between u and v , and as many as one wishes before u and a f t e r v . Given an age & and a f i n i t e r e l a t i o n A belonging t o &, , f o r each A-aqe speci-

f i c a t i o n o f @, , there e x i s t s a t l e a s t one maximal i s s t i l l a s p e c i f i c a t i o n o f &, 1. 0 We could apply the maximal i dea l axiom: see ch.4 5 5. However t h i s axiom, equiva-

l e n t t o the axiom o f choice, i s unnecessary: ZF su f f i ces , s ince the considered e le - ments are f i n i t e re la t i ons which can be based on the se t o f na tura l numbers. 0

A-age i s obtained by au tho r i z ing an a r b i t r a r y

A-ase which includes i L (and

3.6. Given two f i n i t e r e l a t i o n s A and B extension o f A , and a maximal B - a s @ , then the A-aqe induced by @ i s maximal.

0 Cal l 4 the induced A-age, and supPose t h a t i s maximal y e t 4 is n o t maximal. L e t C

A-age inc lud ing 4 and conta in ing the element C . Le t @' be a B-age spec i f i ca- t i o n o f 4' and inc lud ing @ : see 3.4 above. Since ($ i s maximal, then 63' i s i d e n t i c a l t o ($ and hence C i s an A - r e s t r i c t i o n o f an element o f ($ ; hence C i s an element o f y'z : con t rad i c t i on . 0

be an A- re la t ion which does no t belong t o & , and l e t , /$ 'be an

Chapter 10 285

On the o ther hand, a s p e c i f i c a t i o n o f a maximal rel-aqe i s not necessar i l v maximal.

0 Take two elements a , b and l e t A be the chain o f the s ing le ton o f a , and B the chain a < b . The B-age formed o f the f i n i t e chains, extensions o f B i n

which a and b remain consecutive, i s a s p e c i f i c a t i o n o f the A-age o f a l l f i n i t e

chains conta in ing a . The l a t t e r i s obviously maximal; y e t the B-age under consi-

dera t ion i s no t maximal, s ince one can add elements between a and b . 0

However, given two f i n i t e r e l a t i o n s A and B extension o f A , then every maximal

A-age which contains B has a t l e a s t one maximal B-age as a spec i f i ca t i on . This

fo l lows from 3.3 and 3.5 above.

3.7. INCOMPATIBILITY LEMMA FOR REL-AGES

Le t 6?. be an age and A a f i n i t e r e l a t i o n belonging t o & . Given a maximal A-% &, and a f i n i t e A- re la t ion B which does no t belong t o & , there e x i s t s

a r e l a t i o n B ' belonging t o l f z , such t h a t no A-age, spec i f i ca t i on o f %, , con-

ta ins both B and B ' . 0 No A-age, s p e c i f i c a t i o n o f &, contains both B and a l l the elements o f L.f . Suppose t h a t our conclusion i s f a l se . Take a sequence o f re la t i ons

belonging t o & , i n which every element o f (Ze be embeddable. For each i , l e t Ci be a common extension o f B and B; , which belongs t o the age fi , and i s obviously an extension o f A . We can always suppose t h a t B;+l i s an extension

o f B; f o r each i . Moreover, f o r i f ixed, we can suppose t h a t f o r a l l j> i we have the same r e s t r i c t i o n Di = C j / ( l B ( v l B ; l ) , by tak ing a su i tab le i n f i n i t e sequence ex t rac ted from the j . Then the Di c o n s t i t u t e an A-age, spec i f i ca t i on o f & , which contains B and includes '4 : cont rad ic t ion . 0

B; ( i i n tege r )

3.8. MAXIMALIST SUBSET, MAXIMALIST RELATION

Given a r e l a t i o n R , a f i n i t e subset F o f i t s base i s sa id t o be maximal ist

(mod R), i f the (R/F)-age represented by R i s maximal. Equivalent ly, f o r each r e l a t i o n S o f the same age as R , every 1-morphism from R i n t o S w i t h domain 1-isomorphism (see 1.8 above). Equivalent ly, f o r each extension S o f R w i t h the same age, and f o r every r e s t r i c t i o n B o f S t o a f i n i t e super-

s e t o f F , there e x i s t s an isomorphism from B onto a r e s t r i c t i o n o f R , which i s the i d e n t i t y on F (see 3.2, genera l i z ing 1.7 above). Example. Consider the chain Q+ c3 , where Q i s the chain o f ra t i ona ls ; then any

f i n i t e subset o f Q i s maximal ist , modulo the age o f a l l f i n i t e chains; t h i s i s

no t the case f o r the p a i r o f in tegers {0,1) i n the f i n a l i n t e r v a l

Every subset o f a maximal ist f i n i t e se t , i s maximal ist . This fo l lows from 3.6.

A r e l a t i o n R i s sa id t o be maximalist, i f every f i n i t e subset o f i t s base i s maximal ist (mod R).

F , i s a

.


For example, the chain Q o f the r a t i o n a l s i s maximal ist . A second example. L e t t i n g C denote the consecut iv i t y r e l a t i o n on p o s i t i v e and nega- t i v e integers, consider each r e l a t i o n formed o f components isomorphic w i t h

w i t h d i s j o i n t bases, and w i t h value ( - ) f o r every ordered p a i r o f elements taken

i n two d i s t i n c t components. For each p o s i t i v e i n tege r p , the r e l a t i o n formed o f

p components i s a maximal ist denumerable r e l a t i o n . S i m i l a r l y the consecut iv i t y re- l a t i o n formed o f countably many such components; a l l are o f the same age.

A t h i r d example. S t a r t w i t h the consecu t i v i t y r e l a t i o n on the na tura l numbers, say C . To each unary r e l a t i o n U on these in tegers , associate the r e l a t i o n Cu

which d i f f e r s from C on the diagonal, w i t h Cu(x,x) = U(x) f o r each in tege r x . Take a denumerable se t o f the value ( - ) , and which s a t i s f y the fo l l ow ing dens i ty cond i t ion . For each f i n i t e

sequence s o f values (+) and ( - ) , take a component Cu w i t h U beginning by s . We now def ine the b i r e l a t i o n formed o f the preceding b inary r e l a t i o n , and a unary r e l a t i o n 0 , tak ing the value (t) f o r the minimum o f each comoonent, and ( - ) otherwise.

Then our denumerable b i r e l a t i o n s are a l l o f the same age. There are continuum many such. Moreover, they are a l l maximal ist . Indeed, every 1-morphism from such a

b i r e l a t i o n R i n t o another, say R ' , associates t o each element o f a component

Cu o f R , an element o f a component C,', o f R ' , isomorphic w i t h Cu , thus corresponding t o the same unary r e l a t i o n U ; the corresponding elements having a same rank i n Cu and i n C,) . Thus t h i s i s a I-isomorphism.

3.9. MAXIMALIST EXTENSION THEOREM

denumerable, maximal ist and o f the same age.

More s t rong ly , modulo the axiom o f choice: f o r every i n f i n i t e r e l a t i o n R , there e x i s t s a maximal ist extension o f R w i t h the same c a r d i n a l i t y and the same age.

0 Le t F be a f i n i t e subset o f the base o f R . It su f f i ces t o prove tha t , by adding countably many elements, then we can ob ta in an extension S o f R such t h a t the (R/F)-age represented by S i s maximal: the r e s t o f the proo f w i l l f o l l ow by i t e r a t i n g t h i s .

Le t &, designate the (R/F)-age represented by R . Take a maximal (R/F)-age, say @ inc lud ing L+$ : see 3.5. Then take S t o be an (R/F)-extension o f R which i s a representat ive o f @ : see 3.2. 0

0 I n the case o f an uncountable base, the preceding proo f i s modif ied on two po in ts . F i r s t by an o rd ina l indexing o f a l l f i n i t e subsets (us ing the axiom o f choice); secondly by the cons t ruc t ion o f a common extension which preserves the age, o r the rel-age (us ing the coherence lemma, ch.2 0 1.3). 0

C ,

Cu w i t h d i s j o i n t bases, which are mutua l l y l i nked by

For every denumerable r e l a t i o n R , there e x i s t s an extension o f R which i s

3.10. A necessary and s u f f i c i e n t cond i t i on f o r a r e l a t i o n R t o be maximal ist , i s

Chapter 10 287

t h a t every extension o f R w i t h the same age, be a 1-extension ( the su f f i c i ency

uses the axiom o f choice; ZF s u f f i c e s i f R i s denumerable).

This p ropos i t ion i d e n t i f i e s the maximal ist r e l a t i o n s as those re la t i ons which are e x i s t e n t i a l l y closed: a no t i on used i n the model-completeness c r i t e r i o n f o r l o g i c a l theor ies; see ROBINSON 1963.

0 Le t R be a maximal ist r e l a t i o n , R ' an extension o f R w i th the same age,

and F a f i n i t e subset o f the base \RI . Since the (R/F)-age represented by R

i s maximal, i t i s i d e n t i c a l t o t h a t represented by R ' . Thus the i d e n t i t y on F i s a 1-isomorphism from R i n t o R ' . I n other'words R ' i s a 1-extension o f R.

Conversely, suppose t h a t R F which i s no t maximal ist (mod R). By the preceding p ropos i t i on (axiom o f choice), there e x i s t s an extension R ' o f R , having the same age and which i s maximalist.

Thus the i d e n t i t y on F i s no t a 1-morphism from R ' i n t o R , hence R ' i s no t a 1-extension o f R . 0

i s not maximal ist . Thus there e x i s t s a f i n i t e subset

s 4 - R I C H RELATION, d - M O R P H I S M

A denumerable n-ary r e l a t i o n i s sa id t o be @, i f every denumerable n-ary re la -

t i o n i s embeddable i n it.

4.1. For every p o s i t i v e i n tege r n , there e x i s t s a denumerable r i c h n - a 2

r e l a t i o n .

0 This i s obvious f o r n = 1 : take a r e l a t i o n w i t h the value (+) i n f i n i t e l y many

times, and the value ( - ) i n f i n i t e l y many times.

We s h a l l prove the p ropos i t i on f o r n = 2 ; the p roo f extends immediately t o the

case o f n s t r i c t l y g rea ter than 2 . As base, take the s e t o f na tu ra l numbers . We sha l l de f ine our b inary r i c h re la -

t i o n i n several "stages". I n stage 1, l e t and l e t R(0,O) = + and R(1,l) = - . I n stage 2, l e t E2 be the se t o f integers from 2 through 17, and de f ine R as fo l lows. From x = 2 t o x = 9 , a t t r i b u t e

t o R a l l the 2 = 8 poss ib le systems o f values (+) and ( - ) f o r R(0,x) and R(x,O) and R(x,x) . S i m i l a r l y , from x = 10 t o x = 17 , a t t r i b u t e t o R a l l

the 8 possible systems o f values (+) and (-) f o r R(1,x) and R(x, l ) and

R(x,x) . I n general, l e t i be a p o s i t i v e in teger , and suppose t h a t El , ... , Ei are

defined, and t h a t R i s a l ready def ined on t h e i r union, so t h a t f o r any binary r e l a t i o n U based on {1,2 ,..., i} , there e x i s t s a sequence u1 ,..., ui where

u belongs t o E . ( j = 1 ,..., i ) ; so t h a t t he t ransformat ion o f 1 ,..., i i n t o

El be the s e t o f the in tegers 0, 1,

3

j 3 u1 ,..., ui i s an isomorphism from U onto the r e s t r i c t i o n R/{ul ,..., ui


Then we take a f i n i t e se t

union El u ... U Ei ; and we p a r t i a l l y de f ine our r e l a t i o n R i n stage i+l , by a t t r i b u t i n g a l l poss ib le systems o f values R(u,x) and R(x,u) and R(x,x) , f o r

a l l elements u i n El u . .. u Ei . We achieve t o de f ine R r e s t r i c t e d t o the

new union El u . . . u Ei+l , by l e t t i n g R take a r b i t r a r y values on ordered p a i r s

no t considered: f o r example ( 0 , l ) i n stage 1 , o r (2,3) i n stage 2 . Now we eas i l y see t h a t R i s r i c h . Indeed, s t a r t w i t h an a r b i t r a r y denumerable

binary r e l a t i o n X w i t h base the p o s i t i v e in tegers . We subs t i t u te t o each p o s i t i v e i n tege r i an element ui i n Ei , so t h a t the considered transformat ion be an

isomorphism from X i n t o a denumerable r e s t r i c t i o n o f R . 0

Ei+l o f in tegers x , a l l l a r g e r than those o f the

4 . 2 . E n t 2 , no t a l l r i c h denumerable n-ary re la t i ons are isomorphic.

0 S t a r t w i t h a r i c h r e l a t i o n R . Add a new element a t o i t s base, and l e t

R(a,x) = + f o r a l l x i n the base I R I . S i m i l a r l y add a new element b w i t h

the value ( - ) ins tead o f (+) . I f the two extensions were isomorphic, then the element a would be taken i n t o an a ' by the isomorphism, and we would have

R(a',b) = + and - simultaneously: con t rad ic t ion . 0

4.3. Le t R be a r i c h n-ary r e l a t i o n w i t h denumerable base E , and l e t F be

a f i n i t e subset o f E . Then the r e s t r i c t i o n R/(E-F) i s r i c h .

0 Suppose t h a t F reduces t o the s ing le ton o f an element a o f E . Take R ' t o be isomorphic w i t h R and having base E ' d i s j o i n t from E ; then take a common extension S o f R and R ' w i t h base E u E ' , completed a r b i t r a r i l y

f o r n- tuples w i t h some terms i n E and some terms i n E ' . Then S i s equimor- ph ic w i t h R , so t h a t there e x i s t s a r e s t r i c t i o n o f R isomorphic w i t h S . Then whether a belongs t o the image o f E o r o f E ' , we have t h a t R i s

embeddable i n the r e s t r i c t i o n o f R t o i t s base minus the element a . 0

4.4. CONNECTION WITH INDIVISIBLE RELATIONS Recall t h a t R w i t h i n f i n i t e base E i s sa id t o be i n d i v i s i b l e , i f f o r every

p a r t i t i o n o f E i n t o two complementary subsets C and D = E-C , then e i t h e r R i s embeddable i n R/C , o r i n R/D :.see ch.6 5 6.

A r i c h b inary r e l a t i o n R i s no t i n d i v i s i b l e , s ince one can p a r t i t i o n i t s base i n t o the se t o f those x f o r which R(x,x) = + and i t s complement: we ob ta in

two r e s t r i c t i o n s i n t o ne i the r o f which R i s embeddable. However, g iven a b inary r e l a t i o n R , give the value (+) t o a l l ordered p a i r s

(x,x) Then every r e f l e x i f i e d r i c h b inary r e l a t i o n i s i n d i v i s i b l e .

0 Let R be such a r e f l e x i f i e d r i c h b inary r e l a t i o n on the in tegers . Take

on the diagonal: we ob ta in the r e f l e x i f i e d r e l a t i o n R .

Chapter 10 289

an w -sequence o f isomorphic copies Ri o f R , w i t h mutua l l y d i s j o i n t bases Ei

( i i n tege r ) . Take the common extension S o f a l l the Ri , such t h a t f o r x belonging t o Ei and y t o E j , we have S(x,y) = R ( i , j ) ( i , j d i s t i n c t in tegers ) .

Then given a p a r t i t i o n o f the base I S ( some Ei one element i n each Ei . I n both cases, t i ons o f S . Since S i s equimorphic.wi th R , then R i s i n d i v i s i b l e . 0

i n t o two complementary subsets, e i t h e r

i s included i n one o f the subsets; o r the o ther subset contains a t l e a s t R i s embeddable i n one o f the two r e s t r i c -

4.5. c( -MORPHISM, o(-OLDER RELATION, -YOUNGER RELATION

Le t R , S be two r e l a t i o n s o f the same a r i t y ; l e t f be a l oca l isomorphism from R i n t o S w i t h domain F ; and l e t d be an o rd ina l .

We say t h a t f i s an o( -morphism from R i n t o S , i n the fo l l ow ing cases: (1) every l o c a l isomorphism i s a 0-morphism;

( 2 ) i f o( i s a successor o rd ina l , say o( = f 3 + 1 , then f i s an 4 -morphism

i f f o r every se t G = F augmented by a f i n i t e number o f elements i n I R I , there

e x i s t s a f i -morphism from R i n t o S , extending f t o the domain G ;

( 3 ) i f cx i s a l i m i t o rd ina l , then f i s an o( -morphism i f i t i s a 13 -morphism f o r every /s s t r i c t l y l ess than o( . For o( = 1 , we f i n d the no t i on o f 1-morphism: see 1.8.

Every g -morphism i s a (3-morphism f o r each /s less than o( . An o( -morphism, when r e s t r i c t e d t o any subset o f i t s domain, i s s t i l l an s' -morphism. The composition o f two d -morphisms i s an o( -morphism.

An isomorphism from R onto S i s an O(-morphism from R i n t o any extension o f S., f o r every o rd ina l 6 . S i m i l a r l y f o r a l o c a l isomorphism from R i n t o S which i s ex tend ib le t o an isomorphism.

I n p a r t i c u l a r , the i d e n t i t y on a subset o f the base o f R , i s an o( -morphism from R i n t o every extension o f R , f o r every o rd ina l o( . We say t h a t a r e l a t i o n S i s o( -older than R , o r t h a t R i s &-younger than S , i f the empty func t i on i s an @ -morphism from R i n t o S . The comparison " o( -o lder " i s r e f l e x i v e and t r a n s i t i v e . An a( -o lder r e l a t i o n i s f i - o lde r f o r every /3 less than o( . I f R i s embeddable i n S , then S i s & - o l d e r than R f o r every o rd ina l o( . The no t ion o f 1-older coincides w i t h the no t i on o f o lder , def ined i n 1.2; s im i l a r - l y f o r 1-younger.

4.6. L e t R , S be two denumerable re la t i ons . If S & u l - o l d e r than R , - then R i s embeddable i n S (uses countable axiom o f choice).

0 Le t f be an W1-morphism from R i n t o S , w i t h f i n i t e domain F , and l e t G be an a r b i t r a r y f i n i t e superset o f F ( inc luded i n the base I R I ) . It su f f i ces

t o prove t h a t f i s ex tend ib le t o an w l-morphism g from R i n t o S , w i t h

domain G . Indeed, s t a r t i n g w i t h the empty se t as F and i t e r a t i n g t h i s , we end up w i t h an isomorphism from the denumerable r e l a t i o n R onto a r e s t r i c t i o n o f S . Since augmented by a f i n i t e number o f elements, and since f o r each countable o rd ina l o( , the func t ion f i s an (o< +l)-morphism, there e x i s t s a b i j e c t i o n g , extending f t o the domain G , which i s an 4 -morphism from R i n t o S . There are only countably many b i j e c t i o n s g w i t h domain G and range included i n

the denumerable base I S 1 . While ul many countable o rd ina ls c( : so there ex i s t s a t l e a s t one g which i s an o( -morphism f o r every countable o rd ina l % , hence

which i s an Cdl-morphism from R i n t o S . Note t h a t t h i s argument uses the f a c t

t h a t a countable union o f countable o rd ina ls i s a countable o rd ina l ; i n o ther words, we use the countable axiom o f choice, as already i n ch.1 5 2.5 and i n ch.2 5 6.1. 0

G = F

4.7. NON-EMBEDDABILITY RANK

For R , S denumerable r e l a t i o n s w i t h same a r i t y , e i t h e r R ,<S (and hence S

i s o( -o lder than R f o r every o rd ina l o( ); o r R i s no t embeddable i n S , and

by the preceding, there e x i s t s a greatest countable o rd ina l o( f o r which S i s CX -o lder than R , and so a l e a s t o rd ina l o( + 1 from which p o i n t on S i s no t

o lder than R . We c a l l o( + 1 the non-embeddability rank o f R S . This rank remains unchan-

ged i f we replace e i t h e r R o r S by an equimorphic r e l a t i o n . For example, s t a r t w i t h the chain Q o f ra t i ona ls , and w i t h the o rd ina l wOi . By induc t ion , we see t h a t d embeddabil ity rank o f Q i n L& i s a t l e a s t equal t o o( . More prec ise ly , we see by induc t i on on o( t h a t a l o c a l isomorphism f from Q i n t o an o rd ina l , w i t h Dom f = {al ,..., ap) (al< ... < a

i s @ - o l d e r than Q . In o ther words, t h e non-

mod Q ) , i s an P

o( -morphism, provided t h a t each o f the i n t e r v a l s (O,fal) , (fal,fa2) , ... (fap-l,fap) , ( fap , + 00 ) i n the o rd ina l , i s a t l e a s t equal t o w o( .

o( +1 These cond i t ions can obviously be s a t i s f i e d w i t h the o rd ina l w

Consequently, our preceding p ropos i t i on 4.6 generalizes the p ropos i t i on i n ch.5

5 3.2, which s ta tes t h a t Q i s embeddable i n a chain, i f every countable o rd ina l i s i t s e l f embeddable. I n our new terminology, Q i s embeddable i n a r e l a t i o n R

i f R i s % - o l d e r than Q f o r every countable o rd ina l o( . This i s the case,

a f o r t i o r i , i f CJ i s embeddable i n R f o r every countable o rd ina l exponent o(.

.

d .

NON- R I CHNESS RANK

I n p a r t i c u l a r , l e t R be a denumerable non-r ich r e l a t i o n .

We def ine the non-richness rank o f R t o be the l e a s t countable o rd ina l o( f o r which R i s no t o( -o lder than a denumerable r i c h r e l a t i o n . This rank does no t depend on the chosen r i c h r e l a t i o n , s ince a l l r i c h r e l a t i o n s are equimorphic.

Chapter 10 29 1

4.8. The non-embeddability rank leads t o two kinds o f problems, according t o whether

we f i x R and consider o( -o lder r e l a t i o n s than R , o r on the cont ra ry i f we f i x S and consider o( -younger r e l a t i o n s than S . F i r s t f i x S and note t h a t i n the usual cases, there e x i s t s a countable o rd ina l

For example, i f S i s the denumerable b inary r e l a t i o n always (+), o r again the chain Q o f ra t i ona ls , then i t su f f i ces t h a t a denumerable r e l a t i o n X be l-youn-

ger than S , f o r X t o be embeddable i n S . I f S

0-younger r e l a t i o n than S , i s embeddable i n S . I f S i s the chain o f na tura l numbers , then a I-younger r e l a t i o n i s necessar i ly

a chain. A any two given elements, o r before any given element. So t h a t i t can only be e i t h e r

a f i n i t e chain, o r a chain isomorphic w i t h S = Gd , hence embeddable i n

Problem 1 . For each countable r e l a t i o n S , does there e x i s t a countable o rd ina l

I n o ther words, i f X i s no t embeddable i n S , then i s the non-embeddability rank of X i n S un i fo rmly bounded by a countable o rd ina l .

Now f i x R and consider the o( -o lder re la t i ons than R . We saw i n the preceding

paragraph, t h a t f o r R = chain Q o f r a t i o n a l s and f o r every countable o rd ina l o( , then &a i s o( -o lder than Q .

Problem 2 . For each countable r e l a t i o n R o f a r i t y a t l e a s t equal t o 2 , and f o r each countable o rd ina l O( , does there e x i s t an Xb R , which i s denumerable and o( -o lder than R . I n o ther words, i s i t poss ib le t h a t the non-embeddability

rank o f R i n an X # R takes a r b i t r a r i l y la rge countable o rd ina l values. A p o s i t i v e answer f o r r i c h re la t i ons , and more genera l l y f o r inexhaust ib le re la - t i ons (de f ined below i n 5 5 ) , w i l l soon be obtained. This i s the case, f o r ins -

tance, o f the chain o f na tura l numbers , and o f the denumerable b inary r e l a t i o n always (+) .

o( such t h a t every 4 - y o u n g e r r e l a t i o n than S i s embeddable i n S .

i s r i c h , then every countable r e l a t i o n , hence already every denumerable,

2-younger r e l a t i o n can have a t most f i n i t e l y many elements between

.

d , such t h a t every countable @-younger r e l a t i o n than S i s embeddable i n S .

5 5 - INEXHAUSTIBLE RELATION AND INEXHAUSTIBLE AGE

A r e l a t i o n R w i t h i n f i n i t e base E i s sa id t o be inexhaust ib le i f , f o r every

f i n i t e subset F o f E , the r e l a t i o n R i s embeddable i n i t s r e s t r i c t i o n

R/(E-F) , hence i s equimorphic t o i t . By 4.3, every denumerable r i c h r e l a t i o n i s inexhaust ib le. Every i n d i v i s i b l e r e l a t i o n (see ch.6 5 6) i s inexhaust ib le . For example, the chain

w , the chain Q , the denumerable b inary r e l a t i o n always (+). However, there e x i s t o ther denumerable inexhaust ib le re la t i ons . For example, the consecut iv i t y r e l a t i o n on na tura l numbers .

5.1. Le t F be a f i n i t e se t . Consider an m-sequence o f denumerable r e l a t i o n s Ri

( i i n tege r ) , a l l o f the same a r i t y >, 2 , each o f whose base includes F ; such t h a t f o r any two d i s t i n c t in tegers i , j the in te rsec t i on o f the bases l R i l and l R j l i s F ; and f i n a l l y w i t h the c o m p a t i b i l i t y cond i t i on Ri/F = R./F . Let H be an inexhaust ib le r e l a t i o n o f the same a r i t y , w i t h H $ Ri f o r each i . :f the Ri i n which H i s no t embeddable.

0 F i r s t there e x i s t s a common extension o f the r e s t r i c t i o n s

H i s not embeddable: use ch.5 5 1.4. Complete t h i s common extension a r b i t r a r i l y , adding F t o i t s base and preserving each Ri . Then since H i s inexhaust ib le , the non-embeddability cond i t i on subsists. 0

J

Ri/(lRi( - F) , i n which

5.2. Le t H be a denumerable inexhaust ib le r e l a t i o n o f a r i t y >/ 2 . For each countable o rd ina l o( and each f i n i t e subset F o f the base I H I , there e x i s t s a countable r e l a t i o n R i n which H i s non-embeddable, and f o r which the

j d e n t i t v on F o( -morphism from H 1- R (uses countable ax. o f choice).

0 For oC = 0 , take R = H/F . Suppose t h a t o( i s a successor o rd ina l q = A+l and t h a t our p ropos i t i on holds f o r f i . Then t o each f i n i t e subset Fi o f the base I H 1 , inc lud ing F , associa-

t e a r e l a t i o n Ri i n which H i s non-embeddable, and f o r which the i d e n t i t y on Fi i s a (5-morphism from H i n t o Ri . Keep the elements o f F , but eventua l l y change those i n lR i l - F so t h a t f o r i, j d i s t i n c t , the se t d i f fe rences l R i l - F

and common extension R o f the Ri , i n which H i s non-embeddable, and where the

i d e n t i t y on F i s ana-morphism from H i n t o R ( t ake one R i f o r each F i by c.a.c.).

Now suppose t h a t o( i s a non-zero l i m i t o rd ina l , and l e t /ji ( i i n tege r ) be an increasing a-sequence o f o rd ina l s w i t h supremum o( . For each in tege r i , l e t

Ri be a r e l a t i o n i n which H i s non-embeddable, and such t h a t the i d e n t i t y on F

i s a fii-morphism from H i n t o Ri . Suppose t h a t the d i f f e rence se ts l R i l - F are mutual ly d i s j o i n t , and take a common extension R o f t he Ri , i n which H i s non-embeddable. Then the i d e n t i t y on F i s an 4 -morphism from H i n t o R . 0

Then our problem 2 i n 4.8 above, has a p o s i t i v e so lu t i on i n the p a r t i c u l a r case o f an inexhaust ib le r e l a t i o n : i t su f f i ces t o take F empty. The non-embeddability o f H i n R means e i t h e r t h a t R i s p rooer ly embeddable

i n H , o r t h a t R and H are incomparable ( w i t h respect t o embeddabi l i ty) .

I n ce r ta in cases, f o r instance i f H = Q (chain o f r a t i o n a l s ) , the preceding propo- s i t i o n can be strengthened t o R < H . 0 We have seen i n 4.7 t h a t U'( , which i s < Q , i s o( -o lder than Q . And more general ly t ha t , f o r every f i n i t e subset F o f the base 191 , there e x i s t s an

I R . 1 - F are always d i s j o i n t . By the preceding propos i t ion , there e x i s t s a J

o( -morphism from Q i n t o cdat', w i t h domain F . 0

Chapter 10 293

However, the preceding p ropos i t i on cannot always be so strengthened. For H , take the denumerable inexhaust ib le b inary r e l a t i o n always (+) . Then the

cond i t ion R < H would imply t h a t R i s f i n i t e and always (+); and such a re la -

t i o n R i s no t even 1-older than H . 0

5.3. FAITHFUL EXTENSION WITH RESPECT TO AN INEXHAUSTIBLE RELATION Let H be a denumerable inexhaust ib le r e l a t i o n w i t h a r i t y >/ 2 . For every denumerable r e l a t i o n R i n which H i s non-embeddable, there ex i s t s a denumerable R+> R i n which H i s s t i l l non-embeddable. Compare w i th ch.5

5 1.3.(1) f o r a r b i t r a r y re la t i ons , and ch.5 5 4.5 f o r chains; note t h a t here we

have the add i t i ona l requirement t h a t R . Our proof uses countable axiom o f choice. 0 Le t o< be the non-embeddability rank o f H i n R . By the preceding 5.2, there

ex i s t s a denumerable r e l a t i o n S i n which H i s non-embeddable, and which i s

o( -o lder than H ( take F empty). By ch.5 5 1.3.(2), there ex i s t s a denumera- b l e Rt which i s a common extension o f R and S , up t o isomorphism, and which

respects the non-embeddability o f (see 4.7 f o r the non-embeddability rank). 0

I n p a r t i c u l a r , f o r every denumerable non-r ich r e l a t i o n

rab le non-r ich r e l a t i o n which i s s t r i c t l y g rea ter than

dabi 1 i t y ) . Consequently, a denumerable r e l a t i o n which admits an embedding o f every denumera- b l e non-r ich r e l a t i o n , i s necessar i l y r i c h . Compare w i t h the case of denumerable chains admi t t ing an embedding o f every coun- t a b l e o rd ina l : ch.5 5 3.2.

R+ i s denumerable, as i s

H

R , there e x i s t s a denume- R ( w i t h respect t o embed-

5.4. INEXHAUSTIBLE AGE

We say t h a t an age & i s inexhaust ib le , i f there e x i s t s a r e l a t i o n representat ive

o f d% , which i s inexhaust ib le i n the sense o f 5 5 above.

I f an age 6L i s inexhaust ib le , then 6% v e r i f i e s the fo l l ow ing equivalent condi- t ions :

(1) For any two f i n i t e r e l a t i o n s A , B belonging t o &, a-edisjoint, -

there e x i s t s a common extension o f A and B , which belongs t o & ;

( 2 ) For every f i n i t e r e l a t i o n A belonging t o &, and every p o s i t i v e i n tege r p , there e x i s t s an element B o f d% such t h a t A i s embeddable i n the r e s t r i c t i o n - o f B t o i t s base minus p a r b i t r a r y elements;

(3 ) L e t R be a representa t ive o f the age 62 , and E i t s base; then f o r every f i n i t e subset F of E , the r e s t r i c t i o n R/(E-F) i s again a representa t ive

(4 ) For R , S

extension o f R and S , which represents 0% (uses the u l t r a f i l t e r axiom; ZF su f f i ces i f R and S are denumerable).

L o f a ; representat ives o f 0% w i t h d i s j o i n t bases, there e x i s t s a comnon


0 Sta r t i ng w i t h ( l ) , take p+ l copies isomorphic w i t h A and having d i s j o i n t

bases, and a common extension B The cond i t ion (2 ) immediately imp l ies (3); and (3 ) immediately imp l ies (1 ) . Note t h a t i t su f f i ces t h a t there e x i s t s a representa t ive o f &, which s a t i s f i e s (3 )

i n order t h a t every representa t ive s a t i s f y ( 3 ) . F i n a l l y (1 ) imp l ies (4 ) by the coherence lemma (ch.2 5 1.3, equ iva len t t o the u l t r a -

f i l t e r axiom): apply t h i s lemma t o the unions F u G where F i s a f i n i t e subset

o f the base ( R I and G i s a f i n i t e subset o f ( S l . Obviously (4 ) imp l ies (1 ) . 0

Every age which s a t i s f i e s the cond i t ion (1 ) i n the preceding 5.4, i s inexhaus-

t i b l e . Hence each o f the precedinq cond i t ions (1) to (4) i s equ iva len t t o inexhaus-

ti b i 1 i ty . More s t rong ly i f dL s a t i s f i e s (l), then f o r every denumerable r e l a t i o n R repre- _senting a , there ex i s t s a denumerable extension o f and represents 6& (POUZET 1979 p. 343).

0 Suppose t h a t the cond i t ion (1 ) holds. Given a se t o f re la t i ons

d i s j o i n t bases Ei , each a representa t ive o f @ o r o f an age included i n 4$, there e x i s t s a common extension o f the Ri , w i t h base the union o f the

represents & or an age inc luded i n 6?. . To seethis,use the coherence lemma, as precedently i n 5.4 f o r proving (4 ) from ( 1 ) . Now take an ~1 -sequence o f isomorphic copies Ri o f R (a representa t ive o f fi ) y

w i t h d i s j o i n t bases. By the preceding, there e x i s t s a common extension S o f the

Ri , which represents 6$. Le t ai denote the jth element o f Ei ( i ,j in tege rs ) ,

where i t i s understood t h a t there e x i s t s an isomorphism from Ri onto Ri, which

takes a'. i n t o a ( i and i' f i xed , j varying).

Using RAMSEY's theorem (ch.3 5 l . l ) , g iven an i n tege r h , we obta in an w-sequen-

ce ex t rac ted from the sequence o f the in tegers ; such t h a t a f t e r renumbering, the i+l i+l b i j e c t i o n which, f o r each in tege r i

i s a l oca l automorphism o f S . By passing t o the l i m i t which respects each Ri , we obta in a r e l a t i o n S , s t i l l represent ing a , f o r which the preceding holds f o r each in tege r h . Then S i s isomorphic w i t h i t s r e s t r i c t i o n obtained by removing the se t Eo from i t s base. Hence a lso by removing any f i n i t e union o f the

belonging t o the age: we ob ta in (2 ) .

5.5. INEXHAUSTIBLE EXTENSION THEOREM

R which i s inexhaust ib le

Ri w i t h mutual ly

Ei , which

j

i' J j

takes a6 i n t o a. , ... , a: i n t o ah ,

Ei ; thus S i s inexhaust ib le. 0

5.6. Fol lowing POUZET 1979 p . 326, we sha l l prove t h a t every age inc lud ing an inex-

haus t ib le aqe, a lso includes a maximum inexhaust ib le age. More prec ise ly l e t 6% be an age. The maximum inexhaust ib le age inc luded i n d% i s the se t (denumerable up t o isomorphism) o f f i n i t e re la t i ons A f o r which any a r b i t r a r y f i n i t e se t o f isomorphic copies o f A w i t h d i s j o i n t bases, has a common extension i n ba, .

Chapter 10 295

0 It su f f i ces t o see t h a t t h i s se t i s d i rec ted . Le t R be a denumerable r e l a t i o n

represent ing b;L . For each o f the A considered, there e x i s t i n f i n i t e l y many res-

t r i c t i o n s o f R which are isomorphic w i t h A , having d i s j o i n t bases. Thus given

A and an analogous f i n i t e r e l a t i o n B , there e x i s t s a common extension o f A and B , taken w i t h d i s j o i n t bases, which has i n f i n i t e l y many isomorphic copies which are r e s t r i c t i o n s o f R . 0

§ 6 - A RELATION RICH FOR ITS AGE, A RELATION MINIMAL FOR ITS AGE

We say t h a t a denumerable r e l a t i o n R i s r i c h f o r i t s age, i f every countable

r e l a t i o n younger than R i s embeddable i n R . By 2.4 above, i t su f f i ces t h a t every denumerable representa t ive o f the same age as R , be embeddable i n R . I n o ther words, the no t ion o f " r i c h f o r i t s age" i s iden-

t i c a l t o "maximal, w i t h respect t o embeddabili ty, among the denumerable representa-

t i v e s o f i t s age I' . For example, the r i c h denumerable b inary r e l a t i o n (see 4.1) i s r i c h f o r i t s age.

The chain Q o f ra t i ona ls , o r any denumerable chain i n which Q i s embeddable,

o r equ iva len t l y any chain equimorphic w i t h Q , i s r i c h f o r i t s age. Another example. S t a r t w i t h the consecut iv i t y r e l a t i o n on pos i t i veand negative i n te - gers, i . e . the b inary r e l a t i o n tak ing the value (+) f o r those ordered pa i r s where y = x + 1 . Take denumerably many such consecut iv i t y re la t i ons w i t h mutual ly

d i s j o i n t bases, and c a l l these the components; then complete by g i v ing value ( - ) f o r pa i r s o f elements belonging t o two d i s t i n c t components. Then every denumerable

younger r e l a t i o n i s formed o f analogous components which are e i t h e r f i n i t e , o r the consecut iv i t y on na tura l numbers , o r the consecut iv i t y on negat ive integers, o r isomorphic w i t h preceding components: such a r e l a t i o n i s embeddable i n our conse- c u t i v i ty r e l a t i o n .

(x,y)

6.1. There e x i s t s a denumerable p a r t i a l o rder ing which i s r i c h f o r i t s age, and i n which a l l f i n i t e p a r t i a l order ings, thus a l l countable p a r t i a l orderings, are embeddable.

0 Recal l the amalgamation lemma i n ch,2 5 2.2. S t a r t w i t h the order ing reduced t o

a s ing le ton . Take the th ree poss ib le p a r t i a l order ings o f c a r d i n a l i t y 2 , extending our s ing le ton order ing. These are obtained by adding a new element a t the end, o r

a t the beginning, o r making i t incomparable w i t h the unique element o f our given

s ing le ton . By the amalgamation lemma, there e x i s t s a common extension s ing le ton order ing A1 and f o r the th ree preceding extensions o f A1 , which i s i t s e l f a p a r t i a l order ing.

I n general, suppose t h a t we have a p a r t i a l o rder ing Ai ( i i n tege r ) inc reas ing sequence of p a r t i a l order ings w i t h successive c a r d i n a l i t i e s

A2 f o r our

i n which every 1,. . . ,i


i s embeddable. Take a l l poss ib le p a r t i a l l y ordered extensions o f Ai , whose base i s augmented by a new element. There are on ly f i n i t e l y many such extensions, and by the amalgamation l e m a , there e x i s t s a common extension

p a r t i a l order ing. F i n a l l y the common extension o f a l l the union o f the bases, i s a denumerable p a r t i a l o rder ing i n which every countable p a r t i a l o rder ing i s embeddable. 0

Ai+l which i s a f i n i t e Ai ( i i n tege r ) t o the

6.2. There e x i s t s a denumerable t r e e which i s r i c h f o r i t s age, and i n which a l l

f i n i t e t rees , thus a l l countable t rees are embeddable.

Our base w i l l be the se t o f a l l f i n i t e non-empty sequences ve i n tege r ) o f ra t i ona ls . We say t h a t such a sequence i s less than another such sequence ( s l,...,sq) , i f q i s g rea ter than o r equal t o p , and rl = s1 , ... , rp-l = sp-l , and f i n a l l y r c s i s re f l ex i ve , t r a n s i t i v e , antisymmetric, and t h a t any two sequences which are less

than a t h i r d , are comparable: thus we have a denumerable t ree . Le t us show t h a t every countable t r e e i s embeddable i n i t . Le t R be a t r e e w i t h denumerable base E . We can assume t h a t E i s the se t o f na tura l numbers . Take

a subset E ' o f E which contains the element 0 , and such t h a t the r e s t r i c t i o n

I = R / E ' i s a maximal chain: each element o f E-E' is incomparable (mod R) t o a t l e a s t one element o f E ' . Construct a t o t a l l y ordered extension I+ o f I , by adding countably many cu ts

t o the base III . More prec ise ly , f o r each element x o f E-E' , add the cu t , o r i n i t i a l i n t e r v a l o f I , formed o f those elements o f 111 which are less than x (mod R) . Then embed

and i n p a r t i c u l a r the i n tege r 0 , i s taken i n t o a r a t i o n a l ; o r i n o ther words,

i n t o a sequence formed o f a unique r a t i o n a l . As f o r each element x i n E-E' , i t i s taken i n t o a f i n i t e sequence o f ra t i ona ls , the f i r s t o f whose terms i s the

r a t i o n a l r which i s the image o f the c u t o f I prev ious ly associated w i t h x . Let Ur denote the equivalence c lass o f the elements x i n E-E' t o which r i s thus associated.

I n each equivalence class Ur , take the l e a s t na tura l number a subset U,'. o f Ur which contains the element u ( r ) , and f o r which the r e s t r i c -

t i o n Ir = R / U i i s a maximal t o t a l l y ordered r e s t r i c t i o n o f the p a r t i a l o rder ing

R/Ur . As before, construct an extension 1: o f Ir , by adding countably many

cuts t o the elements o f the base I IrI . More prec ise ly , f o r each element x o f

Ur - U,'.

than x (mod R ) . Then embed 1: i n the chain o f the ra t i ona ls . Each element i n

Ub , and i n p a r t i c u l a r the i n tege r u ( r ) , i s taken i n t o a sequence of two r a t i o n a l s

(rl, ..., r ) (p p o s i t i - P

. The reader e a s i l y shows t h a t t h i s comparison P. P

It i n the chain o f the ra t i ona ls . Each element o f E ' ,

u ( r ) , then take

, add the c u t formed o f those elements i n the base l U h l which are less

Chapter 10 291

the f i r s t o f which i s r and the second i s i t s r a t i o n a l image. As f o r each element x i n Ur - LJ; , i t i s taken i n t o a f i n i t e sequence o f a t l e a s t th ree ra t i ona ls , the

f i r s t o f which i s

I: , prev ious ly associated w i t h x . I t e r a t e the preceding cons t ruc t ion by p a r t i t i o n i n g the elements o f each class Ur - U; i n t o subclasses U each associated w i t h the ordered p a i r o f ra t i ona ls

r, s . Note t h a t every element o f the base E i s taken i n t o a f i n i t e sequence o f

ra t i ona ls , s ince we successively take the l e a s t i n tege r i n E , then i n each class Ur , then i n each class U etc., and then we choose a maximal chain going

r,s through t h i s l e a s t in teger . A l s o note t h a t the r e l a t i o n o f being less than, o r g rea ter than, o r incomparable (mod R), i s preserved when rep lac ing each element o f the base E o f R by the f i n i t e sequence o f r a t i o n a l s which i s associated w i t h i t . 0

r and the second i s the image o f the cu t o f Ir , belonging t o

r,s

6.3. There e x i s t s an age no t having any denumerable r i c h representa t ive (SPECKER

1957, unpublished).

0 S t a r t w i t h a l l f i n i t e binary re la t i ons , each having i t s base p a r t i t i o n e d i n t o

equivalence classes; on each class, cons t ruc t a consecut iv i t y r e l a t i o n by t o t a l l y o rder ing the elements, and then g i v ing the value (+) f o r ordered p a i r s o f consecu- t i v e elements and the value (-) otherwise; hence i n p a r t i c u l a r f o r p a i r s whose

terms belong t o d i s t i n c t classes. Now on the diagonal, hence f o r the ordered pa i r s (x,x) where x belongs t o the base, assign an a r b i t r a r y value (+) o r (-). A r e l a t i o n thus constructed sha l l be c a l l e d a modulated consecut iv i t y re la t i on ; these r e l a t i o n s obviously cons t i t u te an age (c losed and d i rec ted under embedda- b i 1 i ty ) . Consider the fo l l ow ing denumerable representat ives o f t h i s age. S t a r t w i t h the

consecut iv i t y r e l a t i o n on the na tura l numbers , modulated on the diagonal, so t h a t every f i n i t e sequence o f (+) and ( - ) i s rea l i zed by a t l e a s t one sequence o f consecutive in tegers . To see t h a t there e x i s t continuum many such representa-

t i ves , i t su f f i ces t o b i j e c t i v e l y associate each representa t ive w i t h an i n f i n i t e se t o f integers. For example, i f we assign a r b i t r a r y values (+) o r (-) t o those pa i r s (x,x) f o r which the i n tege r x i s a square, then there remain a r b i t r a r i l y la rge f i n i t e i n t e r v a l s between two consecutive squares, i n order t o ensure the

existence o f every f i n i t e sequence o f (+) and ( - ) . Suppose t h a t there e x i s t s a denumerable r e l a t i o n R which i s r i c h f o r t h i s age. Then each o f the preceding r e l a t i o n s must be embeddable i n R . Thus R must have components which are obtained by modulation from ei'ther the consecut iv i t y

r e l a t i o n o f the na tura l numbers , o r the consecut iv i t y r e l a t i o n o f the p o s i t i v e and negat ive in tegers . Since R i s denumerable, there are countably many components;

\


and i n each component, t h e r e a r e coun tab ly many modulated c o n s e c u t i v i t y r e l a t i o n s c o n s t r u c t e d on t h e n a t u r a l numbers (up t o isomorphism). Indeed, f o r any such embed-

ding, t h e images o f 1, 2, 3, ... a r e determined as soon as t h e image o f 0 i s chosen. It f o l l o w s t h a t t h e r e a r e cont inuum many modulated c o n s e c u t i v i t y r e l a t i o n s c o n s t r u c t e d f rom t h e n a t u r a l numbers, which a r e n o t embeddable i n R : con t ra - d i c t i o n . 0

6.4. Problem 1 . L e t R be a r e l a t i o n w i t h denumerable base E , which i s r i c h

f o r i t s age, and F be a f i n i t e subset o f E . Then i f R/(E-F) rep resen ts t h e same age as R , i s i t equimorphic w i t h R ( i . e . each one i s i t embeddable i n t h e

o t h e r ) . Problem 2 : f a i t h f u l ex tens ion . I f R i s n o t r i c h f o r i t s age, t hen does t h e r e

e x i s t an e x t e n s i o n s t r i c t l y g r e a t e r t han R under embeddab i l i t y , which rep resen ts t h e same age as R , and which i s n o t r i c h f o r i t s age.

6.5. A RELATION MINIMAL FOR ITS AGE

A denumerable r e l a t i o n R i s s a i d t o be min imal f o r i t s age, i f t h e r e e x i s t s no r e l a t i o n r e p r e s e n t i n g t h e same age, and s t r i c t l y l e s s than R under embeddab i l i t y . Fo r example, t h e cha in W o f n a t u r a l numbers , and i t s converse, a r e t h e o n l y min imal r e l a t i o n s f o r t h e age o f cha ins . The W - c h a i n a b l e r e l a t i o n s , i . e . t h e r e l a t i o n s f r e e l y i n t e r p r e t a b l e i n , a r e min imal f o r t h e i r age: see ch.9 5 5.6.

There e x i s t s an age hav ing no min imal denumerable r e p r e s e n t a t i v e (POUZET 1979, p. 318).

0 S t a r t w i t h t h e s e t o f a l l f i n i t e sequences and O-sequences o f 0 and 1 , which i s e q u i p o t e n t w i t h t h e continuum. F i r s t t a k e t h e unary r e l a t i o n

va lue ( - ) f o r eve ry non-empty sequence. Then t a k e t h e unary r e l a t i o n U w i t h

va lue (+) f o r each non-empty f i n i t e sequence whose l a s t t e r m i s 0 , and w i t h va lue ( - ) o the rw ise .

F i n a l l y , t ake t h e b i n a r y r e l a t i o n R w i t h va lue (+) f o r each o rde red p a i r formed o f a f i n i t e sequence and o f a consecu t i ve sequence, t h e l a t t e r o b t a i n e d by adding

0 o r 1 as a l a s t term; a d d i t i o n a l l y R takes t h e va lue (+) f o r each o rde red p a i r formed o f a f i n i t e sequence s and an U-sequence beg inn ing by t h e i n i t i a l i n t e r v a l s . L e t R t ake t h e va lue ( - ) o the rw ise .

We see t h a t t h e f o l l o w i n g a r e r e p r e s e n t a t i v e s o f t h e same age. Each r e s t r i c t i o n o f t h e m u l t i r e l a t i o n (R,U,B) whose base c o n t a i n s a l l t h e f i n i t e sequences, and

which s a t i s f i e s t h e d e n s i t y c o n d i t i o n , which a s s e r t s t h a t f o r each f i n i t e sequence s be long ing t o t h e base, t h e r e e x i s t s a t l e a s t one element i n t h e base which i s

an Cd-sequence beg inn ing by s .

w i t h va lue (t) f o r t h e empty sequence, and t h e

Chapter 10 299

Note t h a t any two denumerable r e s t r i c t i o n s o f (R,U,H) age as ( R , U , 0 ) , hence whose bases c o n t a i n a l l f i n i t e sequences and which s a t i s f y the d e n s i t y c o n d i t i o n , a r e i d e n t i c a l as soon as they a r e isomorphic . Indeed f o r

each c3 -sequence i t s u f f i c e s t o cons ide r i t s success ive f i n i t e i n i t i a l i n t e r v a l s . F i n a l l y , i f one o f these denumerable dense r e s t r i c t i o n s o f (R,U,0) were min imal

f o r i t s age, t hen i t would s u f f i c e t o t a k e a p r o p e r dense r e s t r i c t i o n , t o o b t a i n a m u l t i r e l a t i o n o f t h e same age, b u t s t r i c t l y l e s s under embeddab i l i t y : con t ra -

d i c t i o n . 0

which rep resen t t h e same

§ 7 - SATURATED RELATION, EXISTENCE CRITERION FOR A RICH RELATION

FOR ITS AGE (POUZET, VAUGHT)

SATURATED SUBSET L e t R be a r e l a t i o n and F a f i n i t e subset o f t h e base. We say t h a t F i s s&-

r a t e d (mod R), i f f o r any r e l a t i o n S r e p r e s e n t i n g t h e same age as R , and any 1-morphism f f rom R i n t o S w i t h domain F , and any f i n i t e s e t G s a t i s f y -

i n g f " ( F ) c G c I S 1 , t h e r e e x i s t s a 1-morphism f rom S i n t o R w i t h domain G , which extends

For example, g i v e n t h e c h a i n Q o f t h e r a t i o n a l s and t h e cha in W o f t h e n a t u r a l numbers, t a k e R = Q + c3 ; then eve ry f i n i t e subset o f t h e i n i t i a l i n t e r v a l Q i s s a t u r a t e d (mod R ) , b u t no subset o f t h e f i n a l i n t e r v a l W i s sa tu ra ted . Th is

example has a l r e a d y been g i v e n i n 3.8 ( m a x i m a l i s t subsets) . Another example. F o r t h e o r d i n a l W2 , t h e empty s e t i s sa tu ra ted .

0 For any c h a i n A and any f i n i t e subset G =

al< ... < ap (mod A) ; f o r a f u n c t i o n g f rom A i n t o C d 2 t o be a 1-morphism, i t s u f f i c e s t o p u t g(al) i n t o t h e second component W o f U 2 , and i n genera l

g(ai) (i = 1, ... ,p) i n t o t h e ( i + l ) s t component, l e a v i n g i n f i n i t e l y many elements

i n each o f t h e i n t e r v a l s d e f i n e d by t h e images g(ai) . 0

f- l . Reca l l t h a t t h e 1-morphism i s d e f i n e d i n 1.8.

al,. . . ,a 4 o f i t s base, w i t h P

7.1. Every s a t u r a t e d f i n i t e subset i s max ima l i s t , i n t h e sense o f 3.8.

I n o t h e r words, i f F i s s a t u r a t e d (mod R) , then t h e (R/F)-age rep resen ted by R

i s maximal; o r e q u i v a l e n t l y , f o r eve ry S r e p r e s e n t i n g t h e same age as R , every 1-morphism f rom R i n t o S w i t h domain F , i s a l s o a 1-isomorphism. The converse i s f a l s e . 0 Take and take F t o be a f i n i t e i n t e r v a l o f t h i s c o n s e c u t i v i t y r e l a t i o n . Then t h e

(R/F)-age rep resen ted by R i s maximal. However, t ake S t o be t h e r e l a t i o n o b t a i - ned f rom two components, each i somorph ic w i t h R , w i th t h e va lue ( - ) f o r p a i r s whose elements be long t o d i s t i n c t components. Then a 1-morphism f f rom R i n t o S w i t h domain F , i s a l s o a 1-isomorphism. B u t by t a k i n g G t o be a f i n i t e

R t o be t h e c o n s e c u t i v i t y r e l a t i o n on t h e p o s i t i v e a n d n e g a t i v e i n t e g e r s ,


superset of f"(F) with elements in bo th of the two components, we see t h a t no local isomorphism from S into R with domain G and which extends f-' , i s a 1-morphism. 0

7 .2 . Let R be a re la t ion, F be a saturated f i n i t e subset of the base I R ) ; l e t G be a subset of F and g be a 1-morphism from R into a relation S representing the same age, with of the same age, and a 1-morphism f from R S+ , extending g to the domain F ( t h i s proposition and i t s following consequence are communicated by HODGES; uses the u l t r a f i l t e r axiom).

0 Let H = Rng g , hence g takes R / G into S/H . Since g i s a 1-morphism, the (S/H)-age represented by S i s included in the image under g of the ( R / G ) - age represented by R . By 3.2 above ( u l t r a f i l t e r axiom), there ex is t s an extension S+ of S such tha t the (S/H)-age represented by S+ i s exactly the image under g of the isomorphism from R onto a certain rest r ic t ion of S+ , which on G i s identical t o g . So tha t th i s isomorphism, when restr ic ted t o F , i s a 1-isomorphism from R into S+ . Consequently, every subset of a saturated f i n i t e subset i s saturated (uses ultra- f i l t e r axiom).

0 Take up the preceding notations, where F i s a saturated f i n i t e subset of the base I R 1 , and G i s a subset of F . Let g be a 1-morphism from R in to S with Dom g = G . Using the preceding extension S+ of S and the 1-morphism f , we see tha t , for every f i n i t e superset K of f"(F) , included in the base I S 1 , there ex is t s a 1-morphism from S+ into R with domain K , hence a for t ior i a 1-morphism from S into R , which extends f - l , hence extends g-' . T h u s G i s saturated. 0

Dom g = G . Then there exis ts an extension S+ of S

(R/G)-age represented by R . We can even require t h a t there ex is t s an

7.3. SATURATED RELATION A relation R i s said t o be saturated, i f every f i n i t e subset of i t s base i s saturated. By the preceding, i t suffices that each f i n i t e subset of the base be included in a f i n i t e saturated subset. For example, the chain of the rationals i s saturated. The rich relation defined in 4 . 1 above, i s saturated. The consecutivity relation with denumerably many components constructed from the chain Z o f positive and negative integers ( 5 6 above); as well as the par t ia l ordering in 6.1 and the t ree i n 6.2, a l l rich f o r the i r ages, are saturated. Every saturated relation i s maximalist, by the preceding 7.1. A non-saturated maximalist relation i s obtained from 3 . 8 with the consecutivity

Chapter 10 30 1

r e l a t i o n on Z denumerably many such components, i s s a t u r a t e d . W i th t h e t h i r d example o f 3.8, we have an age w i t h o u t any r i c h r e p r e s e n t a t i v e : t o see t h i s , t ake up SPECKER's argument i n 6.3. Now by 7.4 below, eve ry r e l a t i o n which

i s n o t r i c h f o r i t s age, i s n o t sa tu ra ted . So t h a t w i t h o u r t h i r d example o f 3.8,

we o b t a i n an age w i t h o u t any s a t u r a t e d r e p r e s e n t a t i v e , b u t w i t h continuum many maxi- m a l i s t r e p r e s e n t a t i v e s ; remark due t o HODGES 1979, unpubl ished.

I n t h e te rm ino logy o f Abraham ROBINSON, a s a t u r a t e d r e l a t i o n i s c a l l e d an e x i s t e n -

t i a l l y u n i v e r s a l model o f a u n i v e r s a l t heo ry . See f o r example HIRSCHFELD, WHEELER 1975 p. 31; o r aga in SIMMONS 1976 p. 384.

Any two denumerable s a t u r a t e d r e l a t i o n s o f t h e same age, a r e isomorphic . More p r e c i s e l y , l e t R , R ' be denumerable s a t u r a t e d r e l a t i o n s o f t h e same age. Then e v e r y 1-morphism f rom R i n t o R ' w i t h f i n i t e domain, i s e x t e n d i b l e t o an isomorphism f rom R o n t o R ' . 0 L e t f be a 1-morphism f rom R i n t o R ' , w i t h f i n i t e domain F and range F ' = f " ( F ) . F o r eve ry f i n i t e superse t G ' o f F ' which i s i n c l u d e d i n t h e base

I R ' I , t h e i n v e r s e f u n c t i o n f-' i s e x t e n d i b l e t o a 1-morphism f rom R ' i n t o R

w i t h domain G ' . By i t e r a t i n g t h i s a l t e r n a t i v e l y f rom R i n t o R ' and back, we o b t a i n an a - s e q u e n c e o f l o c a l isomorphisms, one ex tend ing t h e o t h e r . Tak ing ca re

t o i n c l u d e each element o f t h e bases o f R and R ' i n t h e domains o f l o c a l i s o - morphisms, we end up w i t h an isomorphism f rom R o n t o R ' . 0

( p o s i t i v e and n e g a t i v e i n t e g e r s ) ; o n l y t h e analogous r e l a t i o n w i t h

7.4. Ever.y denumerable s a t u r a t e d r e l a t i o n i s r i c h f o r i t s age.

0 L e t R be a denumerable s a t u r a t e d r e l a t i o n , and S be a denumerable r e l a t i o n

o f t h e same age. Thus t h e empty f u n c t i o n i s a 1-isomorphism f rom R i n t o S . Since R i s sa tu ra ted , f o r any f i n i t e subset G o f t h e base I S 1 , t h e r e e x i s t s a 1-morphism g f rom S i n t o R w i t h domain G . L e t F = g"(G) and f = g . S ince t h e r e l a t i o n R i s m a x i m a l i s t (see 7.1 and 7.3) , t h e (R/F)-age 4 rep resen ted by R i s maximal. S ince g i s a 1-morphism,

t h e (S/G)-age rep resen ted by S i s i n c l u d e d i n t h e r e l - a g e image o f cd under f . By 3 .2 above, t h e r e e x i s t s an e x t e n s i o n T o f S which i s denumerable, which re - p resen ts t h e same age as R and S , and which rep resen ts t h e re l -age image of (R under f . I n o t h e r words, f i s a 1-isomorphism f rom R i n t o T .

I t e r a t i n g t h i s , and s i n c e R i s s a t u r a t e d , we have t h a t g i v e n an a r b i t r a r y f i n i t e superse t G1 o f G i n c l u d e d i n t h e base I S I , we o b t a i n a 1-morphism g1 from T i n t o R , hence f rom S i n t o R , which i s an e x t e n s i o n o f g t o t h e domain

GI ; then we o b t a i n an e x t e n s i o n TI o f S which i s denumerable and o f t h e same

age, and f o r which fl = (gl) i s a 1-isomorphism f rom R i n t o T1 . By i n c l u d i n g eve ry e lement o f t h e base

-1

-1

I S 1 i n t h e domains o f t h e success ive gi ,


we have t h a t t h e u n i o n o f a l l gi ( i i n t e g e r ) i s an isomorphism f rom S o n t o a r e s t r i c t i o n o f R . 0

7.5. L e t R be a r e l a t i o n and F a f i n i t e m a x i m a l i s t subset o f t h e base.

L e t .t% denote t h e (R/F)-age rep resen ted by R . Then F i s s a t u r a t e d i f f , f o r any f i n i t e subset G o f t h e base, which i n c l u d e s F , and f o r any (R/G)-age

@ which i s a s p e c i f i c a t i o n o f L@ , t h e r e e x i s t s a subset G ' o f t h e base,

which i s t h e image o f G under an F-isomorphism f rom R/G o n t o R / G ' , such t h a t t h e (R/G')-age rep resen ted by R i n c l u d e s t h e image of @ . 0 T h i s i s s imp ly t h e t r a n s l a t i o n , i n terms o f re l -age , o f t h e d e f i n i t i o n o f a s a t u r a t e d subset : see 5 7. We s h a l l use i t i n t h e p r o o f o f 7.6 below. 0

7.6. EXISTENCE CRITERION OF A R I C H RELATION OR OF A SATURATED RELATION

Given an age 69 , t h e r e e x i s t s a denumerable r i c h r e l a t i o n f o r t h e age d?, iff f o r each f i n i t e r e l a t i o n A-ages s p e c i f i c a t i o n s o f & . Moreover i n t h i s case, f o r each denumerable r e l a t i o n R r e p r e s e n t i n g a , there e x i s t s a denumerable s a t u r a t e d e x t e n s i o n o f R which rep resen ts ba. (POUZET 1972,

g e n e r a l i z i n g VAUGHT 1961). More s t r o n g l y , f o r eve ry i n f i n i t e r e p r e s e n t a t i v e R o f & , t h e r e e x i s t s a sa tu -

r a t e d e x t e n s i o n of R , o f t h e same c a r d i n a l i t y , which rep resen ts f-R (uses axiom o f cho ice i f R i s uncountable) .

0 Suppose t h a t t h e r e e x i s t s a denumerable r i c h r e l a t i o n R r e p r e s e n t i n g t h e age

&, . Then f o r each f i n i t e r e s t r i c t i o n A o f R and each maximal A-age which i s a s p e c i f i c a t i o n o f 6% , t h e r e e x i s t s an A - r e l a t i o n U r e p r e s e n t i n g ~4 (see 3.1). S ince R i s r i c h , t h e r e e x i s t s a r e s t r i c t i o n isomorphic w i t h U , hence a f i n i t e r e s t r i c t i o n A ' o f R which i s isomorphic w i t h A , such t h a t t h e A'-age rep resen ted by R i n c l u d e s d , hence i s an isomorphic image o f &. F i n a l l y each maximal re l -age which i s a s p e c i f i c a t i o n o f &, , i s d e f i n a b l e , up t o isomorphism, by a f i n i t e r e s t r i c t i o n o f R . Thus t h e r e a r e coun tab ly many such re l -ages.

Conversely, suppose t h a t f o r each e lement A o f t h e age 6?, , t h e r e a r e coun tab ly many maximal A-ages which a r e s p e c i f i c a t i o n s o f R . S t a r t w i t h a denumerable

r e p r e s e n t a t i v e R o f & , which we can always assume t o be max ima l i s t , by 3.9;

and w i t h two a r b i t r a r y f i n i t e subsets F and G i n c l u d i n g F , o f t h e base. Take G i n t o G ' v i a an F - i d e n t i c a l isomorphism h making G ' - F d i s j o i n t from t h e base I R I . L e t ~4 denote t h e (R/F)-age rep resen ted by R (which i s a maximal r e l - a g e ) , and l e t @ be an a r b i t r a r y maximal c i f i e s : see 3.5. Then l e t T be a denumerable r e p r e s e n t a t i v e o f @ , hence

A b e l o n g i n g t o @, , t h e r e a r e coun tab ly many maximal

(h"(R/G))-age which spe-

Chapter 10 303

a representative of the (R/F)-age . By 3 . 2 there exis ts an A-isomorphic copy T ' of T and a common extension R1 of R and T ' , such that i f we denote by GI the imaqe of G ' when passing from T t o T ' , then the (R1/G1)-age represented by R1 includes the image of a. Furthermore we can choose R1 t o be maximalist. Pass from R t o R1 , then i te ra te t h i s , by using a l l possible quadruples ( F , G , ,,A?, , 63 ) . In view of the countability of the s e t of maximal rel-ages, in the

l imit , we obtain a denumerable extension of R , each of whose f i n i t e subsets i s saturated. 0

0 In the case where R i s uncountable, the preceding proof requires the u l t ra f i l - t e r axiom t o take a common extension by 3 . 2 , and more strongly the axiom of choice t o well-order the base, and then t o well-order the se t of f i n i t e subsets. 0

7.7 . Let R be a relation. If there exis ts a saturated f i n i t e subset of the base, then the age of R sa t i s f ies the preceding cr i te r ion , and there ex is t s a saturated relation of the same age as R

0 The empty se t i s saturated by 7 .2 ( u l t r a f i l t e r axiom). Let F be empty in 7.5.

Then for any f i n i t e subset G of the base, and any maximal (R/G)-age 6 , which specifies the age of R , there exis ts a subset G ' which i s the image of G under a local automorphism of R , such that the (R/G')-age represented by R i s isomorphic to a. I t follows that there are countably many maximal rel-ages arising from an arbi t rary f i n i t e res t r ic t ion of R : the cr i ter ion i s sa t i s f ied . 0

(uses the u l t r a f i l t e r axiom).

7.8. A CLASSIFICATION OF AGES Given an age d? :

i f there ex is t s no denumerable rich relation for & , then there ex is t s a s t r i c t l y increasing w1-sequence (under embeddability) of denumerable relations having age &, such that these relations cannot be a l l embedded in any single denumerable relation of age @, (uses the axiom of choice and the continuum hypothesis).

0 Firs t take an U1-sequence of a l l relations Ri ( i countable ordinal) , having age 62 and base the se t o f natural numbers , for example. Let So = Ro ; l e t S1

be a common extension of So and R1 ; then S2 be a common extension of S1 and R2 , e t c . ; where the S are denumerable and represent the age bt . For each countable l imit ordinal o( , l e t Sd be a denumerable common extension of the ( i < o( ) . The since there does not ex is t any rich relat ion. 0

Problem. Can the conclusion be strengthened t o "these relations cannot be a l l embedded in any single denumerable relation (possibly having age larger t h a n 0%)".

Now suppose that there ex is t s a denumerable rich relation with age @ ; then:

-

Si Si ( i countable ordinal) can be chosen t o be s t r i c t l y increasing,


(1) Either a l l denumerable relations which represent 6% are equimorphic. This i s the case for denumerable binary relations always (t), which are a l l isomorphic. This i s the case for equivalence relations with denumerably many classes of cardinality 2 , and possibly classes of cardinality 1 . ( 2 ) Or there ex is t s a s t r i c t l y increasing W -sequence (under embeddability) of denumerable non-rich relations having age ($& , such that every denumerable relation of age & , in which a l l these relations are embeddable, i s rich for @ . This i s the case for relations formed of f in i te ly many components, each isomorphic w i t h the consecutivity relation on Z . ( 3 ) Or there exis ts an W1-sequence satisfying the preceding conditions. This i s the case for chains, with the sequence of the denumerable ordinals ( 4 ) Or there exis ts a relation which represents 6% and which i s immediately less t h a n the denumerable rich relations for & . Problem 1 . The impossibility of case ( 4 ) ; th i s i s , in other words, the problem 2 in 6 .4 . Problem 2 . The incompatibility of cases ( 2 ) and (3) in the same age.

8 - FINITIST RELATION

Given an inf in i te se t E and a f i n i t e subset F of E , a relation R with base E i s said t o be F- f in i t i s t , i f f a r every pair of elements a , b in E-F , the

transposition (a ,b) i s an autonorphism of R . A relation i s said t o be f i n i t i s t , i f there ex is t s an F for which i t i s F- f in i t i s t .

To say t h a t R i s F- f in i t i s t i s equivalent t o saying that every oermutation of the base E which i s the identity on F , i s an automorphism of R . This follows immediately from ch.9 5 1 . 2 . Consequently, t o say tha t R i s F- f in i t i s t i s equivalent t o saying that R i s freely interpretable (ch.9 5 2 ) in the sequence of unary relat ions, each of which takes the value (+) on one element of F (singleton relat ions) . For example, a unary relation i s f i n i t i s t i f f i t takes the value (+) on a f i n i t e subset of the base, or on the complement of a f i n i t e subset.

8.1. KERNEL OF A FINITIST RELATION - If R & F-f in i t i s t and G-f ini t is t (F, G f i n i t e subsets of the base), then R i s (Fn G)-f ini t is t . 0 Take any two elements a , b in E - (FnG) , then an element c in E - (FuG) , and replace the transposition (a,b) by the composition ( a , c ) . ( b , c ) . ( a , c ) . 0

Consequently, i f R i s f i n i t i s t , then there exis ts a minimum f i n i t e s e t F (with respect t o inclusion) for which R i s F-f ini t is t . We cal l th i s F the kernel of R ,

-

Chapter 10 305

A r e l a t i o n R i s constant (see ch.9 5 4) i f f R i s f i n i t i s t w i t h empty kerne l .

I f u belongs t o the kernel o f R , y e t no t v , then (u,v) modif ies R . I f two f i n i t i s t n-ary r e l a t i o n s w i t h the same base, have the same kernel and the

same r e s t r i c t i o n t o the kernel p lus n elements, then they are i d e n t i c a l . Indeed they have the same r e s t r i c t i o n t o each n-element subset.

8.2. Le t R be a r e l a t i o n w i t h denumerable base E . I f R i s f i n i t i s t w i t h a non-empty kernel , then there e x i s t denumerably many

isomorphic copies of R w i t h base E . I f R has empty kernel , hence i s constant, then a l l isomorphic copies of R w i t h base E are i d e n t i c a l t o R . If R 1-

f i n i t i s t , then there e x i s t continuum many isomorphic copies w i t h base

0 The two f i r s t cases immediately fo l l ow from precedinq remarks. Suppose t h a t R i s no t f i n i t i s t . Then f o r each f i n i t e subset F o f E , there e x i s t s a p a i r o f elements x, y i n E-F , such t h a t the t ranspos i t i on (x,y) modi- f i e s R . Le t n be the a r i t y ; then there e x i s t s a (n+l)-element s e t Fo and a p a i r o f elements xo, yo i n Fo , such t h a t the t ranspos i t i on (xo,yo) modif ies the

= R and R,o) = image o f R r e s t r i c t i o n R/FO : see ch.9 5 1.3. Ca l l R

under (xo,yo) . Now there e x i s t s a f i n i t e superset F1 o f Po and a p a i r o f e le -

ments xl, y1 i n F1-Fo , such t h a t the t ranspos i t i on (xl,yl) modif ies both

R/F1 and Rlo)/F1 . Yet (xl,yl) preserves R/FO and Rto)/Fo , since ne i the r

x1 nor y1 belong t o Fo . Ca l l Ri1} = image o f R under (xl,yl) and R40,1)

= image o f R under the composition o f (xo,yo) and (xl,yl) ; then these 4 images o f R a re d i s t i n c t , and even they y i e l d 4 d i s t i n c t r e s t r i c t i o n s t o F1 . I t e r a t i n g t h i s , we obtain, f o r each in tege r i , a f i n i t e se t Fi i nc lud ing Fi-l

and a p a i r o f elements

I o f i n tege r ind ices , we ob ta in an isomorphic copy RI o f R under the composi-

t i o n o f a l l t ranspos i t ions (xi,yi) ( i belonging t o I ) ; the RI being d i s t i n c t f o r d i f f e r e n t sets I , t h i s gives continuum many copies o f R . 0

E .

(empty 1

xi, yi i n Fi - Fi-l . F i n a l l y f o r each countable se t

8.3. CHARACTERIZATION OF THE FINITIST RELATIONS I f R i s a r e l a t i o n w i t h i n f i n i t e base E , then R i s f i n i t i s t w i t h a kernel

o f a t most r elements, i f f there e x i s t a t most r elements x ik E , each o f which has a t l e a s t r+l elements y such t h a t t he t ranspos i t ions (x,y) modify R . 13 I f R i s f i n i t i s t w i t h a kernel of c a r d i n a l i t y less than o r equal t o r , then

our cond i t i on i s obvious. Converse1y;suopose t h a t our cond i t i on holds. Since E i s i n f i n i t e , l e t u be an element which i s d i s t i n c t from the x . Associate t o u

those v such t h a t ( u , v ) modif ies R : there are a t most r many such v ; l e t

THEORY OF RELATIONS 306

F be

kerne ( U J )

the s e t of such v . For any two elements x , y in E-F , the t ranspos i t ions and (u ,y) , hence a l so (x ,y) preserves R . T h u s R i s f i n i t i s t and i t s i s included i n F . 0

8.4. Let R be a f i n i t i s t n-ary r e l a t ion w i t h a kernel of ca rd ina l i t y r , and l e t S be another n-ary r e l a t ion . I f every r e s t r i c t i o n of S w i t h c a rd ina l i t y l e s s than o r equal t o (n+l)(r+1)2 i s embeddable i n R , then S

kernel has ca rd ina l i t y l e s s than o r equal t o r . 0 Suppose t h a t S t i o n , there exist a t l e a s t r+ l elements u , each of which has a t l e a s t r t l elements v such t h a t ( u , v ) modifies S . There a re a t most ( r t l ) ’ many such t ranspos i t ions ; and f o r each t r anspos i t i on , t he re e x i s t s a s e t H of n t l e l e - ments, including u and v , such t h a t ( u , v ) modifies the r e s t r i c t i o n S/H (see ch.9 5 1 .3 ) . This property i s preserved when taking an isomorphism on a r e s t r i c t i o n o f R . T h u s by the preceding propos i t ion , R i s not f i n i t i s t , o r i f so , has a kernel w i t h c a rd ina l i t y s t r i c t l y g rea t e r than r . 0

i s f i n i t i s t and i t s

does not s a t i s f y our conclusion. Then by the preceding proposi-

8 .5 . CHARACTERIZATION B Y (1,p)-EQUIVALENCE We say t h a t two r e l a t ions a r e (1 ,p) -equiva len t , where p i s a natural number, i f f t h e y have the same r e s t r i c t i o n s t o < p We have already defined (1,p)-isomorphism in 1.8 above. Then two re l a t ions a re (1,p)-equivalent i f f the empty function i s a i n to the o ther .

If a r e l a t ion R I s f i n i t i s t , then there e x i s t s an in teger p such t h a t every r e l a t ion w i t h the same base and which i s (1,p)-equivalent t o R , i s isomorphic t o R . Note t h a t the minimum value of p can be s t r i c t l y g rea t e r than the ca rd ina l i t y

of the kernel. For example i f R i s cons tan t , hence w i t h empty kerne l , the m i n i - mum value of p i s the a r i t y of R , by ch.9 § 4.1. The converse of our proposit ion wi l l be proved i n 9.10 below.

0 Let n be the a r i t y and r the ca rd ina l i t y of the kernel of R . Consider a r e l a t ion S w i t h the same base E as R and which i s (1,p)-equivalent w i t h R , where p = ( n + l ) ( r + l ) ’ . By the preceding proposit ion, S i s f i n i t i s t and i t s kernel has ca rd ina l i t y l e s s than o r equal t o r . Now by interchanging R and S , we see t h a t the kernel of S has ca rd ina l i t y exac t ly r . There e x i s t s an isomorphism f which embeds i n t o S the r e s t r i c t i o n of R t o an a rb i t r a ry p-element subset including the kernel. For a ce r t a in choice of this p-element subset, f takes the kernel of R i n t o the kernel o f S : take each element x of the kernel of R , then f o r each x take r+ l elements y such

elements, up t o isomorphism.

(1,o)-isomorphism from one r e l a t ion

~

~

-

Chapter 10 307

t h a t (x,y) m o d i f i e s R , then f o r each such o rde red p a i r (x,y) t ake a r e s t r i c -

t i o n o f R t o n + l elements, i n c l u d i n g x and y , which i s m o d i f i e d by t h e t r a n s p o s i t i o n (x ,y) . Then these c o n d i t i o n s h o l d as w e l l f o r S by t h e isomorphism f , s i n c e p i s s u f f i c i e n t l y l a r g e . So t h a t t h e images f x c o n s t i t u t e t h e k e r n e l o f S . F i n a l l y ex tend f t o a pe rmuta t i on o f E , s t i l l denoted by f . Since p i s a t l e a s t equal t o r t n , o u r pe rmuta t i on f takes R i n t o a r e l a t i o n hav ing t h e same k e r n e l as S and t h e same r e s t r i c t i o n s as S t o i t s k e r n e l p l u s n elements:

by 8.1 above, t h i s image o f R under f i s S i t s e l f . 0

5 9 - ALMOST CHAINABLE RELATION

Given a s e t E , a f i n i t e subset F and a cha in A w i t h base E-F , a r e l a t i o n

R w i t h base E i s s a i d t o be (F,A)-chainable, i f each b i j e c t i o n which i s t h e un ion o f t h e i d e n t i t y on a subset o f F and a l o c a l automorphism o f A , i s a l o c a l automorphism o f R . We say t h a t R i s F-chainable, o r i s a lmost cha inab le .

I n t h e case where F i s empty, we f i n d t h e cha inab le r e l a t i o n : see ch.9 5 5.

9.1. The cons ide red r e l a t i o n R i s F-chainable i f f R i s f r e e l y i n t e r p r e t a b l e i n t h e m u l t i r e l a t i o n formed o f t h e s i n g l e t o n unary r e l a t i o n s o f each element i n F , and o f any t o t a l l y o rde red e x t e n s i o n o f A f o r which F i s e i t h e r an i n i t i a l i n t e r v a l , o r a f i n a l i n t e r v a l , o r t h e un ion o f bo th . Consequently, - i f R & F-chainable, t hen eve ry r e l a t i o n f r e e l y i n t e r p r e t a b l e

- i n R F-chainable. R i s F-chainable i f f t h e r e e x i s t s a c h a i n A w i t h t h e same base as R , such

t h a t t h e b i r e l a t i o n RA i s F-chainable.

9.2. _ I f R & F-chainable, t hen f o r each subset D o f t h e base, t h e r e s t r i c t i o n

R/D ( F n D)-chainable. I n p a r t i c u l a r , i f R w i t h base E i s F-chainable, t hen t h e r e s t r i c t i o n R/(E-F) i s cha inab le , which f o l l o w s immed ia te l y f rom t h e d e f i n i t i o n . The converse i s f a l s e .

0 Take t h e c h a i n Z of P o s i t i v e and n e g a t i v e i n t e g e r s , m o d i f i e d by t h e c o n d i t i o n Z(0,O) = - ( i n s t e a d o f +) . T h i s m o d i f i e d Z i s n o t a lmost cha inab le , s i n c e i t s

p r o f i l e i s n o t bounded: see 9.8 below.

9.3. L e t R be a r e l a t i o n and F a f i n i t e subset o f i t s base. I f f o r each f i n i t e subset X o f t h e base which i n c l u d e s F , t h e r e s t r i c t i o n R / X F-chainable,

- then R & F-chainable (uses t h e u l t r a f i l t e r axiom).

0 To each f i n i t e subset X including F , associate the se t U x of chains Y with base X-F such t h a t R / X i s (F,Y)-chainable. By hypothesis U x i s non- empty for each X ; and i f X ' i s included i n X , then every chain belonging t o U x , when restr ic ted t o U x , . By the coherence lemma (ch.2 0 1.3, equivalent t o the u l t r a f i l t e r axiom), there ex is t s a chain with base R -F , for which R i s (F,A)-chainable. 0

X ' , yields a chain belonging t o A

9 .4 . Let R be an inf in i te relation which i s F-chainable, and R ' the res t r ic . tion of R t o an inf in i te subset including F , and G be a subset of F . Now i f R ' i s G-chainable, then R - i s G-chainable (uses the u l t r a f i l t e r axiom).

0 Let E denote the base of R and E ' the base of R ' . For each f i n i t e subset X of E including F , there ex is t s a bijection which i s F-identical and takes X into a subset X ' of E ' and R/X into R / X ' . Now R / X ' , thus R / X , i s G-chainable. Letting X vary, the preceding proposition shows t h a t R i s G-chainable. 0

- -

9.5. KERNEL OF AN ALMOST CHAINABLE RELATION Let R be an inf in i te relation and F , G be two f i n i t e subsets of the base I R I . - If R F-chainable and G-chainable, then R & (FnG)-chainable (uses the u l t r a f i l t e r axiom). 0 Let E denote the base of R . The restr ic t ion R/ (E-G)uF i s (FnG)-chainable by 9.2. Then R i s (FAG)-chainable by the preceding proposition. 0

Consequently, i f R i s almost chainable, then there ex is t s a minimum f i n i t e s e t F (with respect t o inclusion) for which R i s F-chainable. We call th i s F the kernel of R .

9.6. Let R be an almost chainable relation. Then every R ' which i s younger than R i s almost chainable, and the cardinality of the kernel of R ' i s less than or equal t o the cardinality of the kernel of R (uses the u l t r a f i l t e r axiom).

0 Let E, E ' denote respectively the bases of R , R ' . Let F be the kernel of R . By 9.1, there exis ts a chain A with base E , such t h a t i f for each element a of F , we denote by U a the singleton unary relation of a , then R i s freely interpretable in multirelation formed of a binary relation A ' and of unary relations U i such that the concatenation ( R ' , A ' , U i ,...) i s younger t h a n ( R , A , U a , ...) . The relation A ' i s a chain, since i t i s younger t h a n A . Each Ui i s e i ther a unary relation always ( - ) , or a singleton unary relation of an element a ' of E ' . Let F ' denote the se t of these a ' ; hence F ' i s a subset of E ' and Card F ' ,< Card F . Moreover we can assume t h a t F i s an i n i t i a l interval of A , and so

(A.U , ,... ) . By 1.3'above ( u l t r a f i l t e r axiom), there ex is t s a

Chapter 10 309

F ' i s an i n i t i a l i n t e r v a l o f A ' . F i n a l l y R ' i s f r e e l y i n t e r p r e t a b l e i n (A',U;, ...) , by ch.9 0 2.4. Hence R ' i s a lmost cha inab le w i t h a k e r n e l i n c l u d e d i n F ' . 0

9.7. L e t R be a r e l a t i o n w i t h denumerable base E , and F be a f i n i t e subset

o f E . Then t h e r e e x i s t s a denumerable subset D of E , i n c l u d i n g F and such - t h a t R/D & F-chainable. G e n e r a l i z a t i o n o f ch.9 0 5 .5 . (1 ) .

0 Take a c h a i n A w i t h base E , which begins by t h e i n i t i a l i n t e r v a l F , and then app ly RAMSEY's theorem (ch.3 5 1.1); by c o n s i d e r i n g two f i n i t e subsets

o f t h e base as e q u i v a l e n t , i f b o t h i n c l u d e F and (R,A) /X and (R,A)/X' a re F- isomorphic . I f n des igna tes t h e a r i t y o f R , assumed t o be > 2 , then i t

s u f f i c e s t o cons ide r s e t s o f c a r d i n a l i t y (Card F) t 2n , i n o r d e r t o i n s u r e F - c h a i n a b i l i t y . 0

X, X '

X

9.8. A necessary and s u f f i c i e n t c o n d i t i o n f o r a denumerable r e l a t i o n a lmost cha inab le , i s t h a t t h e p r o f i l e o f R be bounded ( f o r t h e p r o f i l e , see

ch.9 § 7 ; s u f f i c i e n c y uses t h e u l t r a f i l t e r axiom).

R

0 L e t R be a lmost cha inab le , w i t h base E and k e r n e l F . F o r each i n t e g e r p , t h e isomorphism type o f R when r e s t r i c t e d t o p elements, o n l y depends on t h e i n t e r s e c t i o n o f F w i t h t h e s e t o f these p elements. S ince t h e k e r n e l i s f i n i t e , t h e number o f these isomorphism types i s bounded by t h e number o f r e s t r i c t i o n s o f R/F . Conversely, l e t R be a r e l a t i o n w i t h denumerable base E , which i s n o t a lmost

cha inab le . To prove t h a t t h e p r o f i l e i s unbounded, l e t h be an a r b i t r a r y i n t e g e r ; we s h a l l c o n s t r u c t h r e s t r i c t i o n s o f R , a l l o f t h e same f i n i t e c a r d i n a l i t y and

m u t u a l l y non- isomorphic .

S t a r t w i t h a denumerable, cha inab le r e s t r i c t i o n Ro o f R (see 9.7 above ) . The r e l a t i o n R i s n o t younger t h a n Ro : see 9.6, u s i n g t h e u l t r a f i l t e r axiom. Hence t h e r e e x i s t s a f i n i t e subset Fo o f E , such t h a t R/FO i s n o t embeddable

i n Ro . Take a denumerable, Fo-chainable r e s t r i c t i o n R1 o f R . F o r eve ry i n t e -

g e r p 3 Card Fo , no r e s t r i c t i o n o f Ro w i t h c a r d i n a l i t y p i s isomorphic w i t h any r e s t r i c t i o n o f R1 t o a p-element subset i n c l u d i n g Fo . The r e l a t i o n R i s n e i t h e r younger than Ro n o r t h a n R1 . Hence t h e r e e x i s t s a

f i n i t e subset F1 o f E , such t h a t R/F1 i s n e i t h e r embeddable i n Ro n o r i n R1 . Take a denumerable, F1-chainable r e s t r i c t i o n R2 o f R . For eve ry i n t e g e r

p 3 Max(Card Fo,Card F1) , a r e s t r i c t i o n o f Ro w i t h c a r d i n a l i t y p , and a res - t r i c t i o n o f R1 t o a p-element subset i n c l u d i n g Fo , and a r e s t r i c t i o n o f R2

t o a p-element subset i n c l u d i n g F1 , a r e m u t u a l l y non-isomorphic. I t e r a t i n g t h i s h t imes, we o b t a i n h r e s t r i c t i o n s which a r e m u t u a l l y non-isomorphic. 0


9.9. The f o l l o w i n g remark due t o POUZET, proves t h a t t h e p r o f i l e o f an i n f i n i t e r e l a t i o n i s i n c r e a s i n g , w i t h o u t u s i n g t h e i nc idence m a t r i x o r t h e m u l t i c o l o r theorem ( p r o f i l e i nc rease theorem, ch.9 5 7.1, u s i n g ch.3 5 5.3) .

Given a denumerable r e l a t i o n R and an i n t e g e r p , take a f i n i t e subset F o f t h e base, such t h a t eve ry r e s t r i c t i o n o f R w i t h c a r d i n a l i t y p i s embeddable

i n R/F . Then t a k e an F -cha inab le r e s t r i c t i o n o f R : t h e p r o f i l e o f t h i s r e s t r i c - t i o n i s t h e same as t h e p r o f i l e o f R , a t l e a s t up t o t h e va lue p . Hence i t

s u f f i c e s t o prove t h a t t h e p r o f i l e i nc reases i n t h e p a r t i c u l a r case where a lmost cha inab le .

R

Now assume t h a t R i s F-chainable. To each isomorphism type o f a r e s t r i c t i o n

t o p elements, a s s o c i a t e a p-element subset G w i t h R/G hav ing t h i s t ype , and t h e i n t e r s e c t i o n H = F n G hav ing t h e l e a s t p o s s i b l e c a r d i n a l i t y . Then assoc ia te t o t h i s G t h e isomorphism type o f c a r d i n a l i t y p + l , o b t a i n e d by

t a k i n g G p l u s an element n o t b e l o n g i n g t o F . Thus we d e f i n e an i n j e c t i v e func-

t i o n which, t o each isomorphism t y p e o f c a r d i n a l i t y p , assoc ia tes an isomorphism type o f c a r d i n a l i t y p + l . To see t h e i n j e c t i v i t y : i f t h e same isomorphism type o f c a r d i n a l i t y p + l i s o b t a i - ned f rom two e q u i p o t e n t subsets H, H ' o f F , each augmented by e lements n o t

be long ing t o F , then e v e r y isomorphism o f t h e f i r s t r e s t r i c t i o n o n t o t h e second r e s t r i c t i o n , takes H i n t o H ' . For o the rw ise H ' would n o t be t h e i n t e r s e c t i o n

o f F w i t h l e a s t p o s s i b l e c a r d i n a l i t y .

9.10. We a r e now i n a p o s i t i o n t o prove t h e converse o f 8.5

L e t R have base E ; i f eve ry r e l a t i o n w i t h base E , which i s (1 ,p ) -equ iva len t

w i t h R , i s i somorph ic w i t h R , then R i s f i n i t i s t . - . .-

~

0 We g i v e an argument f o r t h e case t h a t E

o f each isomorphism type o f a r e s t r i c t i o n o f R t o 6 p elements, and l e t F

be t h e un ion o f t h e bases o f these r e p r e s e n t a t i v e s . S ince F i s f i n i t e , by 9 .7 above, t h e r e e x i s t s a denumerable r e s t r i c t i o n o f R which extends R/F and i s

F-chainable. By hypo thes i s , t h i s r e s t r i c t i o n i s isomorphic w i t h R : hence we can suppose t h a t R i s a lmost cha inab le .

Assume t h a t R i s (F,A)-chainable, where A i s a c h a i n which can a r b i t r a r i l y be e i t h e r isomorphic w i t h G, or i somorph ic w i t h Q . From t h i s p o i n t on, t h e argu-

ment i n ch.9 e x e r c i s e 2, s l i g h t l y m o d i f i e d t o t a k e account o f t h e k e r n e l F ( o r i n c l u d e d i n F ) , proves t h a t f o r e v e r y p a i r o f e lements x, y i n E-F , t h e

t r a n s p o s i t i o n (x,y) p rese rves R . Hence R i s F - f i n i t i s t . 0

i s denumerable. Take a r e p r e s e n t a t i v e

EXERCISE 1 - THE EXISTENCE OF A R I C H RELATION I N THE UNCOUNTABLE CASE

Modulo t h e axiom o f cho ice and t h e continuum hypo thes i s , we s h a l l p rove as f o l l o w s

t h e ex i s tence , f o r each i n t e g e r n , o f an n-ary r e l a t i o n o f c a r d i n a l i t y W 1 ,

Chapter 10 31 1

or equiva len t ly continuum c a r d i n a l i t y , in which every n-ary r e l a t ion of same car- d i n a l i t y i s embeddable. Let E be the base s e t , which we iden t i fy w i t h the ordinal GJ1 . There exist

tdl many countable subsets of E , denoted by assoc ia te a subset C i of E w i t h c a rd ina l i t y W 1 , so t h a t a l l these Ci a r e mutually d i s j o i n t . This is poss ib le , s ince the Cartesian product of i s equipotent with w1 . S t a r t w i t h an a r b i t r a r y n-ary r e l a t ion Ro w i t h base Do . Consider a l l possible manners of extending thus cdl many such manners. To each of these p o s s i b i l i t i e s , assoc ia te an element x i s poss ib le , s ince there a re u1 many elements i n elements which do not belong t o the countable set More genera l ly l e t u be a countable ordinal index. Suppose t h a t f o r each i < u , we have already defined an n-ary r e l a t ion R i cn D i , these Ri being mutually compatible ( i . e . w i t h same r e s t r i c t i o n t o any in t e r sec t ion of t h e i r bases) . More- over suppose t h a t by using the f o r each Ri t o an additional element. Then define R U on D u , i n an a rb i t r a ry manner with the only requirement of compat ib i l i ty w i t h preceding R i extensions. Now consider a l l possible extensions of R u t o an additional element. For each p o s s i b i l i t y , use an element x C u . This i s poss ib le ; indeed a l l the n-tuples t o which a value (+) o r ( - ) has already been assigned, have a l l their terms belonging e i t h e r t o a Di o r t o the corresponding C i ( i < u) . Yet C u i s d i s j o i n t from these Ci , and the union of these remain G) many elements x i n C u , such t h a t no value has been assigned t o any n-tuple having x among i t s terms. F ina l ly we obtain on the e n t i r e base E , an n-ary r e l a t ion R such t h a t each countable r e s t r i c t i o n of R i s a r b i t r a r i l y ex tendib le t o i t s base plus an addi t io- nal element: thus every n-ary r e l a t ion w i t h c a rd ina l i t y W1 i s embeddable in i t .

D i ( i < W1) . To each D i , we

by i t s e l f W 1

Ro t o a new element, there b e i n g a t most continuum many,

of Co , and cons t ruc t the corresponding extension on the base Do u { x] . This Co , hence a l s o W 1 many such

Do .

Ci , we have already ensured a l l possible extensions

and t h e i r

i n

D i i s countable. So t h a t there

EXERCISE 2 - FINITIST STRUCTURE AND FRAENKEL-MOSTOWSKI MODEL, IN CONNECTION WITH A FUNCTION WHOSE DOMAIN IS STRICTLY SUBPOTENT WITH ITS RANGE

1 - S t a r t w i t h the set of natural numbers, which we consider as urelements, i . e . as copies of the empty s e t , each having no element. On the set N o f these in te - gers , consider the s e t of f i n i t i s t r e l a t i o n s , i n the sense of 5 8 above. More genera l ly , t o each ordinal o( , assoc ia te the set of f i n i t i s t s t ruc tu res with rank 4 on N , defined as follows by induction. Each element o f t i , or urelement, i s a s t ruc tu re w i t h rank 0 . A s t ruc tu re of rank o( >/ 1 i s a s e t of f i n i t i s t s t ruc tu res of ranks s t r i c t l y l e s s than o( , such t h a t t he re e x i s t s a f i n i t e subse t F of N , w i t h the condition t h a t every


permutation of N-F preserves the s e t or structure of rank o( . The reader can verify t h a t these f i n i t i s t s;ructures constitute a mcdel of FRAENKEL-MOSTOWSKI's se t theory: see FRAiSSE 1958. More precisely th i s model sa t i s f ies the axioms of ZF excepting extensionality, which must be weakened as follows: any two non-empty sets which have the same elements are identical. One can easily transform the model so as t o sa t i s fy the fu l l extensionality axiom, b u t while abandonning the foundation axiom. Indeed i t suffices t o consider each urelement a as equal t o the singleton of a i t s e l f . In th i s case, we add the empty se t t o the elements of N , and we give t o each of them the rank 0 , the rank 1 being given only t o those sets which contain several elements of N o r one element of N plus the empty s e t .

2 - Construct the s e t o f words, or f i n i t e sequences without repetition on the s e t N of the urelements; note that th i s se t i s f i n i t i s t with F empty. Consider the function f which t o each non-empty word u without repetition associates the word obtained from structure , with F empty. Dom f i s the s e t of non-empty words, Rng f i s the s e t of a l l words (without repe t i t ion) , including the empty sequence. We shall prove that in the considered model, Dom f i s s t r i c t l y subpotent with Rng f 0 Indeed a bijection from Rng f o n t o Dom f cannot be f i n i t i s t . For such a bijection h , l e t t ing 0 denote the empty sequence, consider the a-sequence of successive i te ra tes 0 , h ( O ) , h ( 0 ) = h ( h ( O ) ) , ... . If th i s w-sequence were f i n i t i s t , with a f i n i t e subset F of N , then a l l i t s terms would be sequences of elements of F . Since h i s a bijection, these terms are a l l d i s t inc t , so t h a t F should be inf in i te : contradiction. 0 Problem. Is i t possible t o generalize t o structures the lemmas in ch.9 0 1 .2 and 1.3, so t h a t we could define a f i n i t i s t structure by using only transpositions instead of general permutations of N-F . Consequently the intersection of two f i n i t e subsets F should be an F , so that each f i n i t i s t structure should have a minimum F called i t s kernel.

u by removing i t s l a s t term. This function i s a f i n i t i s t

(example communicated by H O D G E S ) .

2

313

CHAPTER 11

HOMOGENEOUS RELATION, RELATIONAL SYSTEM, CONNECTION WITH PERMUTATION GROUPS, ORBIT

§ 1 - HOMOGENEOUS RELATION; AMALGAMABLE SET AND AMALGAMABLE AGE

L e t p be a n a t u r a l number ; a r e l a t i o n R i s s a i d t o be o-homogeneous, if every l o c a l automorphism o f R d e f i n e d on p e lements i s e x t e n d i b l e t o an auto-

morphism o f R . Every r e l a t i o n i s 0-homogeneous, s i n c e t h e empty f u n c t i o n i s e x t e n d i b l e t o t h e i d e n t i t y mapping on t h e e n t i r e base. Obvious d e f i n i t i o n o f a ( S p)-homogeneous r e l a t i o n . A r e l a t i o n R i s s a i d t o be homogeneous, i f R i s p-homogeneous f o r eve ry natu-

Example. A b i n a r y c y c l e on a t l e a s t 5 elements i s 1-homogeneous b u t n o t 2-homo-

geneous: c a l l a,c i n t o a,d i s a l o c a l automorphism and i s n o t e x t e n d i b l e . The c h a i n on 2 elements i s n o t 1-homogeneous, b u t i t i s t h e o n l y l o c a l automorphism on 2 elements i s t h e i d e n t i t y t r a n s f o r m a t i o n .

The f o l l o w i n g b i n a r y m u l t i r e l a t i o n ( w i t h b i n a r y and unary component r e l a t i o n s ) & 2-homogeneous y e t n o t 3-homogeneous. 0 S t a r t w i t h t h e b i n a r y r e l a t i o n

a,b,c,d,r,s,t,u,i ,j . I n o r d e r t o ensure t h e 2-homogeneity: f o r each ordered p a i r , say (a,s) f o r i n s t a n c e , add a b i n a r y r e l a t i o n A such t h a t A(x,y) = + e x a c t l y f o r those o rde red p a i r s (x,y) which a r e an image o f (a,s) under one o f t h e t h r e e f o l l o w i n g pe rmuta t i ons : ( 1 ) t h e mapDing which preserves a and c

and in te rchanges (b,d), ( r ,u ) , ( s , t ) , (i,j) ; ( 2 ) t h e mapping which preserves b and d and in te rchanges (a,c), ( r , s ) , ( t , u ) , (i,j) ; ( 3 ) t h e compos i t i on o f

mappings (1) and ( 2 ) , which p rese rves i and j and in te rchanges (a,c), (b,d),

(r,t), ( s , ~ ) . Here t h e images o f (a,s) a r e (a , t ) , ( c , r ) , (c,u) and o b v i o u s l y

(a,s) The o b t a i n e d m u l t i r e l a t i o n i s n o t 3-homogeneous: as a l ready ment ioned i n t h e exer-

c i s e , t h e l o c a l automorphism which p rese rves a and b and in te rchanges ( i , j ) i s n o t e x t e n d i b l e t o an automorphism on t h e e n t i r e base. 0

The p reced ing example i s n o t 1-homogeneous: f o r examule map a i n t o b ; y e t i t becomes

(+) f o r a and c , and ano the r t a k i n g (+) f o r i and j .

r a l number p .

a,b,c,d f o u r consecu t i ve elements; t hen t h e mapping which takes

2-homogeneous, s i n c e

R g i v e n i n ch.9 exerc.1, on t h e 10 elements

i t s e l f , t h e i d e n t i t y pe rmuta t i on b e i n g added t o (1 ) , (2 ) , (3 ) .

1-homogeneous i f we add, f o r i ns tance , a unary r e l a t i o n t a k i n g va lue


Example of homogeneous r e l a t ions . The r e l a t ion always (+); another example, the chain Q of the r a t iona l s . Another homogeneous r e l a t ion . The equivalence r e l a t ion w i t h f i n i t e l y many, o r w i t h denumerably many c lasses having ca rd ina l i t y 2 . In con t r a s t with the two preceding examples, here homogeneity does not s u b s i s t when we remove an element from the base. Problem. For each a r i t y n , does there e x i s t a threshold s (n) above which every n-ary r e l a t ion which i s ( 4 s(n))-homogeneous, i s homogeneous.

1.1. Let E be a denumerable s e t . A r e l a t ion R with base E i s homogeneous i f f , f o r any f i n i t e subset F 0-f E , f o r any loca l automorphism f of R w i t h domain F , and f o r any element u iJ E-F , there e x i s t s a local automorphism of R which extends f t o the domain F U {u 1 . 0 I f R i s homogeneous, then i t obviously s a t i s f i e s our condition. Conversely, suppose t h a t the condition holds. Enumerate the elements of E as a i ( i in teger ) . S t a r t w i t h an a r b i t r a r y local automorphism f o of R , with f in i - te domain. Add a. t o the domain, i f i t does not y e t belong t o this domain, thus obtaining a local automorphism f i extending f o . Simi lar ly add a. t o the range of fb , i f i t does not y e t belong t o th i s range, thus obtaining a local automorphism f l extending f; , t h u s extending fo . I t e r a t e t h i s , going from f i t o f i + l ( i i n t ege r ) t o the domain and then t o the range. F ina l ly the common extension of these f i i s an automorphism of R . 0 This proposit ion does not extend t o the case of any uncountable r e l a t i o n . 0 Consider the chain Q+R , where Q i s the chain of r a t iona l s and R i s isomorphic t o the chain of r e a l s . Then the condition i n our preceding statement i s sa- t i s f i e d ; y e t mapping an element of the i n i t i a l in te rva l Q i n t o an element of the f ina l in te rva l R , we cannot extend t h i s loca l automorphism t o the e n t i r e base. 0

by adding a i

1 . 2 . Note t h a t the r ich denumerable r e l a t ion R defined in ch.10 5 4.1 s a t i s f i e s the following condition. Each f i n i t e r e s t r i c t i o n of R i s a r b i t r a r i l y extendible t o i t s base plus an additional element, which we can f ind i n the base of R . We immediately see t h a t R s a t i s f i e s the condition i n our preceding statement. Hence f o r each in teger n , there e x i s t s a denumerable n-ary r e l a t ion which i s r ich and homogeneous.

1.3.(1) Any two denumerable homogeneous r e l a t ions , with same a r i t y and same age, a r e isomorphic. ( 2 ) Let R be a denumerable homogeneous r e l a t ion . Then every denumerable r e l a t ion which is younger than R , i s embeddable i n R . 0 (1) Let R and R ' be denumerable homogeneous r e l a t ions w i t h respec t ive bases

Chapter 1 1 315

E and E ' , and the same age. Enumerate the elements of E as a i , the elements of E ' as a; ( i in teger) . Embed the restr ic t ion R/{ao) into R ' , thus obtaining a local isomorphism f o from R into R ' with domain { ao\ . Augment the range of fo by adding ah . By the preceding proposition, fo i s extendible t o a local isomorphism go from R in to R ' , whose domain contains a. and whose range contains whose domain contains the elements ao , ..., ai and which i s extendible t o a local isomorphism gi

ah . I terat ing th i s for each i , we obtain a local isomorohism f i

whose range contains the elements a h , . . . ,a! . The union of the onto R ' . 0 f i (or equivalently the gi ) i s an isomorphism from R

0 ( 2 ) Same argument, b u t only in one direction, using the the younger relation into the homogeneous relation R . 0

Consequently, for the age of chains, the chain Q of the denumerable homogeneous chain, u p t o isomorphism.

ocal isomorphisms from

ationals i s the only

1.4. If we remove an element from the base of a denumerable relation which i s rich and homogeneous, then we obtain an isomorphic relation. Similarly i f we change the value of an arbi t rary n - t u p l e (n = a r i t y ) . 0 Indeed, the f i r s t and second operations bo th preserve age, since th i s age i s constituted by a l l f i n i t e n-ary relations ( u p t o isomorphism). Moreover the condition t h a t each f i n i t e res t r ic t ion be a rb i t ra r i ly extendible, i s preserved. 0

Many other relations are isomorphic t o the i r res t r ic t ion a f t e r removing an arbitrary element. For example the chain Q (which i s homogeneous), the chain W (which i s not homogeneous). We have already seen that the equivalence relation with classes of cardinality 2 , i s homogeneous b u t n o t isomorphic with such a res t r ic t ion.

Problem. Let R be a denumerable n-ary relat ion, which i s isomorphic with any relation obtained by changing the value of an arbi t rary n-tuple; then i s R an homogeneous relat ion. Note that necessarily every n-ary f i n i t e relation i s embeddable in R : i f R i s homogeneous, then i t i s rich.

1 .5 . AMALGAMABLE SET, AMALGAMABLE AGE A s e t 6;L of f i n i t e relations of the same ar i ty i s said t o be amalgamable i f , for any relations A , B, C belonging to'R, for any isomorphism f from A onto a res t r ic t ion of B and any isomorphism g from A onto a res t r ic t ion of C , there ex is t s a relation D belonging t o 61. , an isomorphism f ' from B onto a res t r ic t ion of D and an isomorphism g ' from C onto a res t r ic t ion of D , such t h a t fo r each element x of the base I A I , we have ( f ' , f ) x = ( g ' o g ) x . Obviously the relations in the considered s e t are defined up t o isomorphism. A s e t (R of f i n i t e relations i s said t o be strongly amalgamable i f , fo r any relations B, C belonging t o 6!, and C ' isomorphic w i t h C , with a common


restr ic t ion t o the intersection of the bases l B l and I C ' l , then there exis ts a relation D in &, ( u p t o isomorphism), which i s a common extension of B and C ' . For example, the age of a l l f i n i t e chains, and also the age of a l l f i n i t e par t ia l orderings, are strongly amalgamable: see ch.2 fj 2 . 2 and 2.3. The age of those f i n i t e unary relations which take the value (+) for a t most one element, i s amalgamable yet n o t strongly amalgamable. The age of a l l f i n i t e trees i s not amalgamable: the example given in ch.2 fj 2.3 contradicts ordinary, as well as strong amalgamability.

1.6. AMALGAMATION THEOREM Given an age 61 , there ex is t s a countable homogeneous representative of & iff

d?, i s amalgamable.

0 Let R be a countable homogeneous relation. Let A , B be two f i n i t e res t r ic - tions of R , where B i s an extension of A , and l e t g be an isomorphism from A into a third f i n i t e res t r ic t ion C of R . Then by hypothesis g i s extendible t o an automorphism of R . Let f ' denote the rest r ic t ion of th i s automorphism t o the base I B I . Let f be the identity on I A l , l e t g ' be the identity on I C l and f inal ly l e t 0 be the restr ic t ion R / ( [ C I y ( f ' ) ' ( I B l ) ) . We thus have the condition of amalgamation. Conversely, l e t @, be an amalgamable age. Let B be a f i n i t e relation belonging t o & , and f a local automorphism of B . Let A denote the rest r ic t ion of B t o Dom f . We shall construct an extension D of B , s t i l l belonging t o bt , which has a local automorphism extending f t o the domain B . For t h i s , l e t C = B , and denote by g the identity on I A I = Dom f . Now using the amalgamabi- l i t y condition, obtain an extension 0 of B with an isomorphism g ' from B onto a res t r ic t ion of D , such t h a t for every x in Dom f , we have g ' x =

( g ' , g ) x = ( f ' , f ) x = fx : so that g ' extends f . Star t with an w -sequence of f i n i t e relations Ai ( i in teger) , the s e t of whose restr ic t ions gives the amalgamable age &. Let Bo = A. . Using the preceding, replace Bo by i t s extension B1 belonging t o & , such t h a t every local automorphism of Bo automorphism of B1 and whose domain i s Bo . Moreover we require that A1 be embeddable in B1 , which i s possible since every age i s directed. Iterating t h i s , we obtain an 0 -sequence of elements Bi of 61. , such that each B i + l extends B i , and Ai i s embeddable in Bi for each integer i . Take the common extension R of the Bi t o the union of bases. Then every local automorphism of R with f i n i t e domain i s extendible, with alternatively a domain and a range containing every element in every base proved. 0

(there are only f in i te ly many) has an extension which i s a local

I S i ( : the homogeneity i s

Chapter 11 317

1.7. Given an amalqamable aqe, t h e c r i t e r i o n o f ch.10 5 7.6 i s s a t i s f i e d , and unique denumerable homogeneou? r e p r e s e n t a t i v e o f ou r a q t (up t o isomorphism) i s u e s a t u r a t e d r e l a t i o n o f ch.10

0 L e t R be a denumerable homogeneous r e l a t i o n , S a r e l a t i o n r e o r e s e n t i n g t h e same age, and f a l o c a l isomorphism f rom R i n t o S , w i th f i n i t e domain F . Take an a r b i t r a r y f i n i t e superse t G o f f a ( F ) . There e x i s t s an isomorphism h f rom S/G o n t o a r e s t r i c t i o n o f R . Since R i s homogeneous, t h e l o c a l automorphism h,f i s e x t e n d i b l e t o an automorphism k o f R . Then t h e composi t ion

k-',,h i s an isomorphism f rom S/G o n t o a r e s t r i c t i o n o f R , and extends f - l . Rep lac ing G b y any f i n i t e suoerse t ( i n c l u d e d i n t h e base IS1 ) , t h e same argument shows t h a t k - l 0 h i s a 1-morphism f rom S i n t o R . Hence R i s s a t u r a t e d

(see t h e d e f i n i t i o n s i n ch.10 5 7 and 7.3). 0

5 7.3 and 7.6.

1.8. I t i s proved by HENSON 1972, t h a t t h e r e e x i s t continuum many m u t u a l l y non-

isomorphic denumerable homogeneous r e l a t i o n s . Hence continuum many amalgamable

ages. F o r o t h e r r e s u l t s on homogeneous r e l a t i o n s , see JONSSON 1965 and CALAIS 1967.

2 - RELATIONAL SYSTEM: ORBIT: ADHERENCE OF A PERMUTATION GROUP: THEOREM OF THE INCREASING NUMBER OF ORBITS (LIVINGSTONE, WAGNER)

2.1. RELATIONAL SYSTEM Given a s e t E , we say t h a t a r e l a t i o n a l system w i t h base E , o r based on E , i s any o r d i n a l sequence whose terms a r e r e l a t i o n s E . L e t t i n g ni denote t h e a r i t y o f Ri , t h e o r d i n a l sequence o f i n t e g e r s ni i s s a i d t o be t h e arity o f t h e system, and each i s s a i d t o be a component o f t h e

system. By t a k i n g a f i n i t e sequence o f r e l a t i o n s based on E , we f i n d t h e m u l t i r e l a t i o n

based on E (see ch.2 § 1). The n o t i o n s o f r e s t r i c t i o n o f a system t o a subset o f t h e base, e x t e n s i o n t o a superse t of t h e base, isomorphism, automorphism, l o c a l isomorphism o r automorphism,

a l l ex tend immediate ly t o t h e case o f r e l a t i o n a l systems.

However, we have an i m p o r t a n t d i f f e r e n c e which p r o h i b i t s c e r t a i n g e n e r a l i z a t i o n s . Indeed t h e number o f r e l a t i o n a l systems o f a g i v e n a r i t y and o f a g i ven f i n i t e base, i s i n genera l i n f i n i t e . I n p a r t i c u l a r RAMSEY's theorem, used when p a r t i t i o - n i n g t h e p-element subsets o f t h e base i n t o a f i n i t e number o f c lasses, co r res -

ponding t o d i f f e r e n t isomorphism types, can no l o n g e r be used s y s t e m a t i c a l l y . The same remark ho lds f o r t h e coherence lemma.

Ri ( i o r d i n a l ) based on

Ri

p-HOMOGENEOUS SYSTEM, HOMOGENEOUS SYSTEM

The n o t i o n o f homogeneity i n t r o d u c e d i n 5 1 above, can be extended t o r e l a t i o n a l systems. A system R i s s a i d t o be p-homogeneous, i f eve ry l o c a l automorphism


o f R , d e f i n e d on p elements, i s e x t e n d i b l e t o an automorphism o f R . A system i s s a i d t o be homogeneous, i f i t i s p-homogeneous f o r e v e r y i n t e g e r p .

2.2. ORBIT OF AN n-TUPLE, OR OF AN n-ELEMENT SET

L e t E be a se t , and H a s e t o f pe rmuta t i ons o f E , n o t n e c e s s a r i l y a group. Given an i n t e g e r n and an n - t u p l e o f elements al, ..., an i n E , we c a l l t h e

o r b i t o f t h e n - t u p l e (mod H) t h e s e t o f n - t u p l e images o f al, ... ,an under any pe rmuta t i on be long ing t o H . Given an n-element s e t F , we c a l l t h e o r b i t o f F (mod H) t h e s e t o f n-element

s e t images o f F under any pe rmuta t i on be long ing t o H . I f H i s a group, then t h e e x i s t e n c e o f a permuta t i on be long ing t o H , which takes

one n - t u p l e i n t o another , i s an equ iva lence r e l a t i o n between n - tup les . S i m i l a r l y f o r n-element s e t s .

n-TRANSITIVE GROUP, n-SET-TRANSITIVE GROUP

A group o f pe rmuta t i ons o f E i s s a i d t o be n - t r a n s i t i v e , i f f o r any two n - t u p l e s

t h e r e e x i s t s a pe rmuta t i on be long ing o the r . I n o t h e r words, i f t h e r e e x i s t s o n l y one o r b i t o f n - tup les .

Fo r example, t h e symmetric group formed o f a l l pe rmuta t i ons o f E . Another example, i f t h e base E has f i n i t e c a r d i n a l i t y a t l e a s t equal t o n+2 , then t h e a l t e r n a t i n g group formed o f a l l even pe rmuta t i ons , i s n - t r a n s i t i v e .

An n - t r a n s i t i v e group i s m - t r a n s i t i v e f o r any m 6 n . Reca l l JORDAN'S hypo thes i s , 1893: f o r n 3 6 , every n - t r a n s i t i v e group i s

symmetric o r a l t e r n a t i n g . A f f i r m a t i v e s o l u t i o n ; see f o r example CAMERON 1981 p. 9.

A group o f pe rmuta t i ons on E i s s a i d t o be n - s e t - t r a n s i t i v e , i f any two n - e l e - ment s e t s a r e t rans fo rmab le one i n t o t h e o t h e r by a c e r t a i n pe rmuta t i on o f t h e group. I n o t h e r words, i f t h e r e e x i s t s o n l y one o r b i t o f n-element s e t s .

t o t h e group and t a k i n g one n - t u p l e i n t o t h e

2.3. ADHERENT PERMUTATION, GROUP CLOSED UNDER ADHERENCE L e t E be a s e t , n an i n t e g e r and H a s e t o f pe rmuta t i ons o f E . A permuta- t i o n f o f E i s s a i d t o be n-adherent t o H , i f f o r any e lements al,. . . ,a i n E , t h e r e e x i s t s a pe rmuta t i on o f H t a k i n g al i n t o fal , ... , and an i n t o fan . Every pe rmuta t i on which i s n-adherent t o H , i s m-adherent f o r any m g n . A s e t H o f pe rmuta t i ons o f E i s s a i d t o be c l o s e d under n-adherence, i f e v e r y pe rmuta t i on o f E which i s n-adherent t o H , belongs t o H . I f H i s c l o s e d

under n-adherence, t hen H i s c l o s e d under m-adherence f o r each m>, n . A pe rmuta t i on i s s a i d t o be adherent t o H , i f i t i s n-adherent f o r e v e r y n . A s e t H o f pe rmuta t i ons i s s a i d t o be c l o s e d under adherence, i f eve ry adherent pe rmuta t i on belongs t o H .

n

Chapter 11 319

2.4. (1) L e t R be a r e l a t i o n a l system w i t h base E , formed o f components a x a-4 n . Then t h e group o f automorphisms o f R i s c l o s e d under n-adherence.

( 2 ) Fo r eve ry r e l a t i o n a l system, t h e group of automorphisms i s c losed under adhe-

rence.. Consequences o f ch.9 5 1.7.

2.5.(1) L e t G be a group o f pe rmuta t i ons o f E , which i s c losed under n - m -

- rence. Then t h e r e e x i s t s a components, whose group o f automorphisms i s G (uses axiom o f cho ice when E i s uncountable) .

( 2 ) L e t G be a group o f pe rmuta t i ons o f E , which i s c losed under adherence.

Then t h e r e e x i s t s a homogeneous r e l a t i o n a l system whose group o f automorphisms i s G (same remark) . ( 3 ) Assume t h a t E has f i n i t e c a r d i n a l i t y h . Then f o r e v e r y group G o f permuta- t i o n s o f E , t h e r e e x i s t s a homogeneous m u l t i r e l a t i o n of maximum a r i t y h , whose group o f automorphisms i s G . 0 (1) To each n - t u p l e o f elements o f E , a s s o c i a t e t h e o r b i t , i . e . t h e c l a s s o f

n - t u p l e s which can be o b t a i n e d f rom i t , by t a k i n g i t s image under any pe rmuta t i on i n G . Then t o each o r b i t , a s s o c i a t e t h e n -a ry r e l a t i o n based on E , which takes

t h e va lue (+) f o r those n - t u p l e s i n t h e o r b i t , and ( - ) o the rw ise . Us ing t h e axiom

o f choice, t a k e an o r d i n a l sequence R of t hese r e l a t i o n s , which fo rm a system. Every pe rmuta t i on be long ing t o G i s an automorphism o f R . Conversely, eve ry automorphism g o f R t akes each n - t u p l e i n E , i n t o an n - t u p l e be long ing t o

t h e same o r b i t . Thus g be longs t o G , s i n c e G i s c l o s e d under n-adherence. It remains t o see t h a t R i s (6 n)-homogeneous. Given a l o c a l automorphism f o f R , d e f i n e d on a domain of c a r d i n a l i t y 6 n , t a k e an a r b i t r a r y n - t u p l e c o n t a i - n i n g a l l t h e elements al, ..., an o f F , w i t h p o s s i b l e r e p e t i t i o n s . T h i s n - t u p l e

and i t s image belong t o t h e same o r b i t of G . Thus t h e r e e x i s t s a

pe rmuta t i on o f E which extends f and belongs t o G , hence which i s an automorphism o f R . 0

( 2 ) Analogous p r o o f , b u t where n takes a l l i n t e g e r va lues, G be ing c losed

under adherence, i n s t e a d o f n-adherence. 0

0 ( 3 ) P a r t i c u l a r case, where G i s c l o s e d under h-adherence, and w i t h f i n i t e l y

many o r b i t s . 0

By t h e p reced ing p r o p o s i t i o n s , t o eve ry r e l a t i o n a l system R , t h e r e corresponds

a homogeneous system S on t h e same base, hav ing t h e same automorphisms, and

w i t h o u t augmenting t h e maximum a r i t y o f t h e components. Mote however t h a t , s t a r t i n g

f rom a s imp le r e l a t i o n R , we can end up a t a homogeneous r e l a t i o n a l system S

w i t h i n f i n i t e l y many components. For example, s t a r t w i th R = W ( c h a i n o f n a t u r a l

numbers), whose o n l y automorphism i s t h e i d e n t i t y . Then we end up w i t h t h e

(6 n)-homogeneous r e l a t i o n a l system, formed o f n - u

-

fal, ..., fan


sequence o f s ing le ton unary r e l a t i o n s o f a l l the in tegers .

2.6. Here are two examples o f a group o f permutations which i s closed under (n+ l ) -

adherence bu t no t under n-adherence (n na tura l number ); POUZET 1979, unpublished.

0 Le t E be a se t o f c a r d i n a l i t y n+2 , and f an odd permutation o f E . Le t G

be the group o f even permutations generated by the permutations f,t , where t i s

an a r b i t r a r y t ranspos i t i on interchanging two elements o f E . Le t g be a permuta- t i o n (n+l)-adherent t o G . Then g belongs t o G : indeed, there e x i s t s a t rans- p o s i t i o n t such t h a t g and f,t are i d e n t i c a l on n+ l elements o f E , and

hence are s t i l l i d e n t i c a l on the (n+2)nd and l a s t element o f E . Since f i s odd, i t does not belong t o G : it suffices t o see that f is n-adhe- r e n t t o G . Indeed, f o r any x1 ,..., xn belonging t o E , l e t y, z be two d i s -

t i n c t elements which are a lso d i s t i n c t from the x . Le t t be the t ranspos i t i on (y,z) . Then (f,t)xl = fxl, . . . , (f,t)xn = fxn , which proves the n-adherence

o f f . n This example extends t o the case where E i s i n f i n i t e , by tak ing n+2 elements i n E and repeat ing the preceding cons t ruc t ion .

Another example. Le t E be the se t o f a l l (n+ l ) - tup les o f ra t i ona ls , which we sha l l c a l l the r a t i o n a l vector space o f dimension n+ l (n i n tege r >/ 2). Le t G be the group o f l i n e a r permutations o f E w i t h p o s i t i v e determinants. The group G i s no t c losed under n-adherence. Indeed, l e t f be a l i n e a r permu- t a t i o n o f E w i t h negat ive determinant, hence f does no t belong t o G . Take n a r b i t r a r y po in ts al,. . . ,an i n E , then compose f w i t h the symmetry w i t h

respect t o a hyperplane passing through al, ..., an . We obta in a l i n e a r permutation w i t h p o s i t i v e determinant, hence belonging t o G , which takes al i n t o fal , ... and an i n t o fan . On the o ther hand, G i s c losed under (n+l)-adherence. Indeed, l e t g be a per-

mutat ion (n+l)-adherent t o G . Take n+ l po in ts xl, ... , x ~ + ~ belonging t o E , which are l i n e a r l y independent. The images gxl,. . . ,g~,+~ are by hypothesis the images o f the x under a l i n e a r permutation w i t h p o s i t i v e determinant; hence they are l i n e a r l y independent.

Le t u be any p o i n t i n E . Le t u ~ , . . . , u ~ + ~ be the coordinates o f u r e l a t i v e t o the po in ts x , so t h a t we have u .= u .x + ... + ~ ~ + ~ . x ~ + ~ . Now def ine

v = u . x + ... + un.xn and w = ~ ~ + ~ . x ~ + ~ , so t h a t u = v + w . The po in ts 1 1 xl, ..., xn,v are l i n e a r l y dependent. Since t h e i r images under g are by hypothe- s i s of (n+l)-adherence i d e n t i c a l t o t h e i r images under a c e r t a i n l i n e a r permu-

t a t i o n belonging t o G , the dependence r e l a t i o n i s preserved: + un.(gxn) . S i m i l a r l y w and x ~ + ~ are l i n e a r l y dependent, so t h a t we have

QW = ~ ~ + l . ( g x , + ~ ) ; s i m i l a r l y gu = gv + gw = ul.(gxl) + ... + ~,.,+~.(gx,,+I) .

1 1

gv = ul.(gxl) + ...

Chapter 11 321

L e t t i n g u vary, we see t h a t g i s a l i n e a r permutation which takes each xi i n t o gxi ( i = l , . , . ,n+l) . Hence g has p o s i t i v e determinant, so t h a t g belongs t o G . [7

In the preceding proo f , the hypothesis n >/ 2 i s used t o go from u = v + w t o gu = gv + gw ; for t h i s , the 3-adherence o f g i s required. On the o the r hand, take n = 1 thus n + l = 2 , so t h a t E i s the se t o f the r a t i o - na l po in ts , o r vectors i n the plane, and G i s the group o f l i n e a r permutations o f E w i t h p o s i t i v e determinants. Then the permutation g which takes each vector (0,v) i n t o (0,2v) and which preserves each (u,v) when u # 0 ( w i t h u, v r a t i o n a l s ) , i s 2-adherent t o G ; y e t t h i s g does no t belong t o G . Example, due t o the same author, o f a group o f permutations c losed under adherence, y e t no t under n-adherence f o r any n . 0 For each in tege r n , take a se t En w i t h c a r d i n a l i t y n , where the En are mutual ly d i s j o i n t , and l e t E be t h e i r union. Take G t o be the group o f those permutations o f E whose r e s t r i c t i o n t o En , f o r each n , i s an even permutation o f En . Then a permutation o f E which i s adherent t o G i s necessar i l y an 'e le - ment o f G . However, g iven an a r b i t r a r y i n tege r n , a permutation f i s n-adhe- r e n t t o G provided t h a t i t s r e s t r i c t i o n t o Ei i s an even permutation o f Ei f o r each i & n t l , and an a r b i t r a r y permutation o f Ei f o r each i >/ n t2 . 0

2.7.(1) L e t p be an in teger , and R be a p-monomorphic, p-homogeneous r e l a t i o - na l system. Then the group o f automorphisms o f R 2 p-se t - t rans i t i ve . Indeed, fo r any two p-element subsets a, b o f t he base, there e x i s t s an isomorphism from R/a onto R/b , which i s ex tend ib le t o an automorphism o f R . (2) L e t E be a set , and G be a p -se t - t rans i t i ve group o f permutations o f E . Then the closure G' - o f G under adherence i s again p -se t - t rans i t i ve , and conversely. Moreover, there e x i s t s a homogeneous r e l a t i o n a l system whose automorphism group i s G+ , and such a r e l a t i o n a l system i s always p-monomorphic. Th is fo l lows from 2.5 above; i f E i s uncountable, then t h i s uses the axiom o f choice.

-

2.8. THEOREM ON THE INCREASING NUMBER OF ORBITS ( 1 ) L e t p, q be in tegers , E a f i n i t e se t w i t h c a r d i n a l i t y a t l e a s t equal t c 2p + q , and G a group o f permutations o f E . Then the number o f o r b i t s (mod G)

of the (p+q)-element subsets o f E , i s grea ter than or equal t o the number o f o r b i t s o f the p-element subsets (LIVINGSTONE, WAGNER 1965).

0 Associate t o the group G a homogeneous m u l t i r e l a t i o n R whose automorphism group i s G : see 2 . 5 . ( 3 ) above. Then two p-element se ts a, b belong t o the same o r b i t (mod G) i f f the r e s t r i c t i o n s R/a and R/b are isomorphic: indeed by homogeneity, every isomorphism o f one r e s t r i c t i o n onto another i s ex tend ib le t o an

-


automorphism of R . Same r e s u l t f o r the follows from the f a c t t h a t t he number of isomorphism types of r e s t r i c t i o n s t o elements t r i c t i o n s t o p elements: see ch.9 5 7.1. 0

( 2 ) Let E be a denumerable s e t , and G be a group of permutations of E . To each in teger p assoc ia te the countable number of o r b i t s of p-element subsets of E . Then t h i s number increases w i t h p (POUZET 1976).

0 Consider the c losure G+ of G under adherence, and note t h a t f o r each p , t he o r b i t s of the p-element s e t s (mod G ' ) . Take a r e l a t iona l system R whose automorphism group i s G+ :

see 2 .5 . (2 ) above. The proof terminates as before, using the p r o f i l e increase theorem, ch.9 5 7 . 1 . However we must note t h a t th is theorem extends t o the case of a r e l a t iona l system with denumerably many components. Indeed, the multicolor theorem i n ch.3 5 5.3

ly many isomorphism types f o r ce r t a in values of p . 0 (3 ) In p a r t i c u l a r , l e t E have countable ca rd ina l i t y a t l e a s t equal t o 2 p + q , and l e t G be a group of permutations of E . I f G i s (p+q) - se t - t r ans i t i ve , then G i s p - se t - t r ans i t i ve . 0 Indeed p - se t - t r ans i t i v i ty means t h a t a l l the p-element subsets of E belong t o the same o r b i t . Another proof i s obtained from ch.9 5 6.3 , i n view of the inequal i ty : o r equal t o Min(p+q,(Card E)-p-q) ; by using a l s o 2 .7 . (2) above. 0

(p+q)-element s e t s . T h u s our proposit ion

p+q i s g rea t e r than o r equal t o the number of isomorphism types of the res-

p-element s e t s (mod G ) a r e the same as the o r b i t s of the

includes the case of i n f i n i t e l y many co lo r s , hence here of i n f i n i t e -

p l e s s than

2.9. Let p , q be two in t ege r s , and E be a s e t of ca rd ina l i t y g rea t e r than o r equal t o 2p + q ; l e t G be a group of permutations of E . Then every oermuta- t i on of E which preserves the o r b i t s o f (p+q)-element s e t s (mod G ) , a l so preserves the o r b i t s of p-element s e t s .

0 Let G+ be the c losure of G under adherence, and l e t R be a homogeneous r e l a t iona l system whose automorphism group i s G+ : see 2 .5 . (2) above. Let f be a permutation of E which takes each (p+q)-element s e t i n t o another i n t he same o r b i t (mod G ) , hence again i n the same o r b i t (mod G ' ) . Then f takes each r e s t r i c - t i on of R t o p+q elements i n t o an isomorphic r e s t r i c t i o n . Hence f t akes each r e s t r i c t i o n of R t o p elements i n t o an isomorphic r e s t r i c t i o n : see ch.9 5 7.2. Since R i s homogeneous, f o r any p-element s e t a , there e x i s t s an automorphism of R , hence an element of G+ , hence an element of G , which takes R/a i n t o the isomorphic r e s t r i c t i o n R/f"(a) . 0

2.10. Let E be a s e t of ca rd ina l i t y g rea t e r than o r equal t o 2p + q , and l e t G , H be two groups of permutations of E . I f every o r b i t of (p+q)-element sets

Chapter 11 323

(mod G) (p+q)-element s e t s (mod H ) , then eve ry o r b i t

- o f p-element s e t s (mod G) i s i n c l u d e d i n an o r b i t o f p-element s e t s (mod H) . T h i s r e s u l t was c o n j e c t u r e d by BERCOV, HOBBY 1970, and a weaker v e r s i o n was proved by them; t h e p resen t r e s u l t i s due t o POUZET 1976.

0 L e t a, b be two p-element s e t s be long ing t o t h e same o r b i t (mod G) . There

e x i s t s a pe rmuta t i on g be long ing t o G , such t h a t g" (a) = b . T h i s permuta- t i o n g p rese rves a l l t h e o r b i t s (mod G) , and i n p a r t i c u l a r t h e o r b i t s o f t h e

(p+q)-element s e t s (mod G) , hence by hypo thes i s g preserves t h e o r b i t s o f t he (p+q)-element s e t s (mod H) . By t h e p reced ing p r o p o s i t i o n , g preserves t h e

o r b i t s o f t h e p-element s e t s (mod H) . Hence a and b belong t o t h e same o r b i t (mod H ) . 0

i s i n c l u d e d i n an o r b i t o f

§ 3 - CHAINS COMPATIBLE MODULO A PERMUTATION GROUP; GROUP GENERATED BY CHAINS; DILATED GROUP, CONTRACTED GROUP

COMPATIBILITY MODULO A GROUP

L e t m be a n a t u r a l number . A group G o f pe rmuta t i ons on t h e s e t o f i n t e g e r s

1 ,..., m Two cha ins A, 6 G-compatible, o r compa t ib le modulo G , i f f o r

eve ry s e t o f m elements i n t h e i n t e r s e c t i o n o f t h e bases, say al < a2 < ... < am (mod A) , t h e pe rmuta t i on u which r e o r d e r s these elements acco rd ing t o

au(l) c au(2) < . . . < au(,,,) (mod B) , belongs t o t h e group G . We see t h a t G - c o m p a t i b i l i t y i s r e f l e x i v e and symmetric. I f t h e chains A, B have t h e same base, then G - c o m p a t i b i l i t y i s t r a n s i t i v e , hence i s an equiva lence r e l a - t i o n .

I n t h e genera l case, no te t h a t two cha ins a re G-comoatible when t h e i n t e r s e c t i o n o f t h e bases has c a r d i n a l i t y s t r i c t l y l e s s than m ( a r i t y o f G ) . Hence t h e r e i s no t r a n s i t i v i t y , s i n c e f o r example i f A and A ' have t h e same base and B

has base d i s j o i n t f rom t h e base o f A , then A and B on t h e one hand, and A ' and B on t h e o t h e r hand, a r e G-compatible f o r e v e r y G , which i s o b v i o u s l y

n o t t h e case f o r A and A ' , assumed t o be d i s t i n c t . I f m = 2 and G reduces t o t h e i d e n t i t y on t h e s e t { 1 , 2 ) , then G-compati-

b i l i t y means t h a t t h e r e s t r i c t i o n s o f b o t h cha ins t o t h e i n t e r s e c t i o n o f t h e i r bases i s t h e same. Then we f i n d t h e n o t i o n o f c o m p a t i b i l i t y i n t h e sense o f ch.2 5 1.2; and t h e r e e x i s t s a common e x t e n s i o n which i s a c h a i n based on t h e un ion o f t h e two bases.

I f m i s an a r b i t r a r y i n t e g e r , and G i s t h e m-ary symmetric group, o r group o f a l l pe rmuta t i ons o f

Two cha ins a r e G-compatible i f f t h e i r r e s t r i c t i o n s t o t h e i n t e r s e c t i o n o f t h e bases a r e G-compatible.

i s s a i d t o be an m-ary group, o r a group w i t h arity m . a r e s a i d t o be

1,. . . ,m 1 , then a l l cha ins a r e G-compatible.

Let A, B be two chains with the same base E , and l e t G be an m-ary group; l e t n be such t h a t m 6 n < Card E . Then A and B are G-compatible i f f , for each n-element subset X of E , the rest r ic t ions A/X and B/X are G-compatible.

3.1. Let '7%' be a s e t of chains with the same f i n i t e base Then the following three conditions are equivalent. (1) Given three chains A, B , C - in A , the image of C under the permutation of E which takes A into B , belongs t o & . ( 2 ) There exis ts a images under the permutations belonging to G . ( 3 ) There exis ts an integer qd p and a q-ary g roup H such that 4 i s formed of a chain and a l l chains on E which are H-compatible with i t . 0 (1) and (2) are obviously equivalent; (2) i s a particular case of ( 3 ) ; f ina l ly ( 3 ) easi ly implies (1). 0

E , and l e t p = Card E .

- p-ary group G such t h a t 4 i s formed of a chain and i t s

3.2. DILATED GROUP Let G be an m-ary g roup , and E be an m'-element s e t , with m ' 3 in . Let A

be a chain based on E ; consider the se t of a l l chains on E which are G-compatible with A . By the preceding proposition, these chains are the images of A under a certain m'-ary group G ' : t h i s G ' i s called the m'-ary dilated group of G and denoted G m ' . Let G be an m-ary g roup , H a subgroup of G , and l e t m ' be an integer 3 m . Then the dilated group H m ' i s a subgroup of the dilated group G m ' . Given three integers m < m',< m" and an m-ary group G , then ( G m ' ) m " = Gm" .

3.3. Two d i s t inc t qrouDs of the same ar i tv can have the same dilated qrow of a larger a r i ty .

0 Let m = 4 , and l e t G be the group formed of the identity on 1 , . . . ,4 . Let G ' be formed o f the identity plus the transposition ( 2 , 4 ) . Then G I 5 = G = the group formed o f the identity on 1, ..., 5 . Indeed, s t a r t with the chain 1 2 3 4 5 . A chain which i s GI-compatible with i t , orders the integers 1,2,4,5 according t o 1 2 4 5 or according t o 1 5 4 2 . Similarly, we have one of the chains 1 3 4 5 or 1 5 4 3 , and one of the chains 2 3 4 5 or 2 5 4 3 . The chain 1 5 4 2 requires t h a t 5 be before 4 , hence implies 1 5 4 3 and 2 5 4 3 . Thus 5 must be b o t h before and a f t e r 2 , a contradiction. Thus we have 1 2 4 5 , which implies 1 3 4 5 and 2 3 4 5 , 0

5

3.4. Let G , H be two groups of the same ar i ty m , and l e t n 2 m . Then ( G ~ I H ) ~ = Gn n H n and ( G u H ) ~ 2 Gn w H n ; th i s inclusion i s n o t always an i denti ty . - -

Chapter 11 325

0 The group (GnH)n i s included i n G n and in H n , by 3.2 above. On the o ther hand, i f two chains a r e simultaneously then they a re (GnH)-compatible. The group ( G u H ) ~ includes G n and H n , by 3 .2 . F ina l ly l e t G be the group of cyc l i c permutations on 1,2,3 and H be the group generated by the t ranspos i t ion (1 ,3) . Then G u H i s the symmetric group on 1,2,3 ; hence ( G u H ) 4 i s the symmetric group on 1,2,3,4 . On the o ther hand,

G 4 i s t he group generated by the cycle ted by (1 ,4 ) (2 ,3 ) . So t h a t t he group generated by t h e i r union i s the dihedral group on 1,2,3,4 the r e f l ec t ion ( o r inversion

G-compatible and H-compatible,

(1 ,2 ,3 ,4) ; and H 4 i s the group genera-

formed of 8 permutations generated by the considered cycle and

( 1 , 4 ) ( 2 , 3 ) ) . 0

3.5 . Let G , H be two m-ary groups. If. Hm+' i s s t r i c t l y included i n Gml , then ( G n H ) # G and H i s not necessar i ly included i n G . 0 Suppose t h a t G A H = G . Then To have H non-included i n G , take G t o be the group generated by the cycle (1,2,3,4) , and take H t o be the group on 1,2,3,4 formed of the iden t i ty plus the t ranspos i t ion (2 ,4) . Then H 5 reduces t o the iden t i ty on 1,2,3,4,5 ( see 3.3 above), and G 5 i s generated by the cycle (1,2,3,4,5) . 0

- ,-

G i s included i n H , so Gm+' included in Hmtl.

3 .6. If G has a symmetric d i l a t e d group, then G i s i t s e l f synnnetric.

0 Let m be the a r i t y of G , and l e t E be a s e t w i t h c a rd ina l i t y n am . By hypothesis Gn i s symmetric, hence a l l chains on E a r e mutually G-compatible. Taking t h e i r r e s t r i c t i o n s t o a m-element subset D of E , we obtain a l l chains based on D : hence G i s symmetric. 0

3.7 . G R O U P GENERATED BY A SET OF CHAINS; CONTRACTED GROUP Let & be a s e t of chains w i t h t he same base E , and l e t m be an in teger l e s s than o r equal t o Card E . The l e a s t m-ary group G such t h a t the chains i n 4 a re a l l group generated, i n t he usual sense , by a l l permutations ned by taking any two chains i n & and then any m elements i n E , say a l < a2 < .. . < a m modulo one of the chains, and the o the r chain. I f A i s just a s e t of two cha ins , then we c a l l i t a bichain, and be the S t a r t w i t h an n-ary group G . Take a chain A of ca rd ina l i t y n ; then take A t o be the set formed of A and a l l i t s images under the permutations belon-

ging t o

G-compatible i s ca l l ed the group generated by d . This i s a l s o the u of { 1, . . . ,m] obtai-

a u ( l ) < ... < a,,(,,,) modulo

G i s sa id t o w a r y group generated by this bichain.

G . Then f o r m 4 n , the m-ary aroup generated by & i s sa id t o be

the contracted group o f G t o a r i t y m , and denoted by G m . I f H i s a subgroup of G , then the contracted group Hm i s a subgroup of G m . S t a r t w i t h an m-ary group G , take n >/ m and the d i l a t ed group G n , and then

the contracted group d i s t i n c t from G . 0 S t a r t w i t h G of a r i t y 4 , formed of the i d e n t i t y and the t ranspos i t ion (2 ,4 ) . Then G5 reduces t o the iden t i ty ( see 3.3 above), hence ( G ) 4 as wel l . 0

Two d i s t i n c t groups can have the same contracted group. 0 Indeed, there e x i s t s only one poss ib le unary contracted group, and two poss ib le binary ( o r 2-ary) contracted groups: t he group formed of the i d e n t i t y , and the symmetric group formed of the i d e n t i t y and the t ranspos i t ion r a l l y , the proposit ion follows from the increase i n the number of m-ary groups of a r i t y m . 0

S t a r t w i t h an n-ary group G , take mg n and the contracted group Gm , and

then the d i l a t ed group i s an extension of G , and can be

a DroDer extension.

(G')), . Then (G')), i s a subgroup of G , which can be

5

(1 ,Z ) . More gene-

( G m ) n . Then (G,Jn

0 S t a r t w i t h t he 5-ary group G which preserves 4 and 5 and c i r c u l a r l y permu- t e s 1 ,2 ,3 . Then G4 preserves 4 and symmetrically permutes 1,2,3 . Then

( G 4 ) 5 preserves 4 and 5 and symmetrically permutes 1,2,3 . 0

Let G be an m-ary group, and m"g m ' 6 rn . Then the twice contracted aroup ( G m , ) m , , i s an extension of G,,, . Problem. Are these two groups always iden t i ca l . More s t rongly , given a s e t & of chains a l l with the same base, and another s e t 6.3 of chains w i t h the same base, i f c+ and @ generate the same w a r y group,

then do they generate the same m'-ary group f o r m ' < m .

3.8. Let G , H be two m-ary groups and n < m . T& (GuH), = Gn u H n @ (GnH), c_ Gn n H n ; th i s inclusion can be proper.

0 By the preceding, the f i r s t qroup includes G n and H n , hence includes their union. S imi la r ly the t h i r d group i s included i n the in t e r sec t ion . Moreover, i f a chain A i s taken i n t o a chain B by a permutation belonging t o G u H , then there e x i s t s a f i n i t e sequence of chains going from A t o B , and such t h a t the passage from each one t o the next i s e f f ec t ed by a permutation i n G o r i n H . Hence the r e s t r i c t i o n of A t o a given n-element s e t 0 , i s taken i n t o the r e s t r i c t i o n of B t o 0 , by a f i n i t e sequence o f permutations i n G n and i n equal t o i t .

H n . I t follows t h a t the f i r s t group i s included i n the second, hence i s

Chapter 1 1 321

To see t h a t t h e i n c l u s i o n o f t h e t h i r d group i n t h e f o u r t h

G t o be t h e group o f c y c l i c pe rmuta t i ons on { 1,2,3} and take H t o be formed

o f t h e i d e n t i t y and t h e t r a n s p o s i t i o n i n t e r s e c t i o n G n H , and so ( G n H ) 2 reduces t o t h e i d e n t i t y . Yet G2 and H p , hence t h e i r i n t e r s e c t i o n , a re i d e n t i c a l t o t h e symmetric group on {1,2} . 0

can be p roper , t ake

(1,2) on t h e same s e t { 1,2,3} . Then t h e

3.9. L e t G be an m-ary group, and l e t n + m . The d i l a t e d group Gn i s genera-

- t e d ( i n t h e usual sense, by compos i t i on ) f rom t h e un ion o f those

whose c o n t r a c t e d m-ary group i s i n c l u d e d i n G . Problem. Is t h e d i l a t e d group Gn t h e i n t e r s e c t i o n o f t h e n-ary groups whose

m-ary c o n t r a c t e d group i n c l u d e s G . L e t G be an w a r y group and l e t n < m . The c o n t r a c t e d group Gn i s n o t neces-

s a r i l y generated by t h e un ion o f those n-ary groups, whose m-ary d i l a t e d group i s i n c l u d e d i n G . 0 L e t G c o n s i s t o f t h e i d e n t i t y on {l ,..., 5 ) . Thus G4 c o n s i s t s o f t h e iden-

t i t y . Yet t h e un ion o f those 4 -a ry groups whose d i l a t e d 5-ary group reduces t o

t h e i d e n t i t y con ta ins t h e t r a n s p o s i t i o n (2,4) : see 3.3 above. 0

Problem. L e t t i n g n < m , i s t h e c o n t r a c t e d group Gn t h e i n t e r s e c t i o n o f those

groups, whose d i l a t e d m-ary group i n c l u d e s G .

n-ary groups

3.10. There e x i s t a 5-ary group G and two cha ins on t h e i n t e g e r s 1, ..., 4

which a r e G4-compatible, y e t have no G-compatible ex tens ions . 0 L e t A be t h e c h a i n 1 2 3 4 and B t h e c h a i n 3 2 1 4 . L e t G be t h e

5-ary group which p rese rves 4 and 5 , and c i r c u l a r l y permutes 1,2,3 . The con- t r a c t e d group G4 s y m m e t r i c a l l y permutes 1,2,3 hence takes A i n t o B . L e t A ' , B ' be two ex tens ions of A , B t o t h e i n t e g e r s 1, ..., 5 . For A ' and

B ' t o be G-compatible, i t i s necessary t h a t 5 be a f t e r 1,2,3 (mod A ' and

mod B ' ) ; b u t then we do n o t have G - c o m p a t i b i l i t y . 0

§ 4 - I N D I C A T I V E GROUP, REDUCTION THEOREM (FRASNAY)

A group o f pe rmuta t i ons i s s a i d t o .be i n d i c a t i v e i f f i t i s generated by a

b i c h a i n w i t h i n f i n i t e base (wh ich we can assume t o be denumerable).

+ 4.1. Fo r each i n t e g e r m , t h e r e e x i s t s an m m such t h a t eve ry b i c h a i n on a base w i t h c a r d i n a l a t l e a s t equal t o mt genera tes an m-ary i n d i c a t i v e group.

0 Suppose t h e c o n t r a r y , and cons ide r an w-sequence o f b i c h a i n s Bi ( i i n t e g e r )

w i t h f i n i t e bases Ei where Card Ei i s s t r i c t l y i n c r e a s i n g w i t h i , such t h a t

each B i generates the same non-indicative group G . There are only f in i te ly many, say h permutations in G . So for each integer i , there exis ts a subset Fi of Ei with cardinality p 4 h .m , such that the rest r ic t ion B i / F i generates G . We can suppose t h a t the Fi = 41 ,. . . , p ) . Extract from the sequence of the rest r ic t ion of each B i t o the integers 1, . . . , p . Then keep def ini t ively Bo and extract from the sequence of the a second inf in i te sequence giving the same restr ic t ion of each bichain t o the integers

Bo the integers 1, . . . , p + 2. e tc . In the sequence definitively obtained, replace each Bi by i t s res t r ic t ion B i t o the integers generates G , then each B(i generates G as well. Now the common extension of the B; t o the s e t of a l l integers i s an inf in i te bichain which generates G ; hence G i s indicative: contradiction. 0

E i are f i n i t e se t s of integers, and t h a t

B i

B i ( i 31)

a f i r s t in f in i te sequence giving the same

l,..,,p+l . Then keep definitively and B1 , and i t e ra te th i s t o obtain the same restr ic t ion of each bichain to

1 ,..., pti . Since each B i , and i t s restr ic t ion t o 4 1 ,..., p ) ,

4 . 2 . DESCRIPTION OF INDICATIVE GROUPS; INDICATOR Given an inf in i te bichain, for each integer m i t generates an m-ary indicative group G, . The function G which t o each m associates Gm i s said t o be the indicator of the bichain (called "fiche" in FRASNAY's terminology). Let us describe as follows a l l the indicators, hence a l l indicative groups. The indicator S gives by def ini t ion, for each m , the m-ary symmetric group denoted by Sm . Take for instance the chain of n a t u r a l numbers which for each integer m realizes a l l possible permutations of m consecutive integers. The indicator I gives for each m , the group I,,, consisting only of the identity on m elements. Take a bichain formed of two identical in f in i te chains. The indicator J gives for each m , the group J, formed of the identity and of the reflection The indicator T gives for each m , the group Tm of t ranslat ions, generated ( in the usual sense) by the cyclic permutation (1,2,3, . . . . , m ) . Take for instance two inf in i te chains A , B with d is jo in t bases, then the two sums At6 and B+A . The indicator D gives the dihedral group Dm , which i s generated (usual sense) by the union of Tm and J, . Take for instance two inf in i te chains A, B with dis joint bases, then the sums A+B and A- + B- (sum of converses). Given two integers p , q , the indicator I p y q gives for each m , the group formed of those permutations which preserve the i n i t i a l interval 1, . . . , p and the final interval m-q+l, ..., m and which reduce t o the identity on the integers p + l , p+2, . .. , m-q , assuming that m &p+q . In the contrary case where m < p+q , then ,Iyq denotes the wary symmetric group. Take for instance the chain d 2 ... p followed by an i n f i n i t e chain A followed by the chain 1' 2' ... q ' ;

, and another chain

(l,m)(Z,m-l)(3,m-2). .. : take an inf in i te chain and i t s converse.

I:'q

Chapter 11 329

and on the o ther hand, the chain 2 3 . .. p 1 fo l lowed by A fo l lowed by 2 ’ 3 ’ . . .. q ’ 1’ . Note t h a t 1’” i s i d e n t i c a l t o I . Given an i n tege r r , the i n d i c a t o r Jr g ives the group Jk which i s generated by the union o f Iiyr and Jm . Then the i n i t i a l i n t e r v a l

i n t o the f i n a l i n t e r v a l m - r + l , ..., m and conversely; moreover the median i n t e r - val r+l, ..., m- r i s reversed, assuming t h a t m 3 2 r . I n the opposite case where

m < 2r , then JL denotes the m-ary symmetric group. Take f o r instance the chain

1 2 ... r fol lowed by an i n f i n i t e chain A fo l lowed by 1’ 2 ’ ... r ’ ; and on the o ther hand 2 ’ 3 ’ ... r ’ 1’ fol lowed by the converse A- fo l lowed by 2 3 .. .. r 1 . Note t h a t J1 i s i d e n t i c a l w i t h J .

1, ..., r i s taken

4.3. Le t A, B be two chains w i t h common base o f c a r d i n a l i t y l a r g e r than o r equal t o m . A B are D,-compatible, then e i t h e r A and 6 , or A and the converse B- xe Tm-compatible. Le t m, r be two in tegers , w i t h 2 r 6 m . If. A - and B (w i th common base) are Jk-compatible, then e i t h e r A and B , op A and B- are Ik’r-compatible.

4.4. The d i l a t e d groups o f T3 are the Tm (ma3) . The d i l a t e d groups o f D4 are the. Dm ( m >/ 4) . I f p tq 6 m , then the d i l a t e d groups o f ILyq are the I:yq f o r n >/ m

I f m 2,3 and 2 r s m , then the d i l a t e d groups o f Jk are the J: f o r n + m . Consequently, two chains whose i n t e r s e c t i o n o f bases has c a r d i n a l i t y m 3 3 , are

T,-compatible i f f they are T3-compatible.

- For m 3 4 , they are D,-compatible i f f they are D4-compatible.

For m 3 2 and p+q 6 m , i f the i n te rsec t i on o f bases has c a r d i n a l i t y n + m , then the chains are I:’q-compatible i f f they are I~ ’qycompa t ib le .

For ma3 and 2 r Q m , i f the i n te rsec t i on o f bases has c a r d i n a l i t y n a m ,

then the chains are JL-compatible i f f they are J:-compatible.

I n general, the d i l a t e d o r the contracted group o f one o f the prev ious ly descr i - bed i n d i c a t i v e groups i s simply obtained by keeping the i n d i c a t o r I , J, T, D, S

and by modify ing the a r i t y m , keeping the ind ices p, q o r r , where i t i s understood tha t , i f p+q > m o r i f 2 r > m , then the group o r the group

J; i s i d e n t i c a l w i t h the symmetric group S, . 4.5. It i s proved by FRASNAY 1965, t h a t the on ly i n d i c a t i v e groups are those o f the f i v e preceding fam i l i es l i t t l e more complicated than the proo f concerning the

I , J, T, 0, S . For a sketch o f the proo f , which i s Q- ind ica t i ve groups i n 5 5


below, see propositions 5 .1 through 5.8 and exercise 2 below. By examining each case described in 4 . 2 above, we see that the group generated by the union of any two indjsativegroups i s indicative. Since there are f in i te ly many m-ary groups for each integer m , we have the following statements.

A group i s indicative ~ ~ i f f i t i s generated by an arbi t rary se t ~ of chaiFsmcn the

same denumerable base. If G i s an m-ary indicative group and n 3 m , then G n i s indicative and (G')), = G . Given an integer m , there ex is t s an m m such that every s e t of chains on a base of cardinality a t l eas t equal t o m+ generates an indicative m-ary qroup (generalization of 4 . 1 ) . Given an m-ary group G , the dilated group Gn i s indicative for n >, m' .

t

4.6. R E D U C T I O N THEOREM (FRASNAY 1965) Given an m-ary g roup G , there ex is t s a maximumnndicative m-ary group H ( G ) included in G . Moreover for n 3 m we have H ( G n ) = ( H ( G ) ) n . Taking n m such that G n i s indicative, then we have H ( G ) n = Gn and H ( G ) = (G')), .

§ 5 - Q-BICHAIN, Q-INDICATIVE GROUP; FIVE Q-INDICATIVE GROUP THEOREM; SET-TRANSITIVE GROUP THEOREM (CAMERON)

Q-BICHAIN, Q-INDICATIVE GROUP We shall cal l a Q-bichain, any birelation both o f whose components are chains each isomorphic with Q . We say t h a t a group i s Q-indicative, i f i t i s generated by a Q-bichain. We see that the f ive following groups are Q-indicative, for each ar i ty m :

the identity I,,, ; the group Jm ( ident i ty and ref lect ion); the group Tm of translations; the dihedral group Dm generated by the union of Jm and Tm ; and f inal ly the symmetric group Sm . We call these temporarily the canonical groups. The group generated by the union of two canonical groups i s canonical. 0 Indeed, the only case where two of these groups are non-inclusive, i s the case of Jm and Tm , whose union generates Dm . 0

Consequently, for each group G , there ex is t s a maximum canonical qroup included in G .

-

- THE FIVE Q-INDICATIVE G R O U P THEOREM The five canonical groups are the only Q-indicative groups. To prove t h i s , we shall show that i f G i s Q-indicative, then i t i s equal t o the maximum canonical group included in G . This i s obvious for the case of the symmetric group S, ; so we shall consider the four cases the following propositions 5 .1 t o 5.8.

I,, Tm, Jm, Dm in

Chapter 11 33 1

5.1. Consider a Q-bichain with components A , B . We say that a pair of elements x , y of the base, i s preserved or inverted, accordin9 to whether x and y are in the same order modulo A and modulo B , or in the opposite order. Let AB be a Q-bichain having a t l eas t one preserved pair and one inverted pair. Then e i ther the group m , or Tm i s included in the group, for every m . 0 Let u and v denote two elements such tha t u < v (mod A ) and v < u (mod B ) . Since A i s isomorphic with Q , there ex is t inf ini te ly many elements x between u and v (mod A ) . For each such x , we have v < x or x < u (mod B ) , hence there ex is t inf ini te ly many x sat isfying, for example, x < u (mod B ) . Using RAMSEY's theorem (ch.3 g l . l ) , ei ther there e x i s t in f in i te ly many of these form mutually inverted pairs , in which case the aroup m-ary group generated by x forming preserved pairs. Then since u i s less t h a n (mod A) and greater t h a n (mod B ) these elements, the group Tm of translations i s included in the group generated by AB , for each m . 0

-- Jm i s included in the group generated b-y AB , for every

x which

Jm i s included in the AB , for each m . Or there ex is t inf ini te ly many of these

5 . 2 . Let G be the m-ary g roup generated by a given Q-bichain. If the maximum canonical group included in G I,,, , t& G i s identical t o I,,, . 0 Either there only ex is t preserved pairs in the given bichain, in which case G = I m there ex is t s a t l eas t one pair of each kind, in which case Jm or Tm i s included in G . Only the f i r s t case i s possible under our assumption that I,,, i s the maximum canonical group included in G . 0

. Or there only exis t inverted pairs , in which case G = Jm . Or f inal ly

5.3. Let AB be a Q-bichain w i t h base E . Suppose that there exis ts an inf ini te ~- subset U Lf E a l l of whose pairs are preserved, hence A/U = B / U . Let m be a positive integer; suppose tha t the group Tm i s included in the m-ary group generated by the bichain. Then ei ther G = Tm or G = Sm , the symnetri c g roup .

Suppose t h a t mg 2 , since the case where m = 1 i s obvious. The se t U i s different from E , since Tm i s included in G . For each element x of E-U , l e t xA denote the cut defined on A/U by the i n i t i a l interval of those elements of U less than x (mod A ) , and the complementary f inal interval . Let xB denote the cut analogously defined with B . F i r s t suppose that for each x in E-U , we have xA = xB . By hypothesis, there exis ts an inverted pa i r , say { x,y} which i s thus included in implies that xA = xB = yA = yB . I f , before t h i s cut , we have inf ini te ly many elements of U , then G contains the transposition ( m - 1 , m ) , which together with T, generates S, . Similarly, i f there ex is t inf ini te ly many elements

E-U , which then


greater than t h i s cut , then G contains the t ranspos i t i on (1,2) , which together w i t h Tm generates Sm . From t h i s po in t on, we are i n the case where there e x i s t s an x i n E-U w i t h xA # xB . Suppose t h a t a t l e a s t one o f these cuts i s ne i the r the i n i t i a l c u t l y i n a before A/U , nor the f i n a l cu t . Then there e x i s t a t l e a s t two i n t e r v a l s bounded on U by xA and xB , and a t l e a s t one o f these i n t e r v a l s i s i n f i n i t e . I f i t i s the i n t e r v a l between xA and xB , and i f the o ther non-empty i n t e r v a l l i e s before it, then G contains the permutation which preserves 1 and which i s

def ined by the cyc le (2,3 ,..., m) . By composition w i t h (1,Z ,..., m) , we generate Sm . S i m i l a r l y i f the i n t e r v a l between xA and xB i s i n f i n i t e and the o ther

non-empty i n t e r v a l l i e s a f t e r i t . I f the i n t e r v a l between xA and xB i s f i n i t e

and the i n t e r v a l l y i n g before i s i n f i n i t e , then G contains the t ranspos i t i on (m-1,m) which, together w i t h the t r a n s l a t i o n (1,2, ..., m) , generates Sm . Same r e s u l t i n the case o f an i n f i n i t e i n t e r v a l l y i n g a f t e r . Suppose now tha t , i f xA and xB are d i s t i n c t , then they are extremal ( i . e . one o f them i s the i n i t i a l cut , and the o ther i s the f i n a l c u t ) . We can requ i re t h a t

any two elements x, y which give the same non-extremal c u t xA = xB = yA = yB , form a preserved p a i r . Indeed otherwise, we ob ta in again the t ranspos i t i on (1,2) o r (m-1,m) . Thus augment U by a l l these x corresponding t o non-extremal

cuts. From t h i s p o i n t on, every x i n E-U y i e l d s two extremal cuts.

E i the r there e x i s t x, y i n E-U w i t h xA and yB i n i t i a l c u t and xB and yA f i n a l cu t . Then G contains the t ranspos i t i on (1,m) and hence again i s i d e n t i c a l w i t h Sm . O r we are i n the case where, f o r instance, xA i s i n i t i a l and xB f i n a l f o r each

x i n E-U . Then e i t h e r there e x i s t s an i nve r ted p a i r among these; i n which case G contains the permutation which transforms 1 2 3 ... m-1 m i n t o 3 4 ... m 2 1 and t h i s together the t r a n s l a t i o n generates E-U are preserved, i n which case A i s the sum A/ (E -U) + A/U and B i s the sum A/U + A / (E -U) , hence G = Tm . 0

Sm . O r f i n a l l y a l l the p a i r s i n

5.4. Le t G be generated by a Q-bichain. I f the maximum canonical group inc luded - i n G 2 T m , = G = T m .

Because o f RAMSEY's theorem, there e x i s t s an i n f i n i t e subset U o f E , a l l o f whose p a i r s are preserved, o r an i n f i n i t e subset U , a l l o f whose p a i r s are inver - ted . I n the second case, the group J m i s included i n G , and since by hypothesis

Tm i s included i n G as we l l , we have the dihedral group Dm included i n G , con t rad i c t i ng the assumption t h a t i s the maximum canonica.1 group included i n

G . Thus by the preceding statement, G = Tm o r G = S, , t h i s l a s t case contra-

d i c t i n g our assumptions. 0

Tm

Chapter 1 1 333

5.5. L e t AB be a Q-b i cha in w i t h base E . Suppose t h a t t h e r e e x i s t s an i n f i n i t e s-t U of E , a l l o f whose p a i r s

a l l o f whose p a i r s a r e i n v e r t e d . Then f o r each AB i s t h e symmetric group Sm .

m , the m-ary ~ group generated by

0 Note f i r s t t h a t U and V have a t most one e lement i n common. Moreover, t h e

e x i s t e n c e o f V shows t h a t t h e group Jm o f t h e r e f l e c t i o n , i s generated. Using 5.3 above, i t s u f f i c e s t o prove t h a t t h e group Tm o f t r a n s l a t i o n s i s generated, s i n c e then t h e e n t i r e generated group cannot be reduced t o t h e s i n g l e group

and i s t hen i d e n t i c a l w i t h

F i r s t suppose t h a t f o r each i n t e g e r h , t h e r e e x i s t s an x i n V f o r which t h e

c u t s xA and xB d e f i n e d by x on t h e c h a i n A/U = B/U , a r e separated by a t l e a s t h e lements o f U . I n t h i s case, t h e t r a n s l a t i o n ( l , Z , ..., in) i s obta ined, hence o u r p r o p o s i t i o n ho lds .

Suppose now t h a t t h e r e e x i s t s an i n t e g e r h such t h a t f o r each x i n V , we have a t most h e lements i n U between t h e c u t s xA and xB . Take an w-sequence o f elements i n V , which i s f o r example dec reas ing (mod A ) , hence decreas ing

(mod B ) . Then f rom some p o i n t on, t h e c u t s xB become i d e n t i c a l . Thus t h e r e e x i s t i n f i n i t e l y many elements i n U which a r e

e i t h e r a l l g r e a t e r t han these c u t s , o r a l l l e s s than these c u t s . T h i s y i e l d s f o r

i ns tance , f o r each m and each p d m , t h e pe rmuta t i on which p rese rves 12, ... ..,p and which i n te rchanges (p+l,m), (p+Z,m-l), e t c . ; t h i s s u f f i c e s t o generate t h e symmetric group

Tm

Sm .

xA become i d e n t i c a l , as w e l l as t h e

Sm .

5.6. L e t AB be a ( I -b ichain hav ing a t l e a s t one p rese rved p a i r and one inve r ted , - p a i r . Then f o r each m , t h e group Tm i s i n c l u d e d i n t h e group generated by AB . - 0 Consider two e lements u < v (mod A) w i t h v u (mod B ) . I n t h i s case, by RAMSEY's theorem, e x t r a c t an i n f i n i -

t e subset o f t hese e lements, a l l o f whose p a i r s a r e preserved, o r a l l o f whose p a i r s a r e i n v e r t e d . I n t h e case where t h e p a i r s a r e preserved, by o u r cho ice o f t h e element v , we o b t a i n t h e group T, . I n t h e case where t h e p a i r s a re i n v e r - t ed , t h e group Jm i s obta ined; by t h i s r e f l e c t i o n and by o u r cho ice o f u , we o b t a i n aga in Tm . O r t h e r e e x i s t i n f i n i t e l y many e lements < v (mod B ) , and s t i l l between u and v (mod A) . Then t h e p reced ing argument s t i l l works. O r f i n a l l y t h e r e e x i s t i n f i n i t e l y many elements between u and v (mod A) and between v and u (mod B) . Then i n t h e case o f p rese rved p a i r s , we o b t a i n t h e

group prove t h a t i f Tm i s n o t a l r e a d y obta ined, t hen we o b t a i n a l s o an i n f i n i t e s e t o f

p rese rved p a i r s : hence by t h e p reced ing s tatement , we o b t a i n t h e symmetric group S,.

Tm . There remains t h e case o f an i n f i n i t e s e t o f i n v e r t e d p a i r s . We s h a l l

To t h i s end, cons ide r two new e lements s t i l l denoted by u and v , such t h a t

u < v (mod A and mod B ) . Take aga in i n f i n i t e l y many elements between u and v (mod A) . The same argument as b e f o r e proves t h a t , i f t h e r e e x i s t i n f i n i t e l y many o f these elements which a r e l e s s than u (mod B ) , o r i n f i n i t e l y many which a re

g r e a t e r t han v (mod B) , then we again o b t a i n t h e group Tm . F i n a l l y suppose t h a t t h e r e e x i s t i n f i n i t e l y many o f these elements between u and v , modulo A and modulo B as w e l l . I n t h e case o f an i n f i n i t e s e t o f i n v e r t e d

p a i r s , we o b t a i n t h e group o f t h e r e f l e c t i o n . Tak ing t h e images, under t h i s r e f l e c t i o n , o f u, v and o u r i n f i n i t e i n v e r t e d s e t , we aga in o b t a i n Tm . There

remains t h e case o f i n f i n i t e l y many p rese rved p a i r s . Together w i t h t h e p r e v i o u s l y

ob ta ined i n f i n i t e s e t o f i n v e r t e d p a i r s , t h e y g i v e t h e symmetric group

view o f 5 .5 above. 0

Jm

S, , i n

5.7. L e t G be an m-ary group generated by a Q-b i cha in . I f t h e maximum canon ica l

group i n c l u d e d i n G i_i Jm ( r e f l e c t i o n ) , m. G = Jm . 0 By hypo thes i s , t h e r e e x i s t s an i n v e r t e d p a i r . I f t h e r e e x i s t s as w e l l a p rese rved

p a i r , t hen by t h e p reced ing 5.6, t h e group

Jm i s a l s o i n c l u d e d i n G , hence t h e d i h e d r a l group Dm i s i n c l u d e d i n G ; t h e

maximum canon ica l group i n c l u d e d i n G i s a t l e a s t Dm : c o n t r a d i c t i o n which proves

t h a t a l l t h e p a i r s a re i n v e r t e d , so t h a t

Tm i s i n c l u d e d i n G . By hypo thes i s

G = Jm . 0

5.8. L e t G be an w a r y group generated by a Q-b i cha in . I f t h e maximum canon ica l group i n c l u d e d i n G 12 Dm ( d i h e d r a l group) , then G = Dm . 0 L e t AB be t h e cons ide red b i c h a i n , and E be i t s base. By RAMSEY's theorem, t h e r e e x i s t s an i n f i n i t e subse t U o f E , a l l o f whose p a i r s a r e preserved, o r a l l o f whose p a i r s a r e i n v e r t e d . I f they a r e preserved, t hen s i n c e Tm i s i n c l u d e d

i n G , we have t h a t G = Tm o r G = S, by 5.3, c o n t r a d i c t i n g o u r assumption t h a t

Dm i s t h e maximum canon ica l group i n c l u d e d i n G . Hence we have an i n f i n i t e s e t U w i t h i n v e r t e d p a i r s . L e t B- denote t h e converse

c h a i n o f B , and l e t G ' be t h e m-ary group generated by AB- . We s h a l l prove t h a t G ' i s i n c l u d e d i n G . The group G ' i s generated by some pe rmuta t i ons s , each o b t a i n e d by t a k i n g m e lements which we denote by 1,2,.. .,m i n t h e o r d e r i n g A and s ( l ) , s ( 2 ) , ..., s(m) i n t h e o r d e r i n g B- . For such an s , t h e compos i t i on

rmos o f t h e r e f l e c t i o n rm o f t h e i n t e g e r s l , Z , ..., m w i t h s , i s an e lement o f G . Since rm belongs t c G , then s belongs t o G as w e l l .

By t h e preceding, t h e b i c h a i n AB- has p rese rved p a i r s . Then e i t h e r a l l i t s p a i r s

a re preserved, and so A = B- , hence G = Jm , c o n t r a d i c t i n g o u r assumptions. O r AB- has a l s o an i n v e r t e d p a i r , and t h e n Tm i s i n c l u d e d i n G ' by 5.6. F u r t h e r -

more G ' # Sm , s i n c e by hypo thes i s Dm i s t h e maximum canon ica l group i n c l u d e d i n G . Hence by 5.3 we have G ' = T, . Then G , which i s generated by t h e

Chapter 1 1 335

compositions rmos where rm i s the m-ary r e f l ec t ion and s belongs t o G ' , s a t i s f i e s the inc lus ion

Now the f i v e Q- indica t ive group theorem i s proved.

G 5 Dm , and t h u s G = Dm . 0

5.9. Given a chain A , we have introduced the cyc l i c r e l a t ion associated w i t h A , in ch.2 8.6. Here we addi t iona l ly need the notion of dihedral r e l a t ion of A , which i s defined as the quaternary r e l a t ion taking value (+) f o r ( x , y , z , t ) i f x < y < z < t (mod A) , and f o r a l l 4-tuoles obtained from the preceding by any

permutation belonging t o the dihedral group D4 ( t r ans l a t ions and r e f l ec t ions ) . Note t h a t the dihedral r e l a t ion of A expresses t h a t x and z a re s i t ua t ed in the two opposite i n t e rva l s defined by y and t on the c y c l i c r e l a t ion associated w i t h A . T h u s we can say t h a t the dihedral r e l a t ion i s a betweenness, o r an intermediacy, on the cyc l i c r e l a t ion .

Let R be a r e l a t iona l system which i s f r e e l y in t e rp re t ab le i n the chain Q of the r a t iona l s . Then the group of automorphisms of group, o r the group of increas ing b i j ec t ions , i . e . of automorphisms of Q , or the group of increas ing b i j ec t ions and decreasing b i j ec t ions , i .e. automorphisms of the betweenness r e l a t i o n , o r the group of automorphisms of the cyc l i c r e l a t ion associated with Q , or f i n a l l y the group of b i j ec t ions which preserve o r which inverse the cyc l i c r e l a t i o n , i . e . automorphisms of the dihedral r e l a t ion of Q (POUZET 1979).

0 Consider a pos i t i ve in t ege r m and a s t r i c t l y increasing m-sequence x l < x 2 < ... < xm (mod Q ) . Transform t h i s by an automorphism f of R , and l e t s denote the m-ary permutation such t h a t f ( x s ( l ) ) < f ( x s ( 2 ) ) < ... < f(xs(,, ,)) (mod Q ) . Let Gm be the m-ary group generated by these s , when f va r i e s . This group does not depend on the choice of the increasing sequence x l , ..., xm . Moreover, we can suppose t h a t the images fx a re the x themselves, permuted by s . Thus

Gm Notice t h a t i t can be a proper subgroup: f o r ins tance , take the r e l a t ion R such t h a t R(x,y,z) = + i f f x < z and y < z and x # y . To each automorphism f of R , assoc ia te the bichain (Q,f"(Q)) and notice t h a t t he m-ary group generated by this bichain i s included in Gm . Indeed, the inverse s-l i s nothing e l s e but the reordering of the x l , ... ,xm by fo(Q) . Moreover, as f var i e s , the entire group G, i s generated, s ince each s i s obtained from an automorphism f . From the f i v e Q- indica t ive group theorem, i t then follows t h a t the sequence of the groups G, , as m va r i e s , i s one of the sequences, o r ind ica tors I , J , T , 0 , S . Each of these f i v e cases gives one o f the f i v e conclusions of our statement.

R i s e i t h e r the symmetric

i s a subgroup of the group of automorphisms of the r e s t r i c t i o n R/{xl, ... ,xm).


E . Then G i s n-set - under adherence i s one o f

- - 5.10. SET-TRANSITIVE GROUP THEOREM

L e t G be a group o f pe rmuta t i ons o f a denumerable s e t

t r a n s i t i v e f o r eve ry i n t e g e r n i f f t h e c l o s u r e o f G t h e f i v e groups o b t a i n e d as f o l l o w s , s t a r t i n g f rom a c h a i n Q on E , isomorphic

w i th t h e c h a i n o f t h e r a t i o n a l s : symmetric group; group o f automorphisms o f Q ; group o f automorphisms o f betwenness (mod Q ) ; group o f automorphisms o f t h e c y c l i c

r e l a t i o n assoc ia ted w i t h Q ; group o f automorphisms o f t h e d i h e d r a l r e l a t i o n o f Q (CAMERON 1976).

-

0 Each o f t h e f i v e mentioned groups i s o b v i o u s l y n - s e t - t r a n s i t i v e f o r each n , and s i m i l a r l y so f o r any group y i e l d i n g one o f these f i v e groups under adherence.

Conversely, l e t G be a group o f pe rmuta t i ons o f a denumerable s e t E , which i s n - s e t - t r a n s i t i v e f o r e v e r y t i o n a l system R whose automorphism group i s t h e c l o s u r e G+ o f G under adhe-

rence. S ince G , and so G+ as w e l l , i s n - s e t - t r a n s i t i v e f o r eve ry n , t h e system R i s n-monomorphic f o r e v e r y n ; i n o t h e r words, R i s monomorphic: see 2.7.(2) above. It f o l l o w s t h a t R i s cha inab le : see ch.9 5 6.2, which can be

extended t o r e l a t i o n a l systems which a r e O-sequences o f r e l a t i o n s w i t h base E , by s imp ly t a k i n g t h e l i m i t . T h i s h o l d s i n s p i t e o f t h e use o f RAMSEY's theorem i n

t h e beginning, which supposes f o r each i n t e g e r n , a f i n i t e number o f isomorphism types o f m u l t i r e l a t i o n s o f c a r d i n a l i t y n . L e t A be a cha in i n which R i s f r e e l y i n t e r p r e t a b l e . Take t h e c h a i n Q o f t h e r a t i o n a l s , t hen t h e system S f r e e l y i n t e r p r e t a b l e i n Q , which i s o b t a i n e d as f o l l o w s : t a k e each l o c a l isomorphism f f rom A i n t o Q w i t h a f i n i t e domain F , then t a k e t h e image f " (R /F ) , and f i n a l l y t h e common ex tens ion o f these images. The systems R and S have t h e same age. It s u f f i c e s t o prove t h a t S i s homogeneous. Indeed by 1.3. (1) above, which e a s i l y extends t o r e l a t i o n a l systems, R

w i l l be isomorphic w i t h S , hence R w i l l be f r e e l y i n t e r p r e t a b l e i n an isomor-

p h i c copy of Q . Thus o u r p r o p o s i t i o n w i l l f o l l o w f rom t h e p reced ing 5.9.

Now l e t us p rove t h e homogeneity o f S . L e t f be a l o c a l automorphism o f S , w i t h f i n i t e domain F and range F ' , and l e t u be an a r b i t r a r y e lement i n t h e base. Take an isomorphism h f rom t h e f i n i t e c h a i n Q / ( F u { u ) ) o n t o a r e s t r i c t i o n

o f A , and an isomorphism h ' f rom Q/F ' o n t o ano the r r e s t r i c t i o n o f A . L e t v = hu . Then t h e b i j e c t i v e compos i t i on

Since R i s homogeneous, by 1.1 t h e r e e x i s t s a l o c a l automorphism o f R which extends t h e p reced ing l o c a l automorphism t o t h e domain h"(F) augmented by v . L e t g denote t h i s ex tens ion . Due t o t h e f a c t t h a t Q i s dense and w i t h o u t end-

p o i n t s , t h e r e e x i s t s a l o c a l isomorphism f rom A i n t o Q , which i s an e x t e n s i o n o f h ' - ' t o t h e domain h"'(F') augmented by gv . L e t h" denote t h i s ex tens ion .

F i n a l l y t h e b i j e c t i v e compos i t i on h",g,h i s a l o c a l automorphism o f S and extends f t o i t s domain F augmented by u . Then S i s homogeneous by 1.1. 0

n . By 2.5. (2) above, t h e r e e x i s t s a homogeneous r e l a -

hl,f,(h-l) i s a l o c a l automorphism o f R .

Chapter 11 337

§ 6 - EXTENSIVE SUBSET, PSEUDO-HOMOGENEOUS RELATION

Given a relation R with base E , we say that a subset F of E i s extensive (mod R) , i f each local automorphism of R with domain F i s extendible t o an automorphism of R . If F i s extensive and G i s a subset of E with R / G isomorphic t o R/F , then G i s extensive. Note t h a t R i s homogeneous i f f every f i n i t e subset of the base of R i s extensive. A relation R with base E i s said t o be pseudo-homogeneous, i f each f i n i t e subset of E i s included in an extensive (mod R ) f i n i t e subset. For example, the relation of consecutivity on the positiveand negative integers, i . e . on Z , i s pseudo-homogeneous yet n o t homogeneous. 0 Complete each f i n i t e s e t F of these integers by the leas t interval including F : we obtain a f i n i t e extensive superset of F . I n general F i t s e l f i s n o t extensive: take F = {0,25 and the local automorphism which preserves 0 and takes 2 in to 3 . This consecutivity relation associated with Z i s n o t rich for i t s age: the saturated, and thus rich denumerable relation representing th i s age, i s obtained by taking denumerably many components each isomorphic with our consecutivity relation: see ch.10 5 6 and 7.3. Another example: the saturated t r e e , hence rich for i t s age, which i s described in ch.10 5 6.2, i s pseudo-homogeneous yet n o t homogeneous. 0 Repeating our argument of ch.2 5 2.3, take three incomparable elements with an element d < a and < b , yet d I c , and three other incomparable ele-

a , b , c

ments a ' , b ' , c ' with an element e ' < b ' and < c ' , yet local automorphism which takes a , b , c into a ' , b ' , c ' i s an automorphism, since the image of d would be < b ' , thus and f inal ly less t h a n c ' , unless t h a t e ' be less than a ' To see that our relation i s pseudo-homogeneous, complete each

e ' I a ' . Then the n o t extendible t o comparable with e '

f i n i t e s e t F t o G in a manner t h a t any two elements of F have a common predecessor in G . 0

6.1. Every extensive subset i s maximalist. Consequently, every pseudo-homogeneous relation i s maximalist (see ch.10 5 3.8) . The converse i s fa lse: take a consecutivity relation formed of two, or several components, each isomorphic with the consecutivity relation on maximalist, non-pseudo-homogeneous relat ion.

Z ; th i s i s a

6.2. In the case of a pseudo-homogeneous relat ion, the proposition 1.1 becomes: Let E be a denumerable s e t . Then a relation R based on E i s pseudo-homogeneous i f f there exis ts a s e t of f i n i t e subsets F of E such t h a t every f i n i t e


subset i s included in an F ; and furthermore, f o r any local automorphism f of R

having an F as domain and f o r any f i n i t e subset G of E including t h i s F , there e x i s t s a local automorphism extending f t o G .

6.3. The proposit ion 1 .3 . (1 ) can be extended: Any two denumerable pseudo-homogeneous r e l a t i o n s , w i t h same a r i t y and same age, a r e isomorphic.

0 Let R and R ' be two denumerable pseudo-homogeneous r e l a t ions of the same age. S t a r t with a local isomorphism f from R i n t o R ' , whose domain F i s an exten- s ive f i n i t e subse t (mod R) . Let F ' = f " (F ) be the range, and not ice t h a t f o r any f i n i t e subset G ' of the base I R ' ( , including F ' , the inverse function f - l i s extendible t o a local isomorphism from R ' i n t o R w i t h domain G ' . Indeed, by hypothesis there e x i s t s a local isomorphism g from R ' i n t o R w i t h domain G ' . Then the composition g,f i s a local automorphism of R with ex tens ive domain F , hence g,f i s extendible t o an automorphism h of R ; and f i n a l l y h-',, g i s the desired local isomorphism from R ' i n t o R with domain G ' . Now i t su f f i ces t o choose G ' t o extend a l t e rna t ive ly the loca l isomorphism from R i n t o R ' , then the local isomorphism from R ' i n to R , by taking successively each element of I R ( i n the domain of one, and each element of I R ' I in the domain of the o ther . 0

t o be extensive (mod R ' ) ; then by i t e r a t i n g t h i s ,

6.4. The proposit ion 1 .3 . (2 ) cannot be extended t o pseudo-homogeneous r e l a t ions : see the example already given, of the consecut iv i ty r e l a t ion on There even e x i s t s an age which i s represented by a pseudo-homogeneous r e l a t i o n , y e t by no homogeneous r e l a t ion , and by no r ich r e l a t ion .

0 Let C be the consecutivity r e l a t ion on the natural numbers . Denote by 0 the s ing le ton unary r e l a t ion taking the value (+) f o r t he in t ege r 0 only. Let A be a unary r e l a t ion such t h a t , f o r every f i n i t e sequence of (t) and ( - ) , there e x i s t s a sequence of consecutive in t ege r s giving t o Then the t r i r e l a t i o n ( C , O , A ) i s pseudo-homogeneous, s ince each in te rva l beginning w i t h zero and going up t o an a r b i t r a r y in t ege r , i s extensive with the i d e n t i t y a s the unique local automorphism. However, no denumerable r e l a t ion i s r i ch f o r this age, because of the ex is tence of continuum many possible components (cons t ruc ted f o r ins tance , by s t a r t i n g from the consecut iv i ty r e l a t ion on added, without strengthening the represented age (example communicated by POUZET; not ice the analogy w i t h SPECKER's argument i n ch.10 5 6 .3 ) . 0

Z .

A t h i s sequence of values.

2 ) which can be

6.5. Let R be a pseudo-homogeneous r e l a t ion . Then the s e t of the r e s t r i c t i o n s - of R t o extensive f i n i t e subse ts (mod R ) i s amalaamable ( see 1 .5 ) . 0 Let A , B, C be three r e s t r i c t i o n s of T t o extensive f i n i t e subsets of the

Chapter 11 339

base. L e t f be an isomorphism f rom A o n t o a r e s t r i c t i o n o f B , and g be an

isomorphism f rom A on to a r e s t r i c t i o n o f C . The images o f t h e base I A I under f and under g a r e e x t e n s i v e s e t s . Thus t h e r e e x i s t s an automorphism f ' o f R , ex tend ing f - l ; and s i m i l a r l y an automorphism g ' o f R , ex tend ing g-' . Res-

t r i c t f ' t o t h e domain l B l , and g ' t o t h e domain I C l ; then l e t D be t h e f i n i t e r e s t r i c t i o n o f R t o t h e un ion f ' O ( I B 1 ) u g ' " ( l C 1 ) . The amalgamation p r o p e r t y i s s a t i s f i e d , s i n c e f o r each x i n t h e base I A l , we have ( f ' , f ) x = x

and (g',g)x = x . [3

We l e a v e i t t o t h e reader

t i o n s o f R , which generates t h e age o f R under embeddab i l i t y , i s (up t o isomorphism) a subset o f t h e s e t o f r e s t r i c t i o n s o f R t o e x t e n s i v e f i n i t e s e t s .

t o prove t h a t eve ry amalgamable s e t o f f i n i t e r e s t r i c -

6.6. PSEUDO-AMALGAMABLE AGE

An age d?, i s s a i d t o be pseudo-amalgamable, i f t h e r e e x i s t s a subset o f & which i s amalgamable, and fu r the rmore generates 6a. under embeddab i l i t y ; hence t h i s sub- s e t i s d i r e c t e d .

PSEUDO-AMALGAMATION THEOREM

Given an age @,, , t h e r e e x i s t s a coun tab le pseudo-homogeneous r e p r e s e n t a t i v e o f (F!, i f f 6?, i s pseudo-amalgamable (CALAIS 1967, t o whom t h e n o t i o n o f pseudo-amalgamable age i s due).

0 Suppose t h a t t h e r e e x i s t s a oseudo-homogeneous r e l a t i o n R which rep resen ts 6& . Then t h e s e t o f r e s t r i c t i o n s o f R t o e x t e n s i v e f i n i t e subsets o f t h e base, i s amalgamable, by t h e p reced ing 6 .5 . And by d e f i n i t i o n o f t h e pseudo-homogeneous r e l a t i o n s , t h i s s e t covers eve ry f i n i t e subset (under i n c l u s i o n ) ; t hus o u r age

i s pseudo-amalgamable.

Conversely , suppose t h a t t h e age 6% i s pseudo-amalgamable. L e t Ai (i i n t e g e r ) be

a f i n i t e , o r an W-sequence o f f i n i t e r e l a t i o n s , which under embeddab i l i t y g i ves & , and which fu r the rmore forms an amalgamable s e t .

We s h a l l c o n s t r u c t two sequences o f f i n i t e r e l a t i o n s Bi and Ci ( i i n t e g e r ) , each of which i s isomorphic t o an A . ( j i n t e g e r ) ; such t h a t f o r each i we have t h e

e m b e d d a b i l i t y A i d Bi , and Bi i s a r e s t r i c t i o n o f Ci , and Ci i s a r e s t r i c - t i o n o f Bi+l . Moreover, f o r any two r e s t r i c t i o n s o f Bi which a re isomorphic t o a same A

d i b l e t o an isomorphism f rom Bi o n t o ano the r r e s t r i c t i o n o f Ci . Wi th t h i s cons-

t r u c t i o n , t h e common e x t e n s i o n o f t h e Bi s a t i s f i e s t h e c o n d i t i o n o f 6.2 above, hence i s pseudo-homogeneous and rep resen ts

t h e g i v e n age. F o r t h i s sketched c o n s t r u c t i o n , s t a r t w i t h Bo = A. . Suppose t h a t Bi i s a l ready

obta ined, and l e t fi,O , ... , fi ,p ween r e s t r i c t i o n s isomorphic t o a same

A j

-

J

, we r e q u i r e t h a t eve ry isomorphism f rom one o n t o t h e o t h e r , be exten- j

( o r t h e Ci ) t o t h e un ion o f t h e i r bases,

be a l l t h e l o c a l automorphisms o f Bi be t - ( j v a r i a b l e ) . By amalgamation, by

3 40 THEORY OF RELATIONS

s t a r t i n g f rom t h e b i j e c t i o n f . and f rom t h e i d e n t i t y on t h e base I A . 1 , which

bo th t r a n s f o r m A . i n t o a r e s t r i c t i o n o f Bi , we o b t a i n an e x t e n s i o n Di,o o f

Bi , which belongs t o o u r age &, , and such t h a t fi,O i s e x t e n d i b l e t o an isomorphism f rom Bi on to ano the r r e s t r i c t i o n o f Di,o . I t e r a t e t h i s t o o b t a i n t h e sequence, w i t h f i n i t e l e n g t h p , o f success ive ex tens ions D i ,I 9 . . . "Ji ,I! co r res -

ponding t o fi,l ,..., fi,p . Then t a k e f o r Ci Di,p which i s

isomorphic t o an A . F i n a l l y t o be sure t h a t t h e e n t i r e age i s represented, t ake

f o r Bitl a common e x t e n s i o n of Ci and Ai+l , which fu r the rmore i s an A j . 0

1 3 0 J

J

an e x t e n s i o n o f

j

§ 7 - PREHOMOGENEOUS RELATION

G- EXTENSIVE SUBSET

Given a r e l a t i o n R and two f i n i t e subsets o f i t s base, say F and G i n c l u d i n g F , we say t h a t F i s G-extens ive (mod R ) , i f f o r eve ry l o c a l automorphism f

o f R w i t h domain G , t h e r e s t r i c t i o n f / F i s e x t e n d i b l e t o an automorphism o f R . I n t h e case where G = F , we f i n d t h e e x t e n d i b l e f i n i t e s e t , i n t h e sense o f 5 6. - I f F 12 G-extensive and i f F ' , G ' a r e t h e images o f F, G under t h e same

l o c a l automorphism o f R , then F ' i? GI-ex tens i ve .

I f F i s G-extensive, eve ry subset o f F i s G-extensive; f u r the rmore F o r each subset o f F i s G I -ex tens i ve f o r eve ry f i n i t e superse t G ' o f G . A r e l a t i o n R i s s a i d t o be Prehomoqeneous, i f each f i n i t e subset F o f i t s base admits a f i n i t e superse t G f o r which F i s G-extens ive.

Every homogeneous o r pseudo-homogeneous r e l a t i o n i s orehomogeneous.

7.1. If F admi ts a f i n i t e superse t G f o r which F G-extensive, t hen F & maximal i s t . 0 Take any e x t e n s i o n S o f R which rep resen ts t h e same age. Given an a r b i t r a r y

f i n i t e subset H o f t h e base IS1 , i n c l u d i n g G , take an isomorphism h f rom

S/H on to t h e r e s t r i c t i o n R/ho(H) . By hypo thes i s h/F i s e x t e n d i b l e t o an automorphism o f R . By composi t ion, t h e r e e x i s t s an F-isomorphism f rom S/H o n t o a r e s t r i c t i o n o f R : we see t h a t F s a t i s f i e s t h e t h i r d d e f i n i t i o n o f a m a x i m a l i s t subset, ch.10 5 3.8. 0

Consequently, eve ry prehomogeneous r e l a t i o n i s max ima l i s t .

The converse i s f a l s e : t a k e up t h e counterexample i n 6.1; i . e . a c o n s e c u t i v i t y r e l a - t i o n formed o f seve ra l components, each i somorph ic w i t h t h e c o n s e c u t i v i t y on 2 .

7.2. Any two denumerable prehomogeneous r e l a t i o n s w i t h t h e same age a r e i somorph ic

( g e n e r a l i z a t i o n o f 1 .3. (1) and 6 .3 ) .

Chapter 11 341

'3 Let R , R ' be two denumerable prehomogeneous relations with bases E , E ' , representing the same age. Let F be a G-extensive f i n i t e subset (mod R ) and l e t f be a local isomorphism from R into R ' , with domain G . Let G ' be the range of f . Take a f i n i t e superset H ' of G ' fo r which G ' i s HI-extensive (mod R ' ) . By the same argument as in ,6 .3 , we see tha t the inverse function (f/F)-' i s extendible t o a local isomorphism g ' from R ' into R with domain H ' . Similarly the inverse function ( g ' / G ' ) - ' phism from R into R ' , and so for th , going alternatively from R in to R ' and from R ' in to R . To terminate the proof and obtain a common extension of our local isomorphisms which i s an isomorphism from R o n t o R ' , i t suffices to note t h a t F can contain an a rb i t ra r i ly given element of E , then that G ' can conta in an a rb i t ra r i ly given element of

i s extendible t o a local isomor-

E ' , and so for th . 0

7.3. RELATION RA Given a chain A , l e t RA be the ternary relation freely interpretable in A , which i s defined by RA(x,y,z) = + i f f x < z and y < z (mod A) and x # y . (1) Let Q be the chain of the rationals; then RQ i s prehomogeneous b u t n o t pseudo-homogeneous. 0 Given a f i n i t e subset F of the base, l e t G be the se t F augmented by one element which i s s t r i c t l y less (mod Q ) than the minimum element of F . Every local automorphism f of R with domain G , preserves the ordering of the elements (mod Q ) , except when concerning the f i r s t two elements of G , where the ordering can be inverted. Hence the ordering of the elements of F i s preserved; so that f/F i s extendible to an automorohism of Q , hence of RQ . Thus F i s G-extensive (mod R ) ; yet F i s n o t extensive. 0

( 2 ) Every denumerable younger relation t h a n R i s of the form R A , where A i s a denumerable chain. 0 To each f i n i t e subset based on F and such t h a t R C be the rest r ic t ion t o F of the given relation. Then apply the coherence lemma ch.2 5 1.3. 0

Consequently RQ i s rich for i t s age. Hence there exis ts an age having a rich representative, yet having no denumerable pseudo-homogeneous representative. 0 Indeed a denumerable pseudo-homogeneous representative of the age under consideration, would a for t ior i be prehomogeneous, hence isomorphic with R Compare with the counterexamples in 6.4 above.

-

Q

Q

Q

F of the base, associate the s e t of those chains C

Q . O

7.4. ISOLATING PAIR, ISOLATED REL-AGE Let b;L be an age, and an extension of A .

A , B be two f i n i t e relations belonging t o &, with B


The o rde red p a i r (A,B) i s s a i d t o be i s o l a t i n g (mod & ) i f f t h e r e e x i s t s one and o n l y one maximal A-age s p e c i f i c a t i o n o f bE which c o n t a i n s t h e e lement B . We say as w e l l t h a t t h i s maximal A-age i s i s o l a t e d by B . For example, t a k e & t o be t h e age rep resen ted by t h e c o n s e c u t i v i t y r e l a t i o n on t h e n a t u r a l numbers . I f A i s t h e c o n s e c u t i v i t y r e l a t i o n o f a f i n i t e cha in , t hen

t h e o rde red p a i r (A,A) i s i s o l a t i n g , and t h e co r respond ing i s o l a t e d maximal A-age i s formed o f a l l t h e c o n s e c u t i v i t y r e l a t i o n s o f f i n i t e cha ins i n c l u d i n g A as an i n t e r v a l , and t h e i r A - r e s t r i c t i o n s . Now denote by

chains, which we s h a l l c a l l t h e two components, w i t h t h e va lue ( - ) f o r those orde-

r e d p a i r s formed o f one element i n each component.

Then t h e r e e x i s t i n f i n i t e l y many maximal f i r s t component b e f o r e o r a f t e r t h e second, w i t h a non-zero f i n i t e number o f i n t e r -

mediate e lements. T h i s can be done w i t h t h e a i d o f a f i n i t e c o n s e c u t i v i t y r e l a t i o n

denoted by B , which i s an e x t e n s i o n o f A . Thus such an o rde red p a i r (A,B) i s i s o l a t i n g . Now w i t h t h e same A , d e f i n e t h e A-age o b t a i n e d by never t a k i n g t h e un ion o f t h e two components w i t h f i n i t e l y many i n t e r m e d i a t e e lements. Then t h e maximal

t hus d e f i n e d i s i s o l a t e d by no o rde red p a i r o f r e l a t i o n s .

I f (A,B) i s i s o l a t i n g (mod 6 L ) , then so i s (A,B') f o r eve ry B ' which i s a f i n i t e e x t e n s i o n o f B and belongs t o t h e age 6% ; hence B ' a l s o belongs t o t h e

A-age s p e c i f i c a t i o n o f & and i s o l a t e d by B . I f (A,B) i s i s o l a t i n g , t hen so i s (A',B) f o r each r e s t r i c t i o n A ' Lf A . 0 L e t A be t h e unique maximal be t h e A'-age induced by (ch.10 5 3 .3 ) . So A ' c o n t a i n s B and i s i t s e l f

maximal, by ch.10 5 3.6. Suppose t h a t t h e r e e x i s t s a maximal from &' and c o n t a i n i n g t h e e lement B . Then t h e r e e x i s t s an A-age s p e c i f i c a t i o n

o f r' , which c o n t a i n s B , and i s maximal. By hypo thes i s f i s i d e n t i c a l

w i t h , and hence f' ( i nduced by f ) i s i d e n t i c a l w i t h c d ' ( i n d u c e d by & ) : c o n t r a d i c t i o n . 0

A t h e common e x t e n s i o n o f t h e c o n s e c u t i v i t y r e l a t i o n s o f two f i n i t e

A-ages, ob ta ined by d e c i d i n g t o p u t t h e

A-age

-

A-age t o which t h e r e l a t i o n B belongs. L e t d'

A'-age f' d i s t i n c t

7.5. L e t R be a r e l a t i o n , & t h e age rep resen ted by R , and F , G i n c l u d i n g

F , two f i n i t e subsets o f t h e base. If F js- G-extens ive - (mod R) , then there e x i s t s a un ique maximal

ment R/G . Moreover, t h e (R/F)-age thus i s o l a t e d by R/G i s t h a t which i s rep resen ted by R . 0 Suppose t h a t F i s G-extensive, y e t t h a t t h e r e e x i s t two maximal (R/F)-age s p e c i f i c a t i o n s o f fi , which c o n t a i n t h e e lement R/G . I n t h e f i r s t , t a k e an e x t e n s i o n U ' o f R/G , and i n t h e second, t a k e an e x t e n s i o n U " , such t h a t no

(R/F)-age s p e c i f i c a t i o n o f & c o n t a i n s b o t h U ' and U " : see ch.10 5 3.7.

(R/F)-age s p e c i f i c a t i o n o f bt , which c o n t a i n s t h e e l e -

-

Chapter 11 343

Now t a k e two f i n i t e subsets H ' , H " o f t h e base, w i t h R / H ' t h e image o f U ' under an isomorphism denoted by f ' , and R / H " t h e image o f U " under f " . Set F' = f ' ' ( F ) and G ' = f ' " ( G ) , and s i m i l a r l y F" = f " " ( F ) and G" = f " " (G) . By t h e hypo thes i s o f G-extensiveness, t h e r e e x i s t s an automorphism h ' o f R

which takes F' i n t o F w i t h h ' / F ' = ( f ' / F ) - ' and ano the r automorphism h" which takes F" i n t o F w i t h h"/F" = ( f " /F ) - ' . Then t h e (R/F)-age represen- t e d by R con ta ins t h e element R /h ' " (H ' ) , which i s t h e image o f U ' under an F-isomorphism; as w e l l as t h e e lement R/h""(H") , which i s t h e image o f U" :

c o n t r a d i c t i o n . Thus we have Droved t h e uniqueness o f t h e maximal (R/F)-age con-

t a i n i n g R/G . We s h a l l now prove o u r second conc lus ion . L e t & denote t h e unique maximal

age which c o n t a i n s t h e e lement R/G . F o r each e x t e n s i o n U o f R/G , be long ing t o d! , t h e r e e x i s t s an isomorphism f rom U o n t o a r e s t r i c t i o n o f R . Since F i s G-extens ive, by compos i t i on w i t h an automorphism o f R , we can r e q u i r e t h a t ou r p r e v i o u s isomorphism be an

age rep resen ted by R . And s i n c e i t i s maximal, i t i s i d e n t i c a l w i t h t h e age o f R .

(R/F)-

F-isomorphism. Thus d! i s i n c l u d e d i n t h e (R/F)- (R/F)-

7.6. L e t R be a r e l a t i o n and d?, t h e age rep resen ted by R . Suppose t h a t R i s max ima l i s t , and l e t F and G i n c l u d i n g F , be two f i n i t e

subsets o f t h e base, such t h a t t h e o rde red p a i r (R/F,R/G) i s i s o l a t i n g (mod @, ) . Then f o r eve ry l o c a l automorphism h o f R w i t h domain G , t h e r e s t r i c t i o n h/F - iz 1-isomorphism f rom R into R i t s e l f .

0 L e t F ' = h " (F ) and G ' = h"(G) , and take t h e image under h - l / F ' o f t h e

(R/F ' ) -age rep resen ted by R . Since R i s max ima l i s t , t h i s (R/F')-age i s maximal. I n t a k i n g i t s image under

t h e element R/G ( u p t o F-isomorphism). S ince t h e o rde red p a i r (R/F,R/G) i s

i s o l a t i n g , we o b t a i n p r e c i s e l y t h e (R/F)-age rep resen ted by R . Thus h-'/F' , hence a l s o t h e i n v e r s e f u n c t i o n h/F , i s a 1-isomorphism f rom R i n t o i t s e l f . 0

h - l / F ' , we o b t a i n a maximal (R/F)-age which con ta ins

7 . 7 . L e t R be a r e l a t i o n w i t h denumerable base E , and l e t be t h e age r e -

p resen ted by R . Suppose t h a t ( 1 ) R i s max ima l i s t , and

( 2 ) f o r each f i n i t e subset F o f E , t h e r e e x i s t s a f i n i t e subset G i n c l u d i n g F , such t h a t t h e o rde red p a i r (R/F,R/G) i s i s o l a t i n g (mod R ) . Then R i s prehomogeneous. More p r e c i s e l y , f o r each of t h e p reced ing ordered p a i r s (F,G) , t h e subset F & G-extens ive (mod R) . 0 L e t f be a l o c a l automorphism o f R w i t h domain G . We s h a l l c o n s t r u c t an

automorphism of R ex tend ing f/F as f o l l o w s . Set F ' = f " ( F ) and G' = f " (G) ,


and l e t F1 be an arbitrary f i n i t e superset of F , and G1 f i n i t e s e t including

F1 , such t h a t (R /F1 ,R /G1) i s isolat ing. By the preceding proposition, f/F i s a

1-isomorphism from R into R . Thus there exis ts a function f l extending f/F t o the domain Iterating t h i s , by taking each element of the base, alternatively in the domain and in the range, we obtain an a-sequence of extensive local automorphisms, whose union i s an automorphism of R extending f . 0

G1 , which i s a local automorphism of R .

7.8. Let & be an age. Then e i ther , for each f i n i t e relation A belonging t o &, there exis ts a f i n i t e extension B of A belonging t o , such that ( A , B ) isolating (mod ) ; or there e x i s t continuum many maximal rel-ages specifications - of m. . - 0 Let A be an element of & , having no extension which together with A forms an isolating pair . Take two d is t inc t maximal by ch.10 5 3.7, take a relation belonging t o the second, such that there exis ts no A-age specification of (R , which contains b o t h A. and A1 . By hypothesis, among the relations belonging t o & and extending A. , there exis ts none which together with A forms an isolating oair . Thus there ex is t two dis t inct maximal A. . Using again ch.10 5 3.7, take an element

AO,l & , which contains b o t h A

ted, we can always require tha t A and A be extensions o f A. . 0 ,O 0 9 1

Dichotomously, in a similar fashion, obtain A and A s tar t ing with A1 . Finally we obtain continuum many A-ages which we can always take t o be maximal. 0

A-ages soecifications of fl , and A. belonging t o the f i r s t and a relation A1

A-ages specifications of R , each of which contains the element

Ao,O belonging t o the f i r s t and

belonging t o the second, such that there exis ts no A-age specification of

0 3 0 and AO,l . Moreover, since each A-age i s direc-

1 ,o 1 9 1

7 . 9 . EXISTENCE CRITERION FOR A PREHOMOGENEOUS RELATION (1) Given an age fi , there ex is t s a denumerable prehomogeneous representative

"f (R i f f f o r each f i n i t e relation A belonging t o & , there ex is t s a f i n i t e extension B of A belonging t o , such that ( A , B ) i s i so la t i ng (mod ) . ( 2 ) Consequently, i f there ex is t s a denumerable representative o f rich for &. , then there exis ts a denumerable prehomogeneous representative of & (POUZET 1972).

0 By the existence cr i ter ion for a rich relation (see ch.10 § 7 . 6 ) , i f there exis ts a relation rich for i t s age bt , then there are countably many maximal rel-ages specifications of & . Hence by the preceding proposition, for each element A of &, there exis ts an extension B of A belonging t o 8 , such t h a t

which i s

-

Chapter 1 1 345

the p a i r (A,B) To prove (l), note f i r s t t ha t , i f there e x i s t s a prehomogeneous representat ive

o f our age, hence i f f o r every f i n i t e subset F o f the base, there ex i s t s a f i n i t e G i nc lud ing F and f o r which F i s G-extensive, then by 7 .5 our c r i t e r i o n i s s a t i s f i e d , since the p a i r (R/F,R/G) i s i s o l a t i n g . Conversely, assume the c r i t e r i o n i n statement (1 ) . S t a r t w i t h an a-sequence o f f i n i t e r e l a t i o n s Ai ( i i n tege r ) belonging t o our age & , each element o f t h i s age being embeddable i n an Ai . Set Bo = A. . L e t Co be an element o f fi such t h a t (Bo,Co)

B1

be such t h a t (B1,C1) i s i s o l a t i n g , and so f o r t h . The common extension R o f the

Bi ( o r equ iva len t l y the Ci ) i s a representa t ive o f our age, and s a t i s f i e s the second hypothesis o f 7.7.

It remains t o show t h a t R

To t h a t end, i t su f f i ces t o see t h a t each f i n i t e subset o f the base I R I ded i n a maximal ist f i n i t e subset. Indeed modify the preceding cons t ruc t ion as

fo l lows. When we obta in Co , take a f i n i t e extension Do belonging t o our age, which i s chosen so as t o make the Bo-age represented by R maximal. Thus choose

Do t o admit an embedding o f every extension o f Bo t o one add i t i ona l element which belongs t o the unique maximal Bo-age i s o l a t e d by Co . When we obta in C1 , take a f i n i t e extension belonging t o our age, which admits an embedding o f

every extension o f Bo t o two add i t i ona l elements, which belongs t o the unique maximal Bo-age i so la ted by Co , and which belongs t o the unique maximal B1-age

i s o l a t e d by C1 ; and so f o r t h . 0

Recal l t h a t there e x i s t s an age having a prehomogeneous, and even a pseudo-homogeneous representat ive, y e t having no r i c h representat ive: see 6.4 above.

i s i s o l a t i n g . Thus our conclusion ( 2 ) fo l lows from (1) . R

i s i s o l a t i n g . Le t

be an element i n 61 which i s a common extension o f Co and A1 ; then l e t C1

i s maximal ist , which i s the f i r s t hypothesis o f 7 .7 . i s i nc lu -

D1

5 8 - SET-HOMOGENEOUS RELATION, SET-PSEUDO-HOMOGENEOUS RELATION

We say t h a t a f i n i t e subset F o f the base o f a r e l a t i o n R i s set-extensive (mod R) , i f f o r every subset G w i t h R/G isomorphic t o R/F , there e x i s t s

an automorphism o f R t ak ing F i n t o G . Yet n o t each isomorphism from R/F onto R/G i s so ex tend ib le .

- If F i s set-extensive and R/G i s isomorphic t o R/F ,then G i s set-extensive.

A r e l a t i o n

i s set-extensive. A r e l a t i o n i s set-pseudo-homogeneous, i f each f i n i t e subset o f i t s base i s i nc lu - ded i n a f i n i t e set-extensive subset.

i s sa id t o be set-homogeneous, i f each f i n i t e subset o f i t s base

8.1. Given the chain Q of the rat ionals , every relation freely interpretable in Q i s set-homogeneous.

0 For every f i n i t e subset F of the base and every G of the same cardinal i ty , the isomorphism from Q/F onto Q / G i s extendible t o an automorphism of Q , hence of the given relat ion, since i t i s freely interpretable in

For example the relation RQ defined in 7 . 2 , taking the value (+) i f f x < z and y < z (mod Q ) and x f y , i s set-homogeneous, yet n o t pseudo-homogeneous.

The consecutivity relation on Z i s pseudo-homogeneous (by associating t o each f i n i t e s e t F of positiveor negative integers, the smallest interval including F ) . Hence i t i s set-pseudo-homogeneous, yet not set-homogeneous.

Problem. Does there ex is t a set-pseudo-homogeneous relation which i s neither set- homogeneous nor pseudo-homogeneous.

Q . 0

8.2. Every f i n i t e set-extensive subset i s maximalist. Consequently, every set-homogeneous or set-pseudo-homogeneous relation i s maximal i s t .

0 Let R be a relation and F be a f i n i t e set-extensive subset of the base. Let

rel-age i s generated by an a-sequence of f i n i t e relations Ai ( i integer) , a l l extending R/F , with each A i + l an extension of Ai . For each i , there exis ts a res t r ic t ion of R which i s the isomorphic image of Ai under a function denoted by f i . Since F i s set-extensive, there ex is t s an automorphism g i of R which takes f j o ( F ) into F . Since F i s f i n i t e , there are only f in i te ly many permutations of F . Hence there ex is t s an o-sequence of indices i for which the composition giofi i s the same permutation of F , say h . I t follows t h a t the (R/F)-age represented by R i s a superset of the image of J% under h . Since i s maximal, i t s image under h i s maximal. Finally the (R/F)-age represented by R i s maximal. 0

be a maximal (R/F)-age specification of the age represented by R . This

8.3. Let R be a relation and F a f i n i t e subset of i t s base. If F i s set-extensive (mod R ) , then there exis ts a f i n i t e superset G of F , such that the pair ( R / F , R / G ) i s isolating modulo the age of R . Moreover, the unique maximal (R/F)-age which contains R / G i s the rel-age represented by R . Consequently, every set-homogeneous or set-pseudo-homogeneous relation i s prehomogeneous, by 7.7 (communicated by P O U Z E T ) .

We know already that F i s maximalist; hence the by R i s maximal. The same holds for each o f the f in i te ly many (R/F)-ages which are images of

-

(R/F)-age A0 represented

under an arbi t rary automorphism of R/F . We shall denote

Chapter 11 347

these re l -ages by 4 phisms o f R/F , which a r e d i f f e r e n t f rom t h e i d e n t i t y , can p rese rve A, and consequent ly p rese rve each d! We s h a l l f i r s t p rove t h a t t h e A i o f t h e age o f R . Indeed, suppose n o t and cons ide r a maximal (R/F)-age @ d i s - t i n c t f rom t h e A i . Take i n 63 a f i n i t e r e l a t i o n R/F and

does n o t be long t o any "4 . Since B be longs t o t h e age o f R , t h e r e e x i s t s

a r e s t r i c t i o n o f R which i s isomorphic w i t h B . Since F i s se t -ex tens i ve , by

compos i t i on w i t h an automorphism o f R , we can t r a n s f o r m B i n t o an ex tens ion

o f R/F which belongs t o one o f t h e '4 : c o n t r a d i c t i o n . Now c o n s i d e r a f i n i t e e x t e n s i o n C o f R/F which belongs t o LAo b u t which

belongs t o no o t h e r dfi . Since td0 i s t h e (R/F)-age rep resen ted by R , t h e r e e x i s t s a f i n i t e superse t G o f F such t h a t C = R / G . B u t s i n c e t h e o n l y maximal (R/F)-ages a r e t h e A , i t f o l l o w s t h a t t h e p a i r (R/F,R/G) i s i s o l a - t i n g : o u r c o n c l u s i o n f o l l o w s . 0

, .. . , 4 p-l ( p i n t e g e r ) . N o t i c e t h a t c e r t a i n automor-

( i = 0,1,. . . ,p-1) . a r e t h e o n l y maximal (R/F)-ages s p e c i f i c a t i o n s

B which extends

Consequently, any two denumerable set-pseudo-homogeneous r e l a t i o n s o f t h e same age a r e isomorphic .

8.4. RELATION CA

Given a c h a i n A , we l e t t h a t - CA(x,y,z) = + i f f x # y and z i s consecu t i ve t o Max(x,y) (mod A) . L e t t i n g Z denote t h e c h a i n o f t h e p o s i t i v e a n d n e g a t i v e i n t e g e r s , t hen Cz 5 prehomogeneous b u t n o t set-pseudo-homogeneous.

0 To see t h e prehomogeneity, n o t e t h a t i f p 2 3 . . . , ap

al, a2 s a t i s f y Max(al,a2) = a3 - 1 , then a l o c a l automorphism o f Cz w i t h

domain { al ,..., a 5 w i l l necessary take each ai ( i = 1 ,..., p ) i n t o bi w i t h

b3, b49 . - a , bp consecu t i ve i n t e g e r s and Max(bl,b2) = b3 - 1 . So t h a t t h i s l o c a l automorphism, when r e s t r i c t e d t o { a3, ... ,ap) , i s e x t e n d i b l e t o an automor- ph ism o f Cz . Now we see t h a t Cz i s n o t set-pseudo-homogeneous. Indeed, e i t h e r t h e domain o f

a g i v e n l o c a l automorphism, i s formed o f consecu t i ve i n t e g e r s , i n which case t h e f i r s t two elements can be t rans fo rmed i n t o two non-consecut ive elements; o r t h e r e

e x i s t i n t h e domain, a t l e a s t two non-consecut ive i n t e g e r s , which can be taken

i n t o two elements such t h a t t h e l e n g t h o f t h e i n t e r v a l s e p a r a t i n g them

i s n o t p rese rved . 0

CA denote t h e t e r n a r y r e l a t i o n on t h e same base, such -

i s a g i v e n i n t e g e r , and a3, a4, i s a sequence o f p-2 consecu t i ve ( p o s i t i w o r n e g a t i v e ) i n t e g e r s , and i f

P


E X E R C I S E 1 - PERMUTATION GROUP GENERATED BY THE FINITE LOCAL AUTOMORPHISMS OF A RELATION L e t R be a r e l a t i o n , o r a r e l a t i o n a l system, w i t h base E . Consider t h e groups GR o f pe rmuta t i ons o f E , which a re c losed under adherence, and such t h a t eve ry

l o c a l automorphism o f R w i th f i n i t e domain i s e x t e n d i b l e t o an e lement o f GR . Note t h a t t h e symmetric group (g roup o f a l l t h e pe rmuta t i ons ) i s a GR f o r e v e r y

system R . The i n t e r s e c t i o n o f a l l these

t e d bv t h e f i n i t e l o c a l automorohisms o f R . GR i s a group, and s h a l l be c a l l e d t h e group genera-

1 - Note t h a t t h e group thus generated, i s c losed under adherence, and t h a t eve ry

automorphism o f R be longs t o i t .

2 - I n t h e case o f a homogeneous r e l a t i o n a l system R , t h e group generated i s

i d e n t i c a l t o t h e group o f automorphisms o f R . Problem 1 . Is t h e group thus generated always a

f i n i t e l o c a l automorphism o f R e x t e n d i b l e t o a pe rmuta t i on be long ing t o a l l

t h e GR . Problem 2 . Take R t o be t h e c h a i n o f t h e n a t u r a l numbers . What i s t h e group

generated. Is i t i d e n t i c a l t o t h e symmetric group, o r i s i t reduced t o t h e i d e n t i t y .

GR . I n o t h e r words, i s eve ry

E X E R C I S E 2 - THE ONLY INDICATIVE GROUPS ARE THE I , J , T, D, S F o l l o w i n g § 5 above, we c a l l canon ica l groups those groups I , J , T, D, S d e s c r i

bed i n 9 4.2.

1 - L e t G be an i n d i c a t i v e m-ary group, i . e . a group generated by an i n f i n i t e

b i c h a i n AB . Suppose t h a t t h e r e e x i s t s an i n f i n i t e subset U o f t h e base I A B l f o r which A/U = B/U . Show t h a t , i f G does n o t c o n t a i n any pe rmuta t i on which, t o g e t h e r w i t h (1,2, ..., m) , generates t h e symmetric group Sm , then G i s

i n c l u d e d i n Tm (group o f t r a n s l a t i o n s , generated by (1,2, ..., m)) .

Fo r each p a i r o f m-ary canon ica l groups, v e r i f y t h a t t h e un ion group, more p r e c i -

s e l y t h e group generated by t h e i r un ion, i s canon ica l ( i f i t seems t o o l ong , t h e

reader w i l l assume t h i s ) . Consequently, f o r each group G , t h e r e e x i s t s a maximum canon ica l group i n c l u d e d i n G . Conclude f rom t h e preceding, t h a t i f G i s an i n d i c a t i v e m-ary group, and i f t h e maximum canon ica l group i n c l u d e d i n G i s Tm , then G = Tm . 2 - L e t G be an i n d i c a t i v e m-ary group. I f t h e maximum canon ica l group i n c l u d e d i n G i s Dm , then G = Dm . Fur thermore, i f AB i s a b i c h a i n g e n e r a t i n g D, , then eve ry subset U o f t h e base f o r which A/U = B/U i s f i n i t e . 3 - L e t G be an i n d i c a t i v e m-ary group generated by an i n f i n i t e b i c h a i n AB , and l e t U be an i n f i n i t e s e t w i t h A/U = B/U . Suppose t h a t G does n o t c o n t a i n

t h e pe rmuta t i on (1,2, ...,m) and t h a t t h e r e e x i s t p , q ( p t q 6 m) such t h a t G con ta ins none o f t h e f o l l o w i n g types o f pe rmuta t i ons : those which t a k e p t l i n t o

Chapter 11 349

an integer ,< p and preserve p t 2 , p+3, ... , m ; and those which take m-q into an integer < m-q and preserve 1, 2 , . . . , m-q-1 . Then G i s included in I i y q

4 - Let G be an indicative m-ary group. If the maximum canonical group included in G i s I i y q (with p+q sm) , then G = I i y q . 5 - If the maximum canonical group included in G i s J I

G = J I . Moreover, i f AB i s a bichain generating JL , then every U fo r which

A / U = B/U , i s f i n i t e . Recall that these resul ts are due t o FRASNAY 1965; see also FRAISSi 1974’ .

(with 2 r d m ) , then

EXERCISE 3 - THE N U M B E R OF m-ARY INDICATIVE GROUPS For m = 0 , 1, 2 , 3, a l l m-ary permutation groups are indicative. For instance the ternary groups are S3 (symmetric group), 1;” ( ident i ty g r o u p ) , 1;’’ (transposition ( 1 , 2 ) ) , and T3 (cyclic group). Show t h a t there ex is t exactly eleven indicative quaternary indicative groups, namely S4 (symmetric grouo) , 1;” ( ident i ty group), I i y 2 (transposition (3 ,4) ) ,

1;’’ (transposition (1,2)), 1;” (union of the two preceding groups),

I i y 3 (symmetric group on the s e t 2,3,4 ) , 1;’’ (symmetric group on 1,2,3 ) ,

J4 ( re f lec t ion) , J4 (generated by the reflection and the two transpositions

( 1 , 2 ) and ( 3 , 4 ) ) , T4 (cyclic g r o u p ) , D4 (dihedral group).

Show t h a t , for m a t leas t equal t o 4 , the number of indicative m-ary groups i s 3 plus the maximum integer 4 m / 2 (communicated by FRASNAY).

1;” (transposition ( 2 , 3 ) ) , J i (transposition ( 1 , 3 ) ) ,

1 2

2

35 1

CHAPTER 12

BOUND OF A RELATION; WELL RELATION; REASSEMBLING THEOREM

5 1 - BOUND OF A RELATION

Given a relation R , a bound of R i s any f i n i t e relation A with same ar i ty , such t h a t A is not embeddable i n R , b u t every proper res t r ic t ion of A i s embeddable i n R . If we consider the par t ia l ordering of embeddability between f i n i t e relations (considered up t o isomorphism), and the i n i t i a l interval formed of a l l f i n i t e res t r ic t ions of R , then we find the bound of th i s i n i t i a l interval , in the sense of ch.4 5 8. I f R has f i n i t e cardinality p , then any bound o f R has cardinality p + l . Examples. For a reflexive binary relat ion, the binary relation of cardinality 1 which takes the value ( - ) , is a bound. For a reflexive, symmetric re la t ion, we have the preceding bound plus the chain o f cardinality 2 . For a binary relat ion always (+) of cardinality 3 1 , we have the two preceding bounds plus the ident i ty re la t ion of cardinality 2 . For an inf in i te relation always (t), we only have these three bounds. For a relation always (+) of f i n i t e cardinality p , we additionally have as a bound the relation always (+) of cardinal i ty p t l . For a reflexive, antisymmetric re la t ion, we have as a bound the relation of cardinal i ty 1 taking the value (-), and the relation always (+) of cardinality 2 . For a reflexive, antisymmetric and comparable relation ( f o r any x , y the relation takes value (t) ei ther for (x,y) or for (y,x) ) , we have the two preceding bounds plus the identity relation o f cardinality 2 . For a chain of cardinality 3 2 , we have the three preceding bounds, plus the reflexive binary cycle of cardinality 3 . If the chain is in f in i te , then these four bounds are the only possible. I f the chain has f i n i t e cardinality p , then i n addition we have as a bound the chain of cardinality p t l . A f i n i t e relation, a chain, a relation always ( t ) , an identity re la t ion, a l l have f i n i t e l y many bounds (up t o isomorphism). A re la t ion in which every f i n i t e relation of the same ar i ty i s embeddable, has no bound ( for example a rich relat ion) . On the other hand, the consecutivity relation on the natural numbers, has infini- t e ly many bounds, among which, f o r each integer p > , 3 , the binary cycle of cardinal i ty p .

1.1. (1) Let R be a relation. For any f i n i t e relation X , we have X embeddable in R i f f no bound of R i s embeddable in X . (2) Let R , S be two relat ions, a t l eas t one of which i s f i n i t e . Then RG S iff no bound of S i s embeddable in R . 0 (1 ) Follows from ch.4 5 8.2, since embeddability between f i n i t e relations i s a well-founded par t ia l ordering. However l e t us give a direct proof. If X < R , then no bound of R i s embeddable in X . Conversely, i f X 3 ; R , then e i ther X i s a bound of R . Or there ex is t s a res t r ic t ion X1

rating t h i s , a f t e r a f i n i t e number h of s teps , we obtain a res t r ic t ion Xh of X which i s a bound of R . 0

0 (2 ) If R i s f i n i t e , we find ( 1 ) . Suppose that R i s in f in i te and S f i n i t e , hence R $ S . Replace R by a res t r ic t ion R ' whose cardinality i s f i n i t e b u t s t r i c t l y greater than the cardinality of S . Then embedding of a bound of S , by our ( 1 ) . This bound i s thus embeddable in R . 0

On the other hand , i f R and S are bo th in f in i te , then statement (2 ) does n o t necessary hold. 0 Take R t o be the chain of the natural numbers. Take a sequence of f i n i t e relations tion is isomorphic t o an Ai . Take S to be the common extension of the Ai , which takes the value (+) for every ordered pair whose terms belong t o the bases of two d i s t inc t Ai . Then S has no bound; hence no bound of S i s embeddable in R , and yet R i s non-embeddable in S . 0

-

of X t o i t s base minus one element, such t h a t XIB R . Ite-

R'4 S , hence R ' admits an

Ai ( i integer) with dis joint bases, and such that every f i n i t e binary rela-

1.2. (1) For R f i n i t e or S f i n i t e , i f every bound of S i s a bound of R , then R Q S . Indeed no bound of S i s embeddable in R . (2) Any two f i n i t e relations having the same bounds, are isomorphic. (3) If R i s f i n i t e and R < S , then there exis ts a bound of R which i s embeddable in S ; hence there ex is t s a bound of R which i s n o t a bound of S . Indeed S i s not embeddable in R , hence S admits an embedding of a t l eas t a bound of R , by the preceding 1 .l. ( 2 ) . (4) If e i ther R or S i s f i n i t e , and i f the s e t of bounds of S i s properly

-

-

included in the s e t of bounds of R , then R < S . Follows from ( 1 ) and ( 2 ) . - 1.3.(1) Every f i n i t e relation of a r i t y >/ 1 ( 2 ) Every non-empty f i n i t e relation of a r i ty >/ 2 has a t l eas t 4 bounds.

0 (1) Let R be f i n i t e ; by the preceding 1.2 . (3 ) , there exis ts a bound A of R . We have AS R . By ch.5 5 1.3.(1), assuming that the a r i ty i s n o t zero, there exis ts an extension S 7 R respecting the non-embeddability A4 S . By the preceding 1.2.(3), there ex is t s a bound B of R with B & S , hence non-isomorphic

has a t l eas t 2 bounds.

-

Chapter 12 353

w i t h A . 0 0 ( 2 ) L e t R be a non-empty f i n i t e r e l a t i o n w i t h a r i t y 2 . Keeping t h e two

bounds A, B a l ready o b t a i n e d and u s i n g ch.5 § 1.3. (3) , t h e r e e x i s t s a p roper e x t e n s i o n R+ o f R r e s p e c t i n g A $ R+ and B * R + . By t h e p reced ing 1 .2 . (3 ) , t h e r e e x i s t s a bound C o f R such t h a t C,< R+ , hence C i s isomorphic n e i t h e r w i t h A n o r w i t h B . I t e r a t i n g t h i s and s t a r t i n g w i t h A, B, C and R , we

o b t a i n ano the r p r o p e r e x t e n s i o n o f R and f i n a l l y a f o u r t h bound D o f R which

i s n o t isomorphic w i t h any o f A, B, C . 0

1.4. (1) L e t p be an i n t e g e r 3 3 , and S be t h e b i n a r y c y c l e o f c a r d i n a l i t y p . There does n o t e x i s t any p-monomorphic e x t e n s i o n o f S o f c a r d i n a l i t y p + l . ( 2 ) L e t S be a (p-1)-monomorphic r e l a t i o n o f c a r d i n a l i t y p , f o r which t h e r e e x i s t s no p-monomorphic e x t e n s i o n o f c a r d i n a l i t y p + l . L e t R be a r e s t r i c t i o n

o f S w i t h c a r d i n a l i t y p - 1 . Then S i s a bound o f R and eve ry bound o f S

i s a bound o f R (communicated by POUZET 1978).

0 (1) S t a r t w i t h t h e c y c l e S and add an element a t o i t s base. To o b t a i n an

e x t e n s i o n T o f S which i s p-monomorphic, i t i s necessary t h a t t h e r e e x i s t two d i s t i n c t e lements b, c o f t h e base I S 1 w i t h T(b,a) = T(a,c) = + . L e t

b ' be t h e element consecu t i ve t o b i n t h e c y c l e S . Then a, b, b ' a re d i s - t i n c t and T (b ,b ' ) = T(b,a) = + , which makes it imnoss ib le f o r any r e s t r i c t i o n

o f T whose base c o n t a i n s a, b, b ' , t o be isomorphic w i t h t h e c y c l e S . 0

Note t h a t t h e c y c l e S o f c a r d i n a l i t y p i s o b v i o u s l y (p-1)-monomorphic; t hus S s a t i s f i e s t h e hypotheses o f o u r ( 2 ) .

0 ( 2 ) S ince S i s (p-1)-monomorphic, eve ry r e s t r i c t i o n o f S w i t h c a r d i n a l i t y p-1 i s isomorphic t o R ; so t h a t S i s a bound o f R . L e t A be a bound o f S . Then A cannot have c a r d i n a l i t y p + l , by o u r prece-

d i n g (1). Hence A has a t most c a r d i n a l i t y p . Every p roper r e s t r i c t i o n o f A

has a t most c a r d i n a l i t y p -1 and i s embeddable i n S , thus embeddable as w e l l i n R , s i n c e S i s (p-1)-monomorphic. F i n a l l y A i s non-embeddable i n R , s i n c e it i s non-embeddable i n S . Thus A i s a bound o f R . 0

The p reced ing s tatement (2 ) completes 1.2. (4) , by g i v i n g an exemple o f R < S , such t h a t t h e s e t o f bounds o f S

o f R , w i t h S f i n i t e . T h i s answers a problem o f GILLAM 1974. Fo r t h e case where R i s f i n i t e , S i n f i n i t e and R < S , w i t h t h e same p roper

i n c l u s i o n o f t h e s e t o f bounds, we have a l r e a d y cons ide red a f i n i t e cha in R and an i n f i n i t e cha in S . F o r R and S i n f i n i t e , we have g i v e n i n 1.1 a counterexample w i t h R = c h a i n o f t h e n a t u r a l numbers and S w i t h o u t any bound, so t h a t t h e empty s e t o f bounds

o f S i s p r o p e r l y i n c l u d e d i n t h e s e t o f bounds o f R ; y e t R non-embeddable i n S .

i s p r o p e r l y i n c l u d e d i n t h e s e t o f bounds

-

1 . 5 . L e t R, S be two r e l a t i o n s o f t h e same a r i t y . Then t h e f o l l o w i n g t h r e e cond i -

t i o n s a r e e q u i v a l e n t . (1) Every f i n i t e r e s t r i c t i o n o f R i s embeddable i n S ; i n o t h e r words, R

younger than S . ( 2 ) No bound o f S i s embeddable i n R . (3 ) Every bound o f S admi ts an embedding o f a bound o f R . 0 Assume t h e f i r s t c o n d i t i o n and l e t A be a bound o f S . I f A,< R , then a l s o

A S S , hence A i s n o t a bound o f S . Conversely, i f t h e r e e x i s t s a f i n i t e res - t r i c t i o n A o f R which i s non-embeddable i n S , then t h e r e e x i s t s a r e s t r i c -

t i o n o f A , hence o f R , wh ich i s a bound o f S . Thus ( 1 ) and ( 2 ) a r e e q u i v a l e n t .

Assume c o n d i t i o n ( Z ) , and l e t A be a bound o f S ; hence A i s non-embeddable i n R . Thus t h e r e e x i s t s a r e s t r i c t i o n o f A which i s a bound o f R . Conversely, i f t h e r e e x i s t s a bound A o f S which i s embeddable i n R , then A admi ts no

embedding o f any bound o f R . Thus ( 2 ) and (3) a r e e q u i v a l e n t . 0

I n p a r t i c u l a r , i f R and S a r e f i n i t e , t hen t h e embeddab i l i t y R < S i s equiva-

l e n t t o t h e c o n d i t i o n t h a t no bound o f S i s embeddable i n R ( t h i s i s a l r e a d y i n l . l . ( Z ) ) , o r aga in e q u i v a l e n t t o t h e c o n d i t i o n t h a t eve ry bound o f S admi ts

an embedding o f a bound o f R . Another consequence:

L e t R, S be two r e l a t i o n s o f t h e same a r i t y . Then R and S have t h e same age: i f f R and S have t h e same bounds;

i f f no bound o f R i s embeddable i n S , n o r i s any bound o f S embeddable i n R . I n p a r t i c u l a r , i f R and S a r e f i n i t e , t hen R and S a r e isomorphic i f f they

have same bounds; i f f no bound o f R i s embeddable i n S n o r i s any bound o f S

embeddable i n R . Now we can achieve t h e re f i nemen t o f 1 . 2 . ( 4 ) as f o l l o w s .

I f R and S a r e i n f i n i t e and R < S , then t h e s e t o f bounds o f R cannot be p r o p e r l y i n c l u d e d i n t h e s e t o f bounds o f S . 0 L e t A be a bound o f S which i s n o t a bound o f R . By 1 . 5 . ( 1 ) and ( 3 ) , t h e r e e x i s t s a r e s t r i c t i o n B o f A which i s a bound o f R . N e c e s s a r i l y B i s a p roper r e s t r i c t i o n o f A , so t h a t B i s embeddable i n S and hence B cannot

be a bound o f S . 0

1 .6 . Given two r e l a t i o n s R, S o f t h e same a r i t y , i f eve ry f i n i t e r e s t r i c t i o n o f

R i s embeddable i n S , then e v e r y bound of R i s a bound o f S o r i s embedda- b l e i n S . 0 L e t A be a bound o f R . Assume t h a t A i s non-embeddable i n S . Yet eve ry

p roper r e s t r i c t i o n o f A i s embeddable i n R , hence i n S . Thus A i s a bound

o f s . 0

Chapter 12 355

The converse i s f a l s e . Take R t o be an i n f i n i t e chain, and S t o be the chain of ca rd ina l i t y 3 . Then every bound of R i s one of the four r e l a t ions w i t h cardina- l i t i e s l , 2 , 3 considered a t the beginning of the present 5 l . Hence every bound of R i s a bound of S . Yet the chain of ca rd ina l i t y 4 i s a bound of S and not a bound of R . Another example w i t h R and S f i n i t e . Take R t o be the binary re f lex ive cycle of ca rd ina l i t y 3 . There e x i s t four bounds of R , u p t o isomorphism. These are: the binary r e l a t ion of ca rd ina l i t y 1 and value ( - ) ; t he binary r e l a t ion always (+) of ca rd ina l i t y 2 ; the i d e n t i t y r e l a t ion of ca rd ina l i t y 2 ; and f i n a l l y the chain of ca rd ina l i t y 3 . Take S t o be the common extension of these four bounds, taken w i t h d i s j o i n t bases, S taking the value (t) f o r every ordered p a i r whose terms belong t o the bases of two d i s t i n c t bounds. Then each bound of R i s embeddable i n S , y e t R i s non-embeddable i n S . The preceding example can be modified so as t o make bounds, a binary i n f i n i t e r e l a t ion always (t). Even we can make both R and S

i n f i n i t e : replace R by i t s extension t o an i n f i n i t e base, w i t h the value (+) f o r a l l new ordered pa i r s : then the re remain f i n i t e l y many bounds of R and the preceding argument holds.

S i n f i n i t e : add t o our four

1 .7 . We have already sa id t h a t f o r a r e l a t ion with f i n i t e ca rd ina l i t y p , the bounds have a t most ca rd ina l i t y p+ l . For example a chain of ca rd ina l i t y p admits as a bound the chain of ca rd ina l i t y p + l . Let us give examples where the maximum ca rd ina l i t y of bounds i s a t most Take the unary r e l a t ion of ca rd ina l i t y p+q , which takes the value (+) on p e le - ments and ( - ) on q ca rd ina l i t y p+l , and the unary r e l a t ion always ( - ) of ca rd ina l i t y q+l . Take the binary cycle of ca rd ina l i t y p . This i s a (p-1)-monomorphic r e l a t ion . Moreover by 1 . 4 . ( 1 ) , the maximum possible ca rd ina l i t y of bounds i s number i s taken on by the consecut iv i ty r e l a t ion associated w i t h a chain of cardi- n a l i t y p : t h i s i s a bound of the given binary cyc le . Take the p a r t i a l ordering formed of p component cha ins , each of ca rd ina l i t y q ( p , q pos i t ive in t ege r s ) . Then as bounds, we have the chain of ca rd ina l i t y q+l , the i d e n t i t y of ca rd ina l i t y p + l , the r e l a t ion of ca rd ina l i t y 1 and value ( - ) , the r e l a t i o n of ca rd ina l i t y 2 always (+ ) . Fina l ly , assuming t h a t p , q 2 , we have the r e f l ex ive cycle of ca rd ina l i t y 3 ; and the th ree r e l a t ions of ca rd ina l i t y 3 , each having as proper r e s t r i c t i o n s , an iden t i ty and two chains of ca rd ina l i t y 2: i n a pos i t ion of convergence, divergence, o r succession. For p = q , the cardina- l i t y of the base i s p2 b u t the maximum ca rd ina l i t y of bounds is p + l (example due t o LOPEZ). A refinement of th i s example i s obtained by replacing each element by an equivalence c l a s s of f i n i t e ca rd ina l i t y r ; hence transforming the p a r t i a l ordering i n t o

p .

elements. The bounds a re the unary r e l a t ion always (+) of

p . This


a pre-ordering. For p = q = r , the ca rd ina l i t y of the base i s mum ca rd ina l i t y of bounds i s p+ l . A second refinement i s obtained by repeating the pre-ordering and making i t i r r e - f l ex ive ( w i t h the value ( - ) on the diagonal). Then take two d i s j o i n t bases f o r the re f lex ive and the i r r e f l e x i v e pre-orderings, and t h e i r extension t o the union of the two bases, with f o r example the value (+) f o r ordered pa i r s whose terms belong t o d i f f e r e n t bases. Now the new base has ca rd ina l i t y ca rd ina l i t y of bounds remaining p + l (example due t o POUZET 1978).

Problem. Obtain a base of f i n i t e ca rd ina l i t y s t r i c t l y l a rge r than the maximum ca rd ina l i t y of bounds i s a t most p + l .

p3 , b u t the maxi-

Z.p3 , the maximum

Z.p3 , while

1 .8 . A b i r e l a t ion - nent R has i n f i n i t e l y many bounds, and the second component S has f i n i t e l y many bounds.

Take S t o be the chain of the natural numbers, and take R t o be the conse- c u t i v i t y r e l a t ion assoc ia ted with S : hence R(x,y) = + i f f y = x + 1 . Then R has as bounds a l l f i n i t e binary cyc les . However, a bound of RS has ca rd ina l i t y a t most 3 ; so t h a t RS has f i n i t e l y many bounds. Indeed l e t A , B be two f i n i t e r e l a t ions such t h a t every r e s t r i c t i o n of AB w i t h c a r d i n a l i t y 3 i s embeddable i n RS . Then A i s a f i n i t e chain; and B takes the value (+) only f o r ordered p a i r s of consecutive elements (mod A ) . T h u s AB i s embeddable in RS ; hence AB cannot be a bound.

Even more curious i s the f a c t t h a t t he re e x i s t two denumerable binary r e l a t ions R , S each having i n f i n i t e l y many bounds; and y e t t h e i r concatenation RS has only f i n i t e l y many bounds (POUZET 1978, unpublished).

For each f i n i t e set F , consider a l l b i r e l a t ions A8 based on F , such t h a t f o r a l l x , y d i s t i n c t , i f A(x,y) # A(y,x) then B ( x , x ) # B(y,y) ; and s imi la r - l y i f B(x,y) # B(y,x) then A ( x , x ) # A(y,y) . Let 'f be the c l a s s of these b i r e l a t ions . For any two f i n i t e d i s j o i n t s e t s t o , one w i t h base F and the o ther w i t h base G , always admit a common extension w i t h base F u G , again belonging t o f . To see t h i s , f o r each element x in F and each y i n G , take the same value

A(x,y) = A(y,x) , and s imi l a r ly B(x,y') = B(y,x) , which i s compatible w i t h the values A ( x , x ) , A(y,y), B ( x , x ) , B(y,y) already imposed by the preceding. Consequently, t he re e x i s t s a b i r e l a t ion RS with denumerable base, i n which every b i r e l a t ion AB of the c l a s s i s embeddable. Moreover, every bound of RS has ca rd ina l i t y a t most 2 . Indeed, given a b i r e l a t ion AB of f i n i t e cardina- l i t y g rea t e r than o r equal t o 3, i f a l l i t s r e s t r i c t i o n s of ca rd ina l i t y 2 belong t o e , then AB i t s e l f belongs t o , and hence i s embeddable i n RS .

RS can have f i n i t e l y many bounds, even though the f i r s t compo-

-

F, G , two b i r e l a t ions belonging

Chapter 12 351

However, because o f the s ign change i n S(x,x) when we pass from an element x t o an element y w i t h R(x,y) # R(y,x) , any f i n i t e cyc le which i s a r e s t r i c t i o n o f R has even c a r d i n a l i t y . Thus R has i n f i n i t e l y many bounds, which are a l l b inary cyc les o f odd c a r d i n a l i t y , w i t h a r b i t r a r y values f o r ,R(x,x) , f o r each element x i n such a cycle. S i m i l a r l y f o r S . 0

1.9. Le t R, S be two re la t i ons , each o f which i s f r e e l y i n te rp re tab le i n the other. Then t o each bound A of R , o f c a r d i n a l i t y s t r i c t l y l a r g e r than the maximum o f the a r i t i e s o f R and S , we can b i j e c t i v e l y associate a bound o f S having the same base as A (POUZET 1978, unpublished).

0 L e t 'j, be a f ree operator t ak ing R i n t o S , and 3 be a f ree operator t ak ing S i n t o R : see ch.9 5 3.1. L e t B = ( A ) . Then every proper r e s t r i c - t i o n o f B i s embeddable i n 9 ( R ) = S . Moreover d, (B) = A , since both have the same proper r e s t r i c t i o n s , hence the same r e s t r i c t i o n s o f c a r d i n a l i t i e s less than o r equal t o the a r i t y o f A . Hence B i s no t embeddable i n S ; f o r otherwise 2 (8) = A would be embeddable i n &?, ( S ) = R . 0

-

1.10. (1) A f i n i t e 0-ary r e l a t i o n A can be a bound i f f A i s non-empty. However, i f p = Card A , then A i s a bound o f R i f f Card R = p-1 . (2) For a r i t y 1 , a f i n i t e unary r e l a t i o n A can be a bound i f f A i s non-empty and i s always (+) o r always ( - ) . Indeed i f p = Card A and i f A i s always (+) f o r instance, then any unary r e l a t i o n ( f i n i t e o r i n f i n i t e ) which takes p-1 the value (+) admits A as a bound. (3 ) For a r i t y n + 2 , a f i n i t e n-ary r e l a t i o n A can be a bound i f f A i s non-empty. Moreover there e x i s t s an i n f i n i t e r e l a t i o n R such t h a t A i s a bound o f R . Indeed the se t o f f i n i t e re la t i ons i n which A i s no t embeddable i s an i n f i n i t e age: see ch.10 5 2.3, using ch.5 5 1.3.(2).

times

5 2 - PRESERVATION OF BOUNDS I N THE PASSAGE FROM A RELATION TO SOME OF ITS EXTENSIONS

Note f i r s t t h a t t he no t i on o f bound i s immediately ex tend ib le t o mu l t i r e la t i ons .

2.1. L e t R be a f i n i t e m u l t i r e l a t i o n and U a bound o f R . Then there e x i s t s an extension o f R t o i t s base augmented by one element, f o r which U i s again a bound. 0 Suppose f i r s t t h a t R i s a unary r e l a t i o n o f c a r d i n a l i t y p+q , tak ing the value (+) on p elements and (-) on q elements. Then U i s , f o r example, the r e l a t i o n always (+) o f c a r d i n a l i t y p+ l . We replace R by i t s extension which takes the value (+) on p elements and (-) on q+l elements.


The reader will immediately extend th i s procedure t o the case of a unary multirelation, i . e . the concatenation of f i n i t e l y many unary relations with a common base. Suppose now that R has a t l eas t one component with a r i ty >, 2 . We shall argue in the case that R being an immediate extension of our argument. Let U be a bound of R . Add to the base of R a new element a , and l e t R a be the extension of R defined by Ra(a ,x ) = Ra(x,a) = Ra(a,a) = + for every x in the base I R 1 . Similarly add b and define Rb by the analogous cond'i- tions w i t h ( - ) instead of (+) . Then U i s e i ther a bound of Ra or a bound of R b . Indeed every proper res t r ic t ion of U i s embeddable in R , hence in Ra and in U i s not embeddable in bo th : use the argument in ch.5 5 1.3.(1) ( fa i thful extension). 0

i s i t s e l f a binary relat ion, the proof in the general case

R b . On the other hand,

2 . 2 . Consider a f i n i t e s e t of f i n i t e relations A1,...,Ah a l l of the same a r i t y , and suppose t h a t there ex is t relations of the same a r i t y , with a rb i t ra r i ly large f i n i t e cardinal i ty , each having the bounds and possibly other bounds. Then there exis ts a denumerable relation having, among other ones, the bounds A1 ,. . . ,Ah . 0 Let R i ( i integer) are s t r i c t l y increasing, such that each Ri A1, ..., Ah . Let pi denote the cardinality of R i ; we can assume that R i has base { 1,2,. . . . . . , p i 1 . Moreover, since for each admits an embedding of a l l proper res t r ic t ions of A1 fo r instance, then le t t ing kl denote the sum of the cardinal i t ies of these proper res t r ic t ions , we can suppose that they are a l l embeddable in the rest r ic t ion of t o i1.2 ,..., kl) . Similarly for A2 , l e t

k2 denote the sum of the cardinal i t ies of a l l proper res t r ic t ions of A2 : they are a l l embeddable in the rest r ic t ion of R i t o { 1 , 2 ,..., ktd; and so for th . There exis ts an inf in i te sequence, extracted from the sequence of the Ri , which i s formed of relations having the same restr ic t ion S1 t o the singleton of 1 . From th is f i r s t extracted sequence, we extract a second sequence, formed of relations a l l having the same restr ic t ion S 2 t o the pair 1,2) . I terat ing t h i s , we obtain, for each integer r , a relation S r based on { 1,2, ..., r} , where each Sr ( r + 2 ) i s an extension of Sr-l . Let S denote the common extension of the Sr , based on a l l positive integers. Then A1 fo r instance, i s a bound of S . Indeed A1 i s n o t embeddable in S , since otherwise i t would be embeddable in SOM S r , hence in some each proper res t r ic t ion of A1 i s embeddable in S , since i t i s embeddable in the rest r ic t ion of each Ri t o { 1 , 2 ,..., kl) , hence in S(kl) . Same argument

for A 2 , . . . ,Ah which are thus also bounds of S . 0

A1, ..., Ah

be an &-sequence of f i n i t e re la t ions, whose cardinal i t ies has a t least the bounds

i , the relation Ri

R i

R i . Moreover,

Chapter 12 359

2.3. Consider again the f i n i t e relations p , there ex is t s a relation with cardinal i tygreater than or equal t o p , whose bounds are exactly Then there ex is t s a denumerable relation whose bounds are exactly A1, ..., Ah (suggested by PABION in 1970).

0 Let Ri sing values of p . We shall modify our construction in the preceding 2 . 2 , as follows. Take a sequence of a l l the f i n i t e relations U . ( j positive integer) with the same ar i ty as the Ri , and l e t k br the f i n i t e cardinality of U

Replacing, i f necessary, each Ri following. For each i , i f U1 i s embeddable in Ri , then U1 i s embeddable in the rest r ic t ion of Ri t o { 1 , 2 , ..., k l ) . Again for each i , i f U 2 i s embeddable in Ri , then U 2 i s embeddable in the rest r ic t ion of Ri t o the se t {1 ,2 ,... , k l + k 2 ) ; and so forth.

Now, construct relations Sr ( r integer) as in the preceding 2 . 2 , and then take the i r common extension S . Then A 1 , . . . , A h are a l l bounds of S . I t remains to prove that S has no other bound. Suppose tha t B i s a bound of S different from A 1 , . . . , A h . Then f i r s t l y , each proper res t r ic t ion of B i s embeddable i n S , hence in the Sr for a l l r greater than some r ( 0 ) ; hence in a l l the Ri which extend S Secondly B cannot be a bound of R i , for i suff ic ient ly large, so t h a t the integer p associated by hypothesis with Ri i s larger than the cardinality

and whose of B . Hence B i s embeddable in a l l the Ri which extend S index i i s suff ic ient ly large. Finally there ex is t s r(1) >/ r (0) such t h a t B i s embeddable in a l l those Ri which extend S From the f i r s t paragraph, there exis ts an integer k for which, i f B i s embeddable in R i , then B i s s t i l l embeddable in the rest r ic t ion Ri/{1,2 ,..., k ) . Hence B i s embeddable in Sm , where m i s the maximum of k and r ( 1 ) . Thus B i s embeddable in S : contradiction. 0

A1, ... ,Ah ; suppose t h a t fo r each integer

A1, . . . ,Ah plus possibly some bounds of cardinal i t ies > p .

( i positive integer) be our f i n i t e re la t ions, which are l is ted by increa-

3

j ' j by an isomorphic copy, we can suppose the

r(O) .

r(O)

r(1) .

§ 3 - WELL MULTIRELATION: REASSEMBLING THEOREM (FRASNAY)

We say that a multirelation when par t ia l ly ordered under embeddability, forms a well partial ordering. In other words, i f any s e t of f i n i t e res t r ic t ions of R , mutually incomparable under embeddability, i s f i n i t e . For example, every chain i s well. Every t ree i s well, by KRUSKAL's theorem (ch.5 5 2.3). The consecutivity relation on the natural numbers i s well. Indeed, each f i n i t e res t r ic t ion can be represented by a f i n i t e sequence of positive integers, each

R i s &, i f the s e t of i t s f i n i t e res t r ic t ions,


integer i representing a component of i consecutive integers. Then the embedda- b i l i t y between two f i n i t e sequences of integers, implies the embeddability between the two corresponding f i n i t e res t r ic t ions. Now i t suffices t o recall t h a t embeddabi- l i t y between f i n i t e sequences, or words of integers, i s a well partial ordering by HIGMAN's theorem ch.4 5 4.4.

Given a natural number p , we say t h a t a multirelation R i s p - e , i f upon concatenating R with any p unary relations with the same base, we obtain a well mu1 t i re1 ation.

3.1. rf R 2 p-well, then every multirelation freely interpretable in. R &

0 Let S be freely interpretable in R , and l e t F, F ' be two f i n i t e subsets of the base. I f S/F augmented by p unary relations A based on F , and S/F'

augmented by p unary relations A ' on F ' , are incomparable with respect t o embeddability, then the same i s true for R/F augmented by the A and R/F'

augmented by the A ' . The consecutivity relation on the natural 0 For each integer i , take a sequence of i+2 consecutive elements, and define a unary relation t o take the value ( - ) for the f i r s t and the l a s t element, and the value (+ ) between. 0

Problems communicated by POUZET in 1972. (1) If a multirelation i s 1-well, then i s i t 2-well, and even p-well for every integer p . ( 2 ) S t a r t with a multirelation R . Take the concatenations RX with X an arbi- t rary unary relation; then the union of the ages of these concatenations, where R

i s fixed and X varies. This union i s not in general an age. If th i s union i s well partial ordered under embeddability, then R i s 1-well. Is the converse t rue. Same question for 2 , 3 , ... unary relations added.

p - w e l l .

numbers i s well, b u t i s n o t 1-well.

3.2. Every chain, and consequently every chainable multirelation, i s p-well for each integer p (POUZET 1972) .

0 I t suffices t o see that the multirelations ( A , B l , . . . , B p ) , where A i s a f i n i t e chain and the i s well par t ia l ly ordered under embeddability. To each multirelation, associate the f i n i t e cardinality h associate a word of length h , obtained by replacing each i = 1, ... ,h by the sequence of values B1(x) , .. . , Bp(x) , where x designates the i th element of the base, ordered modulo A . We say t h a t two of these sequences are considered t o be incomparable i f f they are d is t inc t . T h u s we have a well partial ordering of

ident i ty , with 2p elements, or sequences, mutually incomparable. By HIGMAN's

B are unary relat ions, form, up t o isomorphism, a se t which

of i t s base. Then

Chapter 12 36 1

theorem (ch.4 5 4.4 f i n i t e case, p rovab le i n ZF), t h e s e t o f words formed o f t h e

p reced ing elements, c o n s t i t u t e s a w e l l p a r t i a l o r d e r i n g , under t h e usual embeddabi- l i t y o f words. Th is g i ves a w e l l p a r t i a l o r d e r i n g , under embeddab i l i t y , o f o u r

f i n i t e m u l t i r e l a t i o n s . 0

3.3. THEOREMS ON FINITE NUMBER OF BOUNDS L e t R be a m u l t i r e l a t i o n ; denote by m t h e maximum o f i t s components' a r i t i e s .

- If R g 2m-well, t hen R has f i n i t e l y many bounds (POUZET 1972).

Consequently eve ry cha inab le m u l t i r e l a t i o n has f i n i t e l y many bounds (FRASNAY 1965)

0 L e t U be a bound o f R and E be t h e base o f U . Take an a r b i t r a r y element al i n E , and l e t El = E - {al) . Consider those subsets F o f El f o r which t h e r e e x i s t s an S < R w i t h base E , which c o i n c i d e s w i t h U on El and on

Note t h a t when t h e a r i t y m = 1 , t h e n no subset F , even empty, s a t i s f i e s t h i s c o n d i t i o n . F o r o the rw ise S would be i d e n t i c a l w i t h U , hence U& R . I f t h e r e e x i s t s no such F , then a s s o c i a t e t o t h e bound U t h e f o l l o w i n g m u l t i - r e l a t i o n : (S,V1 ,..., Vm,A1 ,..., A,,,) where S,< R and S c o i n c i d e s w i t h U on

El ; where t h e V and t h e A a r e unary, V1 t a k i n g t h e va lue (+) on El and

o n l y on El , and A1 b e i n g t h e s i n g l e t o n unary r e l a t i o n o f al ; t h e o t h e r V and A t a k i n g always t h e va lue ( - ) . Now suppose t h a t t h e r e e x i s t subsets s a t i s f y i n g o u r p reced ing c o n d i t i o n . Take

such an F o f maximum c a r d i n a l i t y , say E2 . Th is E2 i s a p roper subset o f El ;

f o r i f E2 = El , then we would have U,< R . L e t a2 be an a r b i t r a r y element o f

El - E2 . Consider those subsets F o f E2 f o r which t h e r e e x i s t s an S,< R w i t h base E , which c o i n c i d e s w i t h U on El and on E2 u {al\ and on Fu{al,a2). Note t h a t when t h e a r i t y m = 2 , then no such F , even empty, e x i s t s . F o r o t h e r -

wise, t h i s would y i e l d t h a t S c o i n c i d e s w i th U on t h e union E2u{al,a2) , c o n t r a d i c t i n g t h e m a x i m a l i t y o f

I f t h e r e e x i s t s no such F , then a s s o c i a t e t o t h e bound U t h e f o l l o w i n g m u l t i - r e l a t i o n : (S,V1 ,..., Vm,A1 ,..., Am) where S 4 R and S co inc ides w i t h U on

El and on E2u{al\ ; where V1 takes t h e va lue (+) o n l y on El , and V 2 takes

(+) o n l y on E2u{al\ ; t h e r e l a t i o n A1 i s t h e s i n g l e t o n o f al and A2 i s t h e

s i n g l e t o n o f a2 , and f i n a l l y t h e o t h e r V and A a r e always ( - ) . I t e r a t i n g t h i s , we o b t a i n , a t most, a s t r i c t l y descending sequence E 1 3 E2 3 ... ... 3 Em and elements al i n E-El , a2 i n El-E2 , ... , a, i n Emm1-E, . Thus t h e r e e x i s t s an S,< R w i t h base E , which c o i n c i d e s w i t h U on E, , on

F u t a l i .

F

E2 .

E2 \al) , on E3 u {al,a2) , .. . , on E m u {al ,... ,am-1)- . Yet no m u l t i r e l a - t i o n S o f maximum a r i t y m can s a t i s f y t h e p reced ing c o n d i t i o n s , and moreover


c o i n c i d e w i t h U on { a l,...,am). F o r o the rw ise S would c o i n c i d e w i t h U on

Em u {al,. .. ,a,,,) , c o n t r a d i c t i n g t h e m a x i m a l i t y o f

Associate t o t h e bound U t h e m u l t i r e l a t i o n (S.V1 ,..., Vm,A1 ,... ,Am) where S d R

and S c o i n c i d e s w i t h U on El , on E2 V i a l ) , ... , on Emu{al ,..., am-l) , t h e r e l a t i o n V1 t a k i n g t h e va lue (+) o n l y on El , and V 2 o n l y on E2 u {al) ,

... , and Vm o n l y on Emu{al, ...,am-1} ; and f i n a l l y A1 b e i n g t h e s i n g l e t o n

r e l a t i o n o f al , ... , and Am t h e s i n g l e t o n r e l a t i o n o f am . I n eve ry case, we see t h a t t h e r e e x i s t s an S,(R c o i n c i d i n g w i t h U when V1 o r

V 2 o r . .. o r Vm takes t h e va lue (+) . Yet i t i s imposs ib le t h a t i n a d d i t i o n , S

c o i n c i d e s w i t h U on t h e s e t o f t h e elements g i v i n g t h e va lue (+) t o A1 o r Ap

o r ... o r Am . Suppose now t h a t R i s Pm-well. Then t h e m u l t i r e l a t i o n s (S,V,A) p r e v i o u s l y asso- c i a t e d t o t h e bounds, f o rm a w e l l p a r t i a l o r d e r i n g under embeddab i l i t y . Suopose t h a t t h e r e e x i s t i n f i n i t e l y many bounds. Then because o f t h i s w e l l p a r t i a l orde- r i n g and by RAMSEY's theorem, t h e r e e x i s t s an w - s e q u e n c e o f m u l t i r e l a t i o n s

(S,V,A) embedding o f each p reced ing one. More p r e c i s e l y , we can assume t h a t each (S,V,A) i s an e x t e n s i o n o f t h e p reced ing . Then t h e r e a r e a t most m e lements f o r which

A1 takes t h e va lue (+). So t h e r e e x i s t s an w -sequence e x t r a c t e d

f rom t h e p reced ing one, f o r which a l l t h e bounds U have t h e same r e s t r i c t i o n t o these elements. L e t U be one o f these bounds, E i t s base, and l e t U ' be ano the r bound whose base p r o p e r l y i n c l u d e s E . Then U ' / E 6 R . L e t (S,V1,. . . ,Vm,A1,.. . ,Am) d e s i -

gnate t h e m u l t i r e l a t i o n a s s o c i a t e d w i t h U , and s i m i l a r l y (S ' ,V i , ..., Vh,A i , ... ..., A;) t h e m u l t i r e l a t i o n a s s o c i a t e d w i t h U ' : t h e second m u l t i r e l a t i o n extends t h e f i r s t one. The r e s t r i c t i o n U ' / E c o i n c i d e s w i t h S ' , hence w i t h S and U , on t h e s e t where V1 takes t h e va lue (+) , on t h e s e t where V 2 takes (+), ... , on t h e s e t where Vm takes (+). Moreover, U ' / E c o i n c i d e s w i t h U on t h e s e t

o f those elements g i v i n g t h e va lue (+) t o A1 o r A2 o r ... o r Am : c o n t r a - d i c t i o n . 0

Em .

assoc ia ted t o t h e bounds, and such t h a t each m u l t i r e l a t i o n admi ts an

o r ... o r Am

3.4. REASSEMBLING THEOREM

L e t G be a group and m i t s a r i t y . There e x i s t s an i n t e g e r n 3 m s a t i s f y i n g t h e f o l l o w i n g .

Given a s e t E o f c a r d i n a l i t y >/ n ( f i n i t e o r i n f i n i t e ) ; and f o r each n-element

subset F o f E , g i v e n a c h a i n AF w i t h base F , where t h e AF a r e m u t u a l l y G-compatible; t hen t h e r e e x i s t s a c h a i n based on E , which i s G-compatible w i t h

a l l t h e AF (FRASNAY 1965; t h e a r i t y o f a group and t h e G - c o m p a t i b i l i t y a r e

d e f i n e d i n ch.11 0 3; o u r p r o o f uses u l t r a f i l t e r axiom; ZF s u f f i c e s i f E coun tab le ) .

Chapter 12 363

0 Suppose f i r s t t h a t E i s f i n i t e . L e t A be t h e c h a i n o f t h e n a t u r a l numbers,

and d e f i n e t h e m-ary r e l a t i o n R on these i n t e g e r s , by s e t t i n g R(xl ,. . . ,xm) = + i f f e i t h e r a t l e a s t two o f t h e x a re i d e n t i c a l , o r i f t h e x a r e a l l d i s t i n c t

and t h e pe rmuta t i on which r e o r d e r s t h e sequence belongs t o t h e group G . C l e a r l y eve ry l o c a l automorphism o f t h e c h a i n A i s a l o c a l automorphism o f R ; i n o t h e r words R i s f r e e l y i n t e r p r e t a b l e i n A , hence

R i s A-chainable. Using 3.2 above, we see t h a t R i s p -we l l f o r each i n t e g e r

p ; and by t h e p reced ing theorem 3.3, R has f i n i t e l y many bounds.

L e t n a m be s t r i c t l y g r e a t e r t han t h e c a r d i n a l i t i e s o f bounds o f R . We assume t h a t t h e f i n i t e s e t E has c a r d i n a l i t y a t l e a s t n . F o r each n-element subset F o f E , l e t AF be a c h a i n on F , where t h e AF a r e m u t u a l l y G-compatible

L e t S be t h e m-ary r e l a t i o n based on E , such t h a t S(xl, ..., xm) = + i f f e i t h e r a t l e a s t two x a re i d e n t i c a l , o r i f t h e xl, ... ,xm a r e a l l d i s t i n c t and t h e pe rmuta t i on which r e o r d e r s t h i s sequence acco rd ing t o t h e c h a i n AF , belongs t o t h e group G ; where F des igna tes an a r b i t r a r y n-element subset o f E con- t a i n i n g xl, ... ,xm . Because o f t h e G - c o m p a t i b i l i t y o f t h e AF f o r any two F , t h e va lue o f S does n o t depend on t h e chosen n-element F . F o r each F , t h e r e s t r i c t i o n S/F i s embeddable i n R : i t s u f f i c e s t o take an

isomorphism f rom AF o n t o a r e s t r i c t i o n o f A . The c a r d i n a l i t y o f each bound o f R i s l e s s than n , so no bound o f R i s embeddable i n S : o the rw ise , such a

bound would be embeddable i n a r e s t r i c t i o n S/F , hence i n R . It f o l l o w s t h a t S i s embeddable i n R . For o the rw ise some r e s t r i c t i o n o f S

would be a bound o f R . L e t f be an isomorphism f rom S o n t o a r e s t r i c t i o n o f R . The i n v e r s e f u n c t i o n f - l takes a r e s t r i c t i o n o f t h e c h a i n A i n t o a

c h a i n B , w i t h t h e r e s u l t t h a t S(xl, ..., x ) = + i f f e i t h e r two x a r e i d e n t i - c a l , o r i f a l l t h e x a r e d i s t i n c t and t h e pe rmuta t i on which r e o r d e r s these x

acco rd ing t h e c h a i n B , belongs t o G . It f o l l o w s t h a t B i s G-compatible

w i t h AF f o r each n-element subset F o f E . The p r o o f i s now achieved f o r

t h e case t h a t E i s f i n i t e . Now suppose E i n f i n i t e . Fo r each f i n i t e subset D o f E , t h e r e e x i s t s a non-empty s e t UD o f cha ins based on D , each b e i n g G-compatible w i t h t h e g i v e n chains, s t i l l denoted by AF . F o r a subset D ' o f D , every c h a i n be lon-

g i n g t o UD , when r e s t r i c t e d t o D ' , g ives an element o f UD, . By t h e coherence lemma (ch.2 0 1.3, e q u i v a l e n t t o t h e u l t r a f i l t e r axiom), t h e r e e x i s t s a c h a i n

based on E , whose r e s t r i c t i o n t o each f i n i t e s e t D be longs t o UD . T h i s cha in i s G-compatible w i th a l l t h e AF . 0 3.5. CHAINABILITY THEOREM

L e t m be a p o s i t i v e i n t e g e r . There e x i s t s an i n t e g e r p a m such t h a t e x m-ary r e l a t i o n w i t h c a r d i n a l i t y >/ p , which i s (6 p)-monomorphic, i s cha inab le (FRASNAY 1965; uses u l t r a f i l t e r axiom; ZF s u f f i c e s f o r a coun tab le r e l a t i o n ) .

xl,. . . ,xm i n i n c r e a s i n g o r d e r

m


0 Consider a l l groups of a r i ty m , and l e t n be the maximum of the integers associated t o these groups in the preceding reassembling theorem. By ch.9 5 5 .5 . (2 ) ,

there exis ts an integer pa n such that every m-ary relat ion with cardinality

Let R be a (4 p)-monomorphic m-ary relation with base E o f cardinality a t least equal t o p , Then a l l the rest r ic t ions o f R with cardinality n are isomorphic, hence they are a l l chainable. To each n-element subset F of E , associate a chain AF based on F , such that the rest r ic t ion R/F i s AF-chainable. Moreover, for any two n-element subsets F , F ' of E , take A F , t o be the image of AF under one of the isomorphisms of R/F on to R/F ' . Let H be an m-element subset of the base E , and l e t F , F ' be two n-element subsets, each of which includes H . The permutation of H which takes AF/H in to AF,/H i s an automorphism of R/H . Indeed, take the image H ' of H under the isomorphism from AF on to A F , ; then H ' in to H by preserving the order of elements (mod A F , ) and using chainability by Designate each element o f H phisms of R / H becomes an m-ary permutation group. The preceding isomorphisms show t h a t G depends neither on H nor on the choice of the n-element s e t F including H . By the preceding, A F and A F , are G-compatible for any two n-element sets F and F ' . Now apply the reassembling theorem: there exis ts a chain A based on E , which i s G-compatible with every AF . We shall prove t h a t R i s A-chainable. Let H, H ' be two m-element subsets of E : we shall f i r s t prove t h a t the isomorphism from A / H onto A / H ' takes R / H in to R / H ' . We can assume t h a t na 2m , hence t h a t there exis ts an n-element subset F of E including both H and H ' . The desired isomorphism can be obtained by composing three isomorphisms: from A/H on to A F / H , from AF/H on to AF/H' , then from A F / H ' o n t o A / H ' . The f i r s t and the third isomorphisms belong t o G , once each element i s designated by i t s rank (mod AF/H o r AF/H') . These are respectively an automorphism of R / H and an automorphism of R / H ' . The second i s an isomorphism from R / H onto R / H ' by the definition of Now l e t K, K ' be two r-element subsets of E , with r s t r i c t l y less t h a n m . Suppose f i r s t t h a t there exis ts an m-element subset H of E including both K and K ' . Take an n-element s e t F including H ; transform K and K ' by the isomorphism of A/H onto AF/H (which belongs t o G and hence i s an automorphism of R/H ) ; we see that the isomorphism from A / K on to A / K ' takes R / K into R / K ' . If no such m-element s e t H ex is t s , then we take an m-element s e t H including K and another H ' including K ' , and by the isomorphism from A/H on to A / H ' we are in the preceding case. 0

3 p has a chainable res t r ic t ion with cardinality n .

A F , . by i t s rank (mod AF/H) . Then the group o f automor-

AF .

Chapter 12 365

3.6. Let m be an in t ege r , p t he in teger 3 m defined by the preceding proposi- t i o n , and R an m-ary r e l a t ion of ca rd ina l i t y s t r i c t l y g rea t e r than p . A s u f f i c i e n t (and necessary) condition f o r R t o be chainable is t h a t each res- t r i c t i o n of R t o p + l elements i s chainable, o r even i s ( s p)-monomorphic (uses the u l t r a f i l t e r axiom; ZF su f f i ces i f R the preceding proposit ion and from ch.9 § 6.1.

i s countable). This follows from

§ 4 - R E D U C T I O N THRESHOLD, REASSEMBLING THRESHOLD, G - C H A I N

4.1. REDUCTION THRESHOLD Given an m-ary group G of permutations, we def ine the reduction threshold of G , denoted by s(G) , as the l e a s t i n t ege r s such t h a t the d i l a t ed group Gm+s i s ind ica t ive : see ch.11 5 4.5. In p a r t i c u l a r s(G) = 0 i f f G i s ind ica t ive . Given an in t ege r m , we define the m-ary reduction threshold , denoted by s(m) , as being the maximum of the s(G) f o r a l l m-ary groups G . These de f in i t i ons a re due t o FRASNAY 1965, who obta ins t h a t = 0 , and s ( 4 ) = 2 , and f o r m >/ 5 , the inequa l i t i e s ( Ib id . p . 493-494). The upper bound is improved t o s(m) 6 m-3 by HODGES, LACHLAN, SHELAH 1977. F ina l ly i t i s proved by FRASNAY 1984 t h a t s(m) = m-3 f o r m 3 5 . More p rec i se ly , the value s(G) = in-3 i s reached by taking G t o be the group on { 1 , 2 , ... ,m} which preserves the ex t remi t ies 1 and m . Indeed the f i r s t d i l a t ed group 6"'' preserves 1 , 2 and m , m t l ; then Gm+2 preserves 1 ,2 ,3 and m , m + l , m + 2 and f i n a l l y the d i l a t ed group

preserves 1,2, ..., in-2 and m , m + l , ..., 2m-3 , hence i t i s the iden t i ty G2m-3

group, which i s obviously ind ica t ive .

s ( 1 ) = s ( 2 ) = s ( 3 ) 16 s(m) 5 (3m-8) - m + l 2

(again f o r m a 5 )

4.2. REASSEMBLING THRESHOLD Given an m-ary group G , we define the reassembling threshold of G , denoted by t ( G ) , as the l e a s t i n t ege r t such t h a t n = m + t s a t i s f i e s the reassembling theorem 3.4. I f G i s not i nd ica t ive , then t(G)4 s(G) + 1 (FRASNAY 1965 p . 500).

2 However, f o r the ind ica t ive group J4 on {1,2,3,4) generated by (1 ,4 ) , (2 ,3 ) and the two t ranspos i t ions ( 1 , 2 ) and (3 ,4) , we have s = 0 and t = 2 ( Ib id . p . 500). Given an in t ege r m , we define the w a r y reassembling threshold , denoted by t ( m ) , as being the maximum of the t ( G ) f o r a l l m-ary groups G . We have t(1) = 0 , t ( 2 ) = t ( 3 ) = 1 , t ( 4 ) = 2 , and f o r m a 5 we have 14 t ( m ) 4 s(m) + 1 (FRASNAY 1965 p . 500 and JULLIEN 1966). Hence t(m)4 m-2 f o r m a3 , i n view of the preceding improved upper bound of s(m) . Fina l ly , i t is proved by FRASNAY 1984 t h a t t ( m ) = s(m) + 1 = m - 2 f o r m 3 5 , by

t a k i n g aga in t h e group on 41,. . . ,m} which preserves 1 and m . Problem. F o r each group G , do we have t h a t s(G),< t ( G ) .

4.3. (G,A)-CHAIN, G-CHAIN

Consider a c h a i n A w i t h base E , and an m-ary group G . F o l l o w i n g FRASNAY 1973, i n s p i r e d by CLARK, KRAUSS 1970, we d e f i n e t h e (G,A)-* as b e i n g t h e m-ary r e l a t i o n R based on E and A-chainable, such t h a t R(xl, ..., x,) = t i f f

xl,. . . ,xm a re a l l d i s t i n c t and t h e r e e x i s t s a pe rmuta t i on s be long ing t o G

w i t h

I n t h e p a r t i c u l a r case where G i s t h e b i n a r y group which reduces t o t h e i d e n t i t y , t hen we have aga in t h e usual c h a i n A , more p r e c i s e l y t h e s t r i c t c h a i n < (mod A ) . We say t h a t R i s a G-chain i f f t h e r e e x i s t s a c h a i n A such t h a t R i s t h e (G,A)-chain ( G-rangement i n FRASNAY's t e r m i n o l o g y ) .

S t a r t i n g f rom t h e c h a i n w o f t h e i n t e g e r s , c o n s i d e r t h e (G, w ) - c h a i n R and denote by m t h e a r i t y o f G . Then we see t h a t , f o r each n a m , t h e d i l a t e d

group Gn i s t h e group o f automorphisms o f any r e s t r i c t i o n o f R t o n elements ( o b v i o u s l y we r e p l a c e each element by i t s rank , which i s an i n t e g e r ) . Consequently,

Gn becomes an i n d i c a t i v e group f o r n & s ( G ) ( r e d u c t i o n t h r e s h o l d ) .

More g e n e r a l l y , c o n s i d e r an m-ary r e l a t i o n R which i s LJ -chainable; f o r eve ry

n a m , denote by G(n) t h e n -a ry group o f automorphisms o f any r e s t r i c t i o n o f R t o n elements. Then t h e r e e x i s t s an n such t h a t , f rom t h i s p o i n t on, t h e ~ r o u p G(n) i s i n d i c a t i v e (HIGMAN 1977). Problem, posed by FRASNAY 1984. F o r n >/ s(m) , t h e group G(n) i s i t always

i n d i c a t i v e .

modulo A . s (m) x ~ ( ~ ) < x s ( 2 ) < ... < x

4.4. The reassembl ing t h r e s h o l d l e d FRASNAY 1965 (p . 517) t o t h e f o l l o w i n g r e s u l t (l), and t h e n POUZET 1981 (p . 307) t o t h e f o l l o w i n g ( 2 ) and (3). L e t G be an m-ary group. There always e x i s t s an m-ary r e l a t i o n R which i s

f r e e l y i n t e r p r e t a b l e i n t h e c h a i n o o f t h e i n t e g e r s , such t h a t G i s t h e group

o f automorphisms o f t h e r e s t r i c t i o n o f R t o an a r b i t r a r y m-element se t , when ordered by i n c r e a s i n g va lues ( f o r i n s t a n c e we can t a k e R t o be t h e ( G , G , ) -

c h a i n ) . Then d e n o t i n g aga in by t ( G ) t h e reassembl ing t h r e s h o l d :

(1) t h e maximum b(R) o f c a r d i n a l i t i e s o f a l l bounds o f R s a t i s f i e s

b(R) \< m + t ( G ) ; t h e l a t t e r va lue i s reached i f R i s t h e ( G , w ) - c h a i n : see FRASNAY 1973 ; ( 2 ) t h e r e e x i s t s a bound o f R w i t h c a r d i n a l i t y m ; hence b(R) b m ;

(3) e i t h e r b(R) = m+t(G) or b(R) = m and t ( G ) = 1 , hence m+t(G) = mtl . 0 (1) L e t 9 be a f r e e o p e r a t o r which takes w i n t o R : see ch.9 5 3.1. L e t U be an w a r y r e l a t i o n , bound o f R w i t h c a r d i n a l i t y s t r i c t l y g r e a t e r t han

m t t ( G ) . Thus r e s t r i c t i o n s o f U w i t h c a r d i n a l i t y a t most equal t o m + t ( G )

Chapter 12 361

a r e a l l embeddable i n R . T h u s each (m+t(G))-element subset X of the base I U l

i s the base of a chain C x

C x G-compatible. By the reassembling theorem 3.4, there e x i s t s a chain C w i t h base I U I , which i s G-compatible w i t h each Cx . I t follows t h a t U = ( C ) , hence t h a t U i s embeddable i n R , so U i s not a bound: contra- d i c t ion . 0 0 ( 2 ) Take the r e s t r i c t i o n H of R t o t he f i r s t m i n t ege r s . Preserve the values of H f o r each sequence of m terms f o r which a t l e a s t two terms are equal. However, change the value of H when the m terms a re a l l d i s t i n c t , e i t h e r by taking always the value (+) ( i f i t i s not already the case f o r H ) , or by always taking ( - ) . The r e l a t ion H thus modified i s a bound of R w i t h c a rd ina l i t y m . 0

(3) I t su f f i ces t o prove t h a t n = Max(m+l,b(R)) s a t i s f i e s the inequal i ty n 3 m t t ( G ) , hence s a t i s f i e s the reassembling theorem. Take a s e t E of ca rd ina l i t y a t l e a s t equal t o n , i n which each n-element subset X i s the base of a chain DX , where these chains a re assumed t o be mutually G-compatible. Then the image r e l a t ions 7 ( O x )

i . e . they have a common r e s t r i c t i o n t o the in t e r sec t ion of their bases. To see th i s , use the inequal i ty n >/ m + l : we can always pass from an n-element s e t X t o another Y by a f i n i t e number of intermediate n-element s e t s X i , each of which includes the in t e r sec t ion XnY , and each of which has a t l e a s t m common elements with the preceding one. Thus f o r these successive X i , the images P(DX ) , then y(DX) and y(Dy) have a common r e s t r i c t i o n t o the

in t e r sec t ion Xn Y . Under these condi t ions , there e x i s t s a common extension S of these ? ( O x ) . Moreover, each r e s t r i c t i o n of S t o a t most n elements i s the image of a chain under 9 , hence i s embeddable i n R . Additionally n 3 b(R) , hence S admits no embedding of any bound of R . I t follows t h a t S i s i t s e l f embeddable in R . T h u s t he re e x i s t s a chain D w i t h base E , such t h a t S = P(D) . This D i s G-compatible w i t h each DX , thus the reassembling theorem holds. 0

Problem. Existence of a " t e ra to log ica l " w -chainable m-ary r e l a t ion R whose bounds have maximum ca rd ina l i t y b ( R ) = m and y e t w i t h t ( G ) = 1 where G denotes the group of automorphisms of any r e s t r i c t i o n of R t o an m-element s e t .

s a t i s fy ing t o U / X = ? ( C x ) . By our hypotheses, the a re mutually

a re mutually compatible,

i

4 .5 . Given the in t ege r m , the maximum ca rd ina l i t y of the bounds of a l l chainable r e l a t ions with i n f i n i t e base is equal t o the - hold, in i t s complete form m + t ( m )

m - 3 m-ary reassembling thres - -

(FRASNAY 1973; this r e s u l t i s extended

by POUZET 1981 t o chainable and almost chainable mul t i r e l a t ions w i t h maximum a r i t y equal t o m ) . 0 In view of the preceding propos i t ion , i t su f f i ces t o cons t ruc t an R m + l . Take the ( G , d ) - c h a i n where G i s the iden t i ty ; i n o ther words, take the r e l a t ion x l < x2 < . . . 4 xm (mod LS)) . Then consider the binary cycle C w i t h c a rd ina l i t y m+l ; on the base of C , define the m-ary r e l a t ion U which takes the value (+) i f f x1,x2, ..., xm a re consecutive modulo C : t h i s U i s a bound o f R . 0

m-ary r e l a t ion f r ee ly in t e rp re t ab le i n w , which admits a t l e a s t one bound of ca rd ina l i t y

4 .6 . I f an m-ary r e l a t ion R (m+t(m))-monomorphic and i f i t s ca rd ina l i t y i s i n f i n i t e , o r f i n i t e but s u f f i c i e n t l y l a rge , then p . 508 prop. 12.1.1 and POUZET 1981 p . 311 proo. V.3.8). 0 By the f i n i t a r y form of RAMSEY's theorem (ch.3 5 1 . 3 ) , i f the ca rd ina l i t y of the base i s i n f i n i t e o r f i n i t e b u t s u f f i c i e n t l y l a rge , then there e x i s t s a r e s t r i c t i o n of R which is chainable and of ca rd ina l i t y m + t ( m ) . By monomorphism, a l l r e s t r i c t i o n s with ca rd ina l i t y l e s s than o r equal t o m + t ( m ) a r e chainable. S t a r t i ng w i t h such a r e s t r i c t i o n , and extending the chain i n which i t i s f r ee ly in t e rp re t ab le t o the chain w , we obtain an m-ary r e l a t ion S , which i s f r ee ly in t e rp re t ab le i n G, , and a l l of whose r e s t r i c t i o n s ' to a t most m + t ( m ) a r e embeddable in R . We sha l l show t h a t every f i n i t e r e s t r i c t i o n of R i s embeddable i n S , which wi l l prove t h a t these r e s t r i c t i o n s a re chainable, and consequently t h a t R i s i t s e l f chainable. Suppose t h a t every r e s t r i c t i o n of R t o a t most k elements i s embeddable in S , and suppose k >/ m + t ( m ) . I f there e x i s t s a r e s t r i c t i o n with ca rd ina l i t y k + l which i s not embeddable i n S , then t h i s r e s t r i c t i o n i s a bound of S , cont rad ic t ing the preceding proposit ion. 0

R is chainable (FRASNAY 1965

elements

§ 5 - MONOMORPHISM THRESHOLDS, CHAINABILITY THRESHOLD

5.1. MONOMORPHISM THRESHOLD-PAIR, THRESHOLDS p AND q

Given an in t ege r m , we define the monomorphism threshold-pair as being an ordered pa i r (p ,q) of in tegers such t h a t : (1) every m-ary, t o q i s chainable; ( 2 ) th is i s no longer true i f we replace p by p ' < p o r q by q ' < q . Consequently we have two d i s t i n c t monomorphism thresholds: the threshold p(m) i s the l e a s t i n t ege r p f o r which every m-ary p-monomorphic r e l a t ion w i t h i n f i n i t e o r s u f f i c i e n t l y la rge f i n i t e ca rd ina l i t y i s chainable ( r e c a l l t h a t , f o r a s u f f i c i e n t l y la rge cardinal of the base, p-monomorphism

(6 p)-monomorphic rq l a t ion w i t h ca rd ina l i t y a t l e a s t equal

Chapter 12 369

i m p l i e s t h e t h r e s h o l d q(m) i s t h e l e a s t i n t e g e r q f o r which eve ry m-ary monomorphic r e l a t i o n o f c a r d i n a l & q i s cha inab le . The e x i s t e n c e o f these t h r e s h o l d s f o l l o w s f rom 3.5 above (see FRASNAY 1965 p. 513)

By t h e p reced ing 4.6, we have p(m) \< m + t (m) : t h e t h r e s h o l d p i s bounded by t h e reassembl ing t h r e s h o l d . Moreover

s i n c e i t i s proved by CAMERON 1983 t h a t t h e o t h e r p o s s i b l e va lue 5 i s f a l s e

(see 5.3 below). From t h e upper bound o f HODGES, LACHLAN, SHELAH 1977 (see 4.1),

i t f o l l o w s t h a t p(m) \< 2m - 2 f o r m ) / 3 . F i n a l l y we have p(m) = 2m-2 f o r

m >/ 3 , a r e s u l t due t o FRASNAY 1984, unpubl ished.

F o r t h e t h r e s h o l d q , we have q ( 1 ) = 1, q ( 2 ) = 4 and q ( m ) b m+2 f o r m + 2 (FRASNAY 1965 p. 513).

(4 p)-monomorphism by ch.9 5 6.3) ;

p ( 1 ) = 1, p (2 ) = 3, p (3 ) = 4, p ( 4 ) = 6

5.2. CHAINABILITY THRESHOLD F o l l o w i n g t h e n o t i o n o f t h e t h r e s h o l d r ( m ) : t h e l e a s t i n t e g e r r such t h a t eve ry m-ary r e l a t i o n a l l o f whose r e s t r i c t i o n s o f c a r d i n a l r a r e chainable, and whose base i s i n f i n i t e o r

f i n i t e b u t s u f f i c i e n t l y l a r g e , i s i t s e l f cha inab le .

C l e a r l y r (m) ,< p(m),( m + t ( m ) .

p , we d e f i n e t h e c h a i n a b i l i t y t h r e s h o l d

5.3. The f o l l o w i n g s t r e n g t h e n i n g o f t h e theorem on s e t - t r a n s i t i v e groups (ch.11 5 5.10) i s due t o CAMERON 1976, as i s t h e theorem i t s e l f , and i s more

e a s i l y r e o b t a i n e d by POUZET 1981 p . 323. L e t m be a p o s i t i v e i n t e g e r and G be a pe rmuta t i on group on a denumerable

s e t E . f G & m, m + l , ..., p(m)-set t r a n s i t i v e b u t n o t m - t r a n s i t i v e , t hen t h e c l o s u r e o f G under m-adherence i s u - s e t - t r a n s i t i v e f o r eve ry i n t e g e r u . Then t h e conc lus ion i n ch.11 5 5.10 ho lds , except f o r t h e symmetric group,

s i n c e G i s n o t m - t r a n s i t i v e . More p r e c i s e l y , t h e r e e x i s t s a cha in Q based on E and isomorphic w i t h t h e c h a i n o f t h e r a t i o n a l s , such t h a t t h e c l o s u r e o f G under m-adherence i s e i t h e r t h e group o f automorphisms o f Q , o r t h e

group o f automorphisms o f i n te rmed iacy (mod Q) , o r t h e group o f automorphisms

o f t h e c y c l i c r e l a t i o n assoc ia ted w i t h Q , o r f i n a l l y t h e group o f automorphisms o f t h e d i h e d r a l r e l a t i o n assoc ia ted w i t h Q . These groups a r e a t most 3 - t r a n s i t i v e . So i t f o l l o w s t h a t f o r m 3 5 , every m, m+l, . . . , p ( m ) - s e t - t r a n s i t i v e group which i s 4 - t r a n s i t i v e i s a l s o m- t rans i -

t i v e . The case where in = 4 i s i n t e r e s t i n g . Indeed t h e r e e x i s t s a group whose c l o s u r e i s none o f t h e f o u r groups l i s t e d above, and which i s 4 and 5-set - t r a n s i t i v e y e t n o t 4 - t r a n s i t i v e (CAMERON 1983). Thus p ( 4 ) f 5 ; s i n c e p (4 )

= 5 o r 6 by FRASNAY 1965, hence n e c e s s a r i l y p ( 4 ) = 6 .


§ 6 - FINITELY BOUNDED AGE, UNIVERSAL CLASS, ANTIBOUND

6.1 . B O U N D OF AN AGE Given an age 4, a f i n i t e relation A i s said t o be a =of df i f f A does n o t belong t o (/t yet every proper res t r ic t ion belongs t o i&. Equivalently A a bound of any representative relation of the age.

i s

FINITELY BOUNDED AGE An age & i s said t o be f in i te ly bounded i f f there are only f in i te ly many bounds of 4 . For example, the age of a l l f i n i t e chains i s f in i te ly bounded. The age of a l l f i n i t e consecutivity relations and t h e i r res t r ic t ions i s n o t f in i te ly bounded, since every binary cycle i s a bound: see 5 1 above.

6.2. U N I V E R S A L CLASS Given an arbi t rary f i n i t e s e t U of f i n i t e relations of given a r i t y , the universal class defined by U i s the se t of those f i n i t e relations ( u p t o isomorphism) which do not admit an embedding of any element of U . A f i n i t e or inf in i te relation i f f R does n o t admit an embedding of any element of U . We can always suppose that a l l elements of U are mutually incomparable under embeddability: i t suffices t o remove each relation in U which has a proper res t r ic t ion belonging t o U . A f i n i t e relation A i s said t o be a bound of the universal class ‘4 i f f A does n o t belong t o h! yet every proper res t r ic t ion belongs t o A!. By the previous removing, we reduce U t o the bounds. As a f i r s t example of a universal c lass , take any f in i te ly bounded age; for instance the age of a l l f i n i t e chains. Every f i n i t e intersection of unjversal classes ( o f a given ar i ty ) i s a universal

class. I t suffices t o take the union o f the se t s of bounds, and then t o remove those relations which have a proper res t r ic t ion in t h i s union. Every f i n i t e union of universal classes (of a given a r i t y ) i s a universal c lass . 0 Let & and @ be two universal classes. A f i n i t e relation X belongs t o the union i f f , for each bound A of and each bound B of d3 , e i ther X does n o t admit any embedding of A , or of B . Equivalently, for each minimal common extension C of A and B , the relation X does not admit any embedding of C . Finally note that Card C i s a t most equal t o Card A + Card B , so t h a t

R i s said t o be a representative of the class

Chapter 12 37 1

there ex is t only f in i te ly many such C . 0

Now we have other examples of universal classes, by taking f i n i t e unions and f in i te intersections of f in i te ly bounded ages. By ch.10 5 2 .1 , we see that such unions and intersections are n o t necessarily ages. For example, we leave i t t o the reader t o show t h a t there ex is t ten binary relations with cardinality 2 , up t o isomorphism. Now consider again the age mentioned in ch.10 § 2.1: take an arbi t rary f i n i t e s e t E , then a subset F of E , and take the value (+) for ordered pairs in F , the value ( - ) i f a term in the pair belongs t o E-F . Then among the ten considered relat ions, seven of them are the bounds of the considered age (which i s thus f i n i t e l y bounded).

Given an a r i ty n 3 2 , every universal class i s an intersection of f in i te ly many f in i te ly bounded ages. Consequence of the fac t t h a t , given a f i n i t e n-ary relation A , the universal class having the unique bound A i s an age, by ch.10 5 2.3. For a r i ty 1, the class formed of a l l unary f i n i t e relations always (+) and f i n i t e relations always ( - ) i s neither an age, nor even an intersection of ages.

6.3. TWO ELEMENTARY CHARACTERIZATIONS OF UNIVERSAL CLASSES The reader will see the equivalence between our definition and the following character izat ion of universal classes by VAUGHT 1953:

dinal i ty of bounds) such that : if R i s a representative of & and i f S id relation a l l of whose restr ic t ions t o at.m_Tt p elements are_ embeddable in R , .- then S i s i t s e l f a representative of LA! . Also l e t us mention the or iginal , logical definition due to TARSKI: there exis ts a prenex formula having only universal quant i f iers , and taking the value (t) only for representatives of the class. Note t h a t the minimal possible number of quantifiers i s equal t o the maximum cardinality of bounds. For the meticulous reader, note tha t , when extending the usual truth-value t o the case where the base of relations can be empty, then the logical definition for universal classes f a i l s , and yet the two other mentioned definitions subsist. The exceptional case concerns th i s universal class which reduces t o the singleton of the 0-ary relation (O,+) with empty base (and similarly with the value ( - ) ) : This i s an exceptional case where the representing formula cannot be written in prenex form: see FRAISSE 1982, p . 72.

& i s universal i f f there exis ts an integer p ( i n fact p i s the maximum car-

.. . 6.4 . NON-TRIVIAL EXAMPLES OF UNIVERSAL CLASSES Given an integer m , the class whose representatives are a l l m-ary chainable relations i s universal; similarly for m-ary monomorphic relations (JEAN 1967; easy consequence of FRASNAY's theorems 3.3 t o 3.5 above).


Given a permutation group G , t he c l a s s whose representa t ives a re a l l the- G-chains i s un iversa l ; moreover the maximum ca rd ina l i t y of bounds i s (FRASNAY 1979).

m + t ( G )

6 .5 . ANTIBOUND OF A UNIVERSAL CLASS Given a universal c l a s s L& , a f i n i t e r e l a t ion df i f f A belongs t o d y e t no proper extension of A belongs t o (A.

I n the pa r t i cu la r case of a f i n i t e l y bounded age, only a t r i v i a l antibound can ex i s t : take f o r instance a f i n i t e r e l a t ion A and the age of a l l r e s t r i c t i o n s of A : then A i t s e l f i s the only antibound. In the general case, consider f o r instance the quadr i re la t ion R n = (In,Cn,On,Un) of MALITZ, introduced i n ch.5 § 8.3. t a t i v e s a re those quadr i re la t ions R , a l l of whose r e s t r i c t i o n s t o a t most 3 elements a re embeddable i n Then each R n f o r n a 7 i s an antibound of our universal c l a s s . For f u r t h e r informations, see FRAISSE 1973 p. 98 t o 103.

A i s s a id t o be an antibound of

Take the universal c l a s s whose represen-

R7 , and consequently embeddable in each Rn ( n 2 7 ) . .. .

31 3

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Martin ZIEGLER 1980 - Algebraisch abgeschlossene Gruppen. Word Problems 11, ed. S . I . Adian, W . W . Boone, G. Higman, North-Holland (Amsterdam), p . 449-576.

p . 389-396.

p . 261-281.

M . M . ZUCKERMAN 1973 - Natural sums o f ordina ls . Fundamenta Math. vol. 77 p. 289-294.

381

INDEX

ABBOTT, HANSON, 3-exerc. 1 absorption, 1-3.1; 1-3.3 access ib le ca rd ina l , 2-6.7 addi t ive ly f r e e set , 3-exerc. 1 adherence, n-adherence, adherent permutation, 11-2.3 age, 10-2

almost chainable r e l a t i o n , 10-9 amalgamable age, amalgamable s e t , 11-1.5 amalgamation lemma, 2-2 .2 ; amalgamation theorem (homogeneous re1 a t ions ) , 11-1.6 a leph , 1-6.1; aleph rank, 1-6.4 ALEXANDROFF, 1-exerc. 2 antibound, 12-5.4 an t icha in , 2-2.9 a r i t i e s of an opera tor , 9-3; comparison between a r i t i e s , 9-3.2 a r i t y , n-ary ( r e l a t i o n , mu l t i r e l a t ion ) , 2-1 ARONSZAJN ( t r e e ) , 5-7.1 ass ignable (mul t i r e l a t ion ) , 9-3

automorphism, 2-3; local automorphism, 9-1.7 axiom of a c c e s s i b i l i t y , 2-6.7 axiom of choice, 1-1.7; f o r f i n i t e s e t s , 1-2.10 axiom of dependent choice, 2-1.6 axiom of foundation, 1-2; 1-exerc. 1 axiom of i n f i n i t y , 1-2.4 axiom of maximal chain (HAUSDORFF-ZORN), 2-2.7 axiom of maximal chain of inc lus ion , maximal chain on i n i t i a l i n t e rva l s , 4-1.4 axiom of maximal idea l , 4-5 axiom of ordering, 2-3 axiom of reinforcement, 2-4.2 axiom o f trichotomy, 2-2.5 axiom of well-ordering, 2-2.5 axioms of ZF, 1-2.4

AJDUKIEWICZ, 1-1.2

-

ASSOUS, 8- 1.3

BACHMANN, 1-9 bad sequence, 4-2.2 ba r r i e r , 8-1 b a r r i e r sequence, good and bad b a r r i e r sequence, 8-2 base (nota t ion 1 I ) , 2-1 basic clopen set , 7-2.2 BAUMERT, 3-exerc. 1 BELL, SLOMSON, 2-exerc. 2

BERNAYS, 1-exerc. 1

BERNSTEIN, SCHROOER, 1-1.4; f o r r e l a t ions , 5-1.1 b e t t e r pa r t i a l ordering, q -be t t e r pa r t i a l ordering, _better quasi-ordering, 8-5.1 - b e t t e r pa r t i a l ordering ( w i t h respect t o b a r r i e r s ) , b e t t e r quasi-ordering, 8-6

b e t t e r pa r t i a l ordering of words, 8-6.9 betweenness r e l a t ion = intermediacy, 2-8.6 bichain, 11-3.7 b i j ec t ion = b i j ec t ive func t ion , 1-1.2 binary r e l a t i o n , 2-1; binary cyc le , 2-8.6

BENEJAM, 2-1.5 BERCOV, HOBBY, 11-2.8

BERNSTEIN, 2-8.1

388 Index

b i r e l a t i o n , 2-1 BIRKHOFF. 4-1.8 b i v a l e n t - t a b l e , 9-9 BLASS, 1-7.2; 2-6.8 BOGART, 4-exerc. 2 B O N N E T , 4-5.3; 7-2.1 and 2 .4 ; 7-4.8 and 4 . 9 ; 7-exerc. 4 ; 8-4.5 ; 8-exet-c. 3 BONNET, POUZET, 4-1.3 and 1 . 5 ; bound: i n i t i a l i n t e r v a l , 4-8; bound o f a r e l a t i o n , 12-1

4-9.2; 7-4.8 and 4 .9

CALAIS, 11-1.5; 11-6.4 CAMERON, 11-5.10; 12-5.1 and 5.3 CANTOR, lemma, theorem, 1-1.5; normal C a n t o r ' s theorem f o r p a r t i a l o r d e r i n g s canonica l group, 11-5; 11-exerc. 2 c a r d i n a l , 1-5.4 c a r d i n a l e x p o n e n t i a t i o n ( a b ) , p roduct C a r t e s i a n product ( x ) , 1-1.2; 1-1.8 c h a i n = t o t a l o r d e r i n g , 2-2.3 cha in a s s o c i a t e d t o a t ree , 4-6.2

orm, 1-3 .5 (DILWORTH, GLEASON), 5-2.6

x ) , sum (+), 1-5.5; 2-3.8

cha in meet ing every h e i g h t (POUZET), 4-4.5 cha in sequence, 8-7 .1 G-chain, (G,A)-chain, 12-4.3 c h a i n a b i l i t y t h r e s h o l d , 12-5.2 cha inable r e l a t i o n , 9-5

c h o i c e axiom, choice set and f u n c t i o n , 1-1.7 choice axiom f o r f i n i t e se t s , 1-2.10

c l o s e d c o f i n a l set = c l o s e d unbounded s e t , 1-exerc . 2 c losed under embeddabi 1 i t y , 10-2 c l o s u r e ( t o p o l o g y ) , 7-2.2 c l o s u r e ( t r a n s i t i v e ) , 1-5.1 c o f i n a l , c o - i n i t i a l se t and r e s t r i c t i o n , 2-5.1 c o f i n a l h e i g h t (Cofh) , 2-7.3 the c o f i n a l h e i g h t is a c a r d i n a l (POUZET), 2-7.6 c o f i n a l i t y ( C o f ) , c o - i n i t i a l i t y , 2-5.4 c o f i n a l sum i n an indecomposable o r d i n a l ( z o( c o f i n a l well-founded r e s t r i c t i o n , 2-5.1; 2-5.3; 2-7.1 COHEN, 1-2.4; 1-4.4; 2-exerc. 1 coherence lemma, 2-1.3 c o l o r , 3-1.1; 3-4.1 combina tor ia l lemmas on c o l o r s (POUZET), 3-4.1 t o 4 .3 common e x t e n s i o n o f compat ible r e l a t i o n s , 2-1.2 common 1-extens ion , 10-2.5 commutative or n a t u r a l sum and p r o d u c t , 1-9 comparable e lements i n a p a r t i a l o r d e r i n g , 2-2 comparable r e l a t i o n , 5-2.7 comparison between a r i t i e s , 9-3.2 compatible r e l a t i o n s , 2-1.2; G-compatible c h a i n s , 11-3 component i n a m u l t i r e l a t i o n , 2-1 conjugate p a r t i a l o r d e r i n g s , 4-exerc. 3 conjunct ion o f p a r t i a l o r d e r i n g s , 4-7.3; c o n j u n c t i o n o f r e l a t i o n s , 2-1.7 c o n s e c u t i v i t y r e l a t i o n , 2-8.6 cont inuum-equipotent s e t , 1-4.1; cont inuum-equipotent r e l a t i o n , 2-1 continuum h y p o t h e s i s , 1-4.4; connec t ion w i t h the e q u a l i t y G 2 = al, 1-6.5 c o n t r a c t e d group, 11-3.7

(G-rangement = G-chain)

CHANG, KEISLER, 10-1.4

C H U N G , 3-1.6 C L A R K , KRAUSS, 12-4.3

= Sup o( . ) , 2-5.5 1

Index 389

convergent func t ion , 7-exerc. 1 converse r e l a t i o n , 2-1.7

countable set , countable axiom of choice, 1-2.5 couple = ordered p a i r , 1-1.2 covering by doublets, 6-4.5; f i n i t e number of c l a s ses ( L A V E R ) , 8-4.5 covering by indecomposable chains, 6-4.3 c r i t e r i o n theorem f o r prehomogeneous r e l a t ions ( P O U Z E T ) , 11-7.9 c r i t e r i o n theorem f o r sa tura ted r e l a t ions ( P O U Z E T , VAUGHT) , 10-7.6 c u t , 4-1.1; DEDEKIND's theorem, 1-4.6; 4-1.1 cycle o f consecut iv i ty , 2-8.6 ; cyc l i c te rnary r e l a t ion assoc ia ted w i t h a chain, 2-8.6

COROMINAS, 8-1.4; 8-4.4

3-cycle, g-6:5

decomposable chain = addi t ive ly decomoosable chain, 6-3 decomposable ord ina l , 1-3.6 decomposition of a chain (homomorphic image), 2-3.6 decomposition of an indecomposable chain ( H A G E N D O R F ) , 6-3.3 decomposition of a p a r t i a l ordering i n t o i d e a l s , 4-5.2 and 5.3 decreasing sequence of chains of r e a l s , DUSHNIK, MILLER, 5-5.2; SIERPINSKI, 5-5.3 DEDEKIND, 1-1.3 ( f i n i t e se t ) ; 1-4.6; 4-1.1 de f in i t i on by recurs ion , 1-2.11

dense chain, 5-3.1; a-dense chain, 5-3.4 dense s e t of r e a l s , 1-4.5; dense s e t i n a cha in , 5-6 denumerable s e t , 1-2.5; denumerable r e l a t i o n , 2-1 denumerable subset axiom, 1-2.6 denumerablv SzDilra.in chain. 7-4.4

DE JONGH, PARIKH, 4-9.2 DEMBOWSKI , 9-6.9

dependent choice axjom, 2-1 16 DEVLIN, JOHNSBRATEN, 5-6 d i l a t e d group, 11-3.2 DILWORTH, 4-exerc. 2

dimension of a p a r t i a l ordering, 4-7.3 d i rec ted pa r t i a l ordering = n e t , 4-5 d i rec ted under embeddability, 10-2 d i r e c t product of p a r t i a l o r t o t a l o rder ings , 4-7 d is junc t ion of r e l a t ions , 2-1.7 domain of a function (Dom), 1-1.2 domain chain, 8-7.1 doublet of indecomposable cha ins , 6-4.4 DUSHNIK, MILLER, 3-3.2 and 3.3; 4-7.3; 4-7.8; 5-5.2

DILWORTH, GLEASON, 5-2.6

n-element se t , 1-2.3

embeddability (6 ) between r e l a t ions , 5-1; between sequences, 4-2 A-embeddability, 10-3.1 embeddability theorem f o r s ca t t e r ed chains ( L A V E R ) , 8-4.4 empty function, 2-3 equimorphic, equimorphism (nota t ion I ), 5-1 equipotence, 1-1.3 equivalence r e l a t ion generated by a quasi-ordering, 2-2 (1 ,p)-equivalence, 10-8.5 ERDdS, 3-3.4; ERDOS, RADO, 3-3.5; 3-exerc. 2 and 3; 5-3.7 ex is tence c r i t e r i o n , see "c r i t e r ion" e x i s t e n t i a l l y c losed , see "maximalist" exponentiation between card ina ls o r s e t s , notation ab , 1-1.6; 1-5.5 exponentiation between ord ina ls : extension of a func t ion , 1-1.3; extension of a r e l a t i o n , 2-1.1 1-extension, 10-1.4; common 1-extension, 10-2.5 extensive s e t , 11-6; G-extensive s e t , 11-7

ELLENTUCK, 3-6.1

ba , 1-3.3; 1-exerc. 3

390 Index

extensive b i va len t tab le , 9-9 ex t rac ted sequence, 1-2.2 ex t rac t i on property, 4-9.3 ex t rac t i on theorem f o r we l l p a r t i a l order ings, 4-3.4

f a i t h f u l extension o f a r e l a t i o n , 5-1.3; o f a chain, 5-4.5

f i che = i nd i ca to r , 11-4.2 f i l t e r , 1-7.1 f i n a l i n t e r v a l , 2-2 f i n e r ( f i l t e r ) , 1-7.1 f i n i t e l y bounded age, 12-5.1 f i n i t e l y f r e e p a r t i a l order ing, 4-3 f i n i t e l y generated i n i t i a l i n t e r v a l ( F ) , 4-1.8 f i n i t e number of bounds f o r a chainable r e l a t i o n (FRASNAY), 12-3.3 f i n i t e set , 1-1.1; f i n i t e r e l a t i o n , f i n i t e m u l t i r e l a t i o n , 2-1 f i n i t e t rees (KRUSKAL's theorem), 5-2.3 f i n i t i s t r e l a t i o n , 10-8; charac ter iza t ion , 10-8.5 f i n i t i s t s t ruc tu re , 10-exerc. 2 f i x e d Do in t lemma (KNASTER. TARSKI). 1-1.2

FEFERMAN, 2-1.6 FELGNER , 2-4.2

FODOR,' 1-exerc. 2 ' FOLKMAN, 3-1.6 forerunner ba r r i e r , forerunning, 8-3 and 3.1 forerunner b a r r i e r sequence , 8-3.3 foundat ion axiom, 1-2; 1-exerc. 1

FRAISSf, 10-2.5; 10-8.5; 10-exerc. 2 ; 12-6.3 and 6 .5 FRASNAY, 11-4.5; 12-3.3 t o 3.5; 12-4.1 and 4.2: 12-5 FREDRJCKSON. 3-e&erc. 1 f ree i n t e r p r e t a b i l i t y , 9-2 f ree operator, 9-3 f ree se t ( i n a p a r t i a l o rder ing) , 2-2.9 func t ion , 1-1.2 fundamental rank , 1-5.2

FRAENKEL, 1-1.4; 1-2.4

GALVIN, 3-2.2; 3-6 ; 7-4.8; 8-exerc. 1 GALVIN, MAC KENZIE, 7-4.8 GALVIN, PRIKRY , 3-6.1 GARCIA, 3-1.6 general ized continuum hypothesis, 1-4.4; 2-exerc. 1 generated i n i t i a l i n t e r v a l , 4-1.8; 4-2.5; 4-4.3 GILLAM, 12-1.4 GIRAUD, 3-1.6 GLEASON, GREENWOOD, 3-1.4 GLEASON. see DILWORTH GBDEL, i-2.4; 1-4.7 GOLOMB, see BAUMERT good sequence, 4-2.2 graph = symmetric b inary r e l a t i o n

grea ter sequence , 4-2 GREENWOOD, see GLEASON GRINSTEAD, ROBERTS, 3-1.5

GRAVER, YACKEL, 3-1.5

HAGENDORF, 5-1.2 and 1.3; 5-2.2; 5-4.1 t o 4.7; 5-5.2; 5-exerc. 1 t o 3; 6-3.3 t o 3.5; 6-3.8; 6-6.3; 6-exerc. 3; 8-4.6 and 4.7 HALPERN, LEVY, 2-2.8

Index

HANSON, 3-1.6; see ABBOTT HARTOGS aleph, o r s e t , 1-6.2 HATCHER. 1-2.9

3 91

HAUSDORFF, 6-2.1 HAUSDORFF, ZORN, 2-2.7; 2-exerc. 3 h e i g h t ( H t ) , 2-3.2 and 3.3

h e r e d i t a r i l y f i n i t e s e t , 1-5.1 h e r e d i t a r i l y indecomposable chain, 6-5 h e r e d i r a r i l y indecomposable chains: theorem (LAVER), 6-5.4 and 5.5 h e r e d i t a r i l v t r a n s i t i v e . 1-2.9

HENSON, 11-1.5

HESSENBERG," 1-9 HIGMAN, 4-4.1; 4-4.4 ; 8-5.3 HILL, IRVING, 3-1.6 h-indecomposable c h a i n = h e r e d i t a r i l y indecomposable chain, 6-5 HIRAGUCHI , 4-7.9 HIRSCHFELD, WHEELER, 10-7.3 HOBBY, see BERCOV HODGES, 2-6.8; 4-2.4; 10-7.3; 10-exerc. 2

homogeneous r e l a t i o n , p-homogeneous r e l a t i o n , 11-1 homogeneous system, 11-2.1 homomorphic image o f a cha in , 2-3.6 ; 8-7.2 homomorphic image o f a r e l a t i o n , 9-8

HODGES, LACHLAN, SHELAH, 12-4.1; 12-4.6

i d e a l , 4-5 i d e n t i c a l f i l t e r , 10-1.4 image c h a i n = homomorphic image, 2-3.6; 8-7.2 immediate e x t e n s i o n o f a cha in , 5-4.2; o f a r e l a t i o n , 5-1.2 immed ia te l y g r e a t e r = immediate e x t e n s i o n (up t o isomorphism) i n a c c e s s i b l e c a r d i n a l , 2-6.7 i nc idence m a t r i x , 3-5.1 i n c l u s i o n ( C_ ) , s t r i c t o r p r o p e r i n c l u s i o n ( c ) , 1-1 i n c l u s i o n among i n i t i a l i n t e r v a l s ( 3 ) , 4-1.3

i n c o m p a r a b i l i t y lemma f o r re l -ages (POUZET), 10-3.7 incomparable cha ins o f r e a l s , 5-5.3 incomparable elements i n a p a r t i a l o r d e r i n g ( n o t a t i o n I ) , 2-2 i n c r e a s i n g f u n c t i o n on r e a l s , 2-8.3 i n c r e a s i n g number o f o r b i t s (LIVINGSTONE, WAGNER), 11-2.8 indecomposable chain, 6-3 indecomposable cha in : HAGENDORF's decomposi t ion theorem, 6-3.3 indecomposable o r d i n a l , 1-3.6 indecomposable sequence, 8-5 i ndex ( i n a sequence), 1-2.2 i n d i c a t i v e group (FRASNAY), 11-4; i n d i c a t o r ( = f i c h e ) , 11-4.2

Q - i n d i c a t i v e group, 11-5 i n d i v i s i b l e chain, HAGENDORF and POUZET's r e s u l t s , 6-6.3 i n d i v i s i b l e r e l a t i o n , 6-6 induced age, 10-3.3 induced c u t , 4-1.1 i n d u c t i o n ( t r a n s f i n i t e ) , 1-2.11 i n e x h a u s t i b l e r e l a t i o n , 10-5; i n e x h a u s t i b l e age, 10-5.4 i n e x t e n s i v e t a b l e , 9-9 in f imum ( I n f ) , see "supremum" i n f i n i t e se t , 1-1.1

392 Index

i n f - r e s t r i c t ion of a b a r r i e r sequence, 8-2.2 i n i t i a l in te rva l = i n i t i a l segment, 2-2 i n i t i a l in te rva l generated by a minimal bad sequence, 4-2.5; 4-4.3 i n i t i a l i n t e rva l of a sequence, 4-2.1 i n i t i a l i n t e rva l s of a well p a r t i a l ordering (HIGMAN), 4-4.1 i n i t i a l i n t e rva l s : ordering by inc lus ion (nota t ion 3 ), 4-1.3 i n i t i a l in te rva l theorem (GALVIN), 3-2.2 i n i t i a l l y maximal chain, 4-6.5 in j ec t ab le o rd ina l , 1-6.2 in j ec t ion = i n j e c t i v e func t ion , 1-1.2 in j ec t ive f i l t e r , 10-1.1 in j ec t ive opera tor , 9-3.2 in t ege r = non-negative in teger = natura l number = f i n i t e o rd ina l , 1-2.3 intermediacy = betweenness r e l a t i o n , 2-8.6 i n t e r p r e t a b i l i t y ( f r e e ) , 9-2 in t e rva l , 2-2 inverse function f-' ; inverse transformation ( f - ' ) " , 1-1.2 inverse pro jec t ion modulo a f i l t e r , 10-1 inverted p a i r , 11-5.1 IRVING, 3-1.6 ISBELL, 3-1.6 i so l a t ed element (topology) , 7-2.3 i so l a t ed rel-age, i so l a t ing p a i r of f i n i t e r e l a t ions , 11-7.4 isomorphic, isomorphism, isomorphism type, 2-3 isomorphism (1 ocal ) , 9-1.4 1-isomorphism, (1,p)-isomorphism, 10-1.8 A-isomorphism, 10-3.1

JEAN, 9-6.6 and 6.7; 12-6.4 JECH, 1-2.5 and 2.6; 2-1.6; 5-5

JOHNSBPATEN, see DEVLIN JENSEN, 5-5

JONSSON. 11-1.5 JORDAN, 9-6.7; 11-2.2 JULLIEN, 5-2.1; 5-4.5 and 4.7; 6-3.3; 6-3.6 and 3.7; 7-4.6 and 4.7; 12-4.2

KALBFLEISCH, 3-1.6 KANTOR, 3-5.1

KERY, 3-1.5 KEISLER, see CHANG

kernel of a f i n i t i s t r e l a t i o n , 10-8.1; of a f i n i t i s t s t ruc tu re , 10-exerc. 2 kernel of an almost chainable r e l a t ion , 10-9.5 KLAUA, 1-9 KNASTER, 1-1.2 KONIG, theorem, 1-1.8; lemma, 4-4.5; 5-2.5 KRASNER, 7-exerc. 2 KRAUSS. 10-2 KRAUSSI see CLARK KRUSKAL, 5-2.3 KURATOWSKI, 1-1.2; 2-2.7; 2-8.1 KUREPA, 4-6.1; 5-7.1

LACHLAN, see HODGES

LAVER, 6-5.4 and 5.5; 8-4.4; 8-7.3 t o 7.8 l e f t indecomposable chain, s t r i c t l y l e f t indecomposable chain, 6-3.1 length of a sequence, 1-2.2

LARSON, 6-3.8 LAUCHLI, 2-2.3

Index 393

lesser sequence, 8-7.1 "less than" relation among sequences, 4-2 LEVY, 1-2.5; see HALPERN lexicographically well-ordered set, lexicographic rank, 3-2.1 limit aleph, 1-6.3 limit ordinal, 1-2.4 linear independence lemma (KANTOR), 3-5.1

local isomorphism, 9-1.4; local automorphism, 9-1.7 LOPEZ, 3-6; 5-8.3 and 8.4; 5-exerc. 1; 9-1.2; 9-9; 12-1.7

LIVINGSTONE, WAGNER, 11-2.8

MAC KENZIE, see GALVIN MAC NEILLE completion, 4-exerc. 1

mapping = function, 1-1.2

matrix (incidence), 3-5.1 maximal chain, 2-2.6 maximal chain axiom (HAUSDORFF-ZORN), 2-2.7; 2-exerc. 3 maximal chain of inclusion, 4-1.2; 4-1.4 maximal free set or maximal antichain, 2-2.9 maximal ideal axiom, 4-5 maximal reinforced chain theorem (DE JONGH, PARIKH), 4-9.2 maximal rel-age , 10-3.5 maximalist extension theorem, 10-3.9 maximalist set, maximalist relation, 10-3.8 maximum (Max). maximal element. 2-2

MALITZ, 5-8.3 ; 12-6.5

MATHIAS, 2-4.2

MILLER, see DUSHNIK MILNER, 4-5.3 minimal bad sequence, 4-2.3 minimal bad barrier sequence (NASH-WILLIAMS), 8-2.2 minimum (Min), minimal element, 2-2 monochromatic set, 3-1.1; 3-1.3 monomorphic, p-monomorphic, ( 5 p)-monomorphic relation, 9-6 monomorphism threshold, 12-4.6

1-morphism, (1,p)-morphism, 10-1.8

multicolor theorem (POUZET), 3-5.3 multirelation, 2-1

MOORE, 2-2.7

* -morphism, 10-4.5 MOSTOWSKI, 2-1.6

NASH-WILLIAMS, 3-2.4; 8-1.4; 8-2.1 and 2.2; 8-3.1; 8-6 natural number = integer (non-negative), 1-2.3 natural sum and product, 1-9 negation of a relation, 2-1.7 neighborhood, 4 -neighborhood, 6-2.2 ; neighborhood rank, 6-2.4 net = directed partial ordering, 4-5 von NEUMANN, 1-2 non-embeddability rank, non-richness rank, 10-4.7 normal form (CANTOR), 1-3.5

older relation, 10-1.2 A-older relation, 10-3.2; o( -older relation, 10-4.5 operator: free, 9-3; injective, 9-3.2; partial, 9-3.3

394 Index

o r b i t , 11-2.2 o r d e r a b l e s e t , 2-2.3 ordered p a i r , 1-1.2 o r d e r i n g , 2-2 ; o r d e r t y p e , 2-3 o r d e r i n g axiom, 2-3 o r d i n a l , 1-2.1 ; o r d i n a l w , 1-2.4; o r d i n a l w1 , 1-6.3 o r d i n a l e x p o n e n t i a t i o n , 1-3.3; 1-exerc. 3 ord ina l - indexed sequence = o r d i n a l sequence, 1-2.2 o r d i n a l sum ( ) and product ; o r d i n a l s : 1-3.2 and 3 . 3 ; c h a i n s : 2-3.6 and 3 .7

PABION, 9-2.2; 11-7; 12-2.3 P A I L L E T , 9-2.4 ; 10-2.7 p a i r = unordered p a i r , 1-1 PARIKH, see DE JONGH p a r t i a l o r d e r i n g , 2-2 p a r t i a l o r d e r i n g o f i n i t i a l i n t e r v a l s ( 3 ) , 4-1.1 p a r t i a l o r d e r i n g o f words, 4-2 p a r t i t i o n i n s l i c e s (BONNET, POUZET), 4-1.5 p a r t i t i o n lemmas: OUSHNIK, M I L L E R , 3-3.2; ERDOS, 3-3.4 p a r t i t i o n lemmas and theorems: DUSHNIK, M I L L E R , 3-3.2 and 3.3 p a r t i t i o n lemmas and theorems: ERDOS, RADO, 3-3.4 and 3 .5 ; 3 -exerc . 2 and 3 p e r f e c t b a r r i e r sequence, 8-2 .1 PERLES, 4-exerc. 2 permuta t ion , 1-1.2 PINCUS, 2-2 .3 POIZAT, 4-3.3; 4-3.5 POUZET, 2-5.1 t o 5 .3; 2-7.1 t o 7 .6 ; 3-2.2; 3-4.2; 3-5.3; 4-3.5; 4-4.1 t o 4 .5 ; 4-7.8; 4-10; 4-exerc . 4 ; 7-2.3; 7-3.1 t o 3 .10; 8-1.3 and 1 . 7 ; 8-5.4; 8-6.7 and 6.8; 9-3.5; 9-6.3 t o 6 . 5 ; 9-7; 10-5.5 and 5.6; 10-6.5; 10-7.6; 10-9.9; 11-2.6

POUZET, ROSENBERG, 3-exerc. 4

POUZET. see BONNET

t o 2.10; 11-7.9; 11-8.3; 12-1.7 t o 1 .9; 12-3.1 t o 3 .3; 12-4.3; 12-4.7

POUZET, VAUGHT, 10-7.5

POWELL; 1-2.9 power, see "exponent ia t ion" ; power se t ( T ) , 1-1

. I , p r e d e c e s s o r , 1-2 prehomogeneous r e l a t i o n , 11-7; c r i t e r i o n theorem, 11-7.9 pre-order ing = q u a s i - o r d e r i n g preserved p a i r , 11-5.1 PRIKRY, see G A L V I N p r i n c i p a l chain o r p r i n c i p a l t o t a l o r d e r i n g , 2-5.2 product : o r d i n a l s , 1-3.2; c a r d i n a l s ( x ) , 1-5.5; c h a i n s , 2-3.7 p r o f i l e o f a r e l a t i o n , 9-7 p r o j e c t i o n f i l t e r , 10-1 proper embedding = s t r i c t embedding ( < ) , 5-1 proper i n i t i a l i n t e r v a l , 2-2 pseudo-amalgamable age , 11-6.6 pseudo-amalgamation theorem ( C A L A I S ) , 11-6.6 pseudo-homogeneous r e l a t i o n , 11-6

Q = chain of the r a t i o n a l s , 2-2.3 q u a n t i f i e r ( u n i v e r s a l , e x i s t e n t i a l ) , 9-8 q u a s i - o r d e r i n g , 2-2

Index

RAOO, 2-1.5; Rado's well p a r t i a l o rder ing , 4-4.2; 8-5.2 RADO, see EROOS

395

RAMSEY. 3-1.1 and 1.3 Ramsey number, 3-1.4; 3-3.1,. Ramsey set o r sequence (EROOS-RADD, followed by LOPEZ) , 3-6.1 rang& (Rng) , 1-1 :2 rank: fundamental, 1-5.2; lexicographic, 3-2.1; neighborhood, 6-2.4 ranking func t ion , 8-3.3 ra t iona l and real number, 1-4.5 reassembling theorem (FRASNAY), 12-3.4; th reshold , 12-4.2 recurs ion , 1-2.11 reduced t r e e , 4-6.4 reduction theorem (FRASNAY), 11-4.6; th reshold , 12-4.1 regress ive func t ion , 1-exerc. 2 regular aleph o r ca rd ina l , 2-6.1 reinforcement, 2-4.1; reinforcement axiom, 2-4.2 reinforcement of a well-founded ordering, 2-4.4 reinforcement of a well pa r t i a l ordering, 4-3.5; 4-exerc. 5 rel-age, 10-3.1 r e l a t i o n , 2-1 r e l a t iona l system, 11-2.1 re1 a t i ve r e s t r i c t i o n , isomorphism, embeddabi 1 i t y , 10-3.1 representa t ive r e l a t ion ( i n an age ) , 10-2 r e s t r i c t i o n : func t ion , 1-1.3; r e l a t i o n , 2-1.1 r e t ro -o rd ina l , 2-1.7 r i ch r e l a t i o n , 10-4; r i ch f o r i t s age, 10-6 r i ch f o r i t s age: ex is tence c r i t e r i o n (POUZET, VAUGHT), 10-7.6 r i g h t indecomposable, s t r i c t l y r i g h t indecomposable chain, 6-3.1 ROBERTS, see GRINSTEAD ROBINSON. 10-3.10: 10-7.3 ROSENBERG, see PO~JZET ROSENSTEIN, 2-8.2; 5-5.3; 8-3.4 and 3.5; 8-6.10 RUBIN, 1-1.7; 2-3.8

SABBAGH , 5-4.7 sa tu ra t ed r e l a t i o n , 10-7.3; ex is tence c r i t e r i o n , 10-7.6 sca t t e red chain, 6-1 scheme of foundation, 1-2.8 scheme of induction f o r f i n i t e s e t s , 1-1.1 SCHRODER, see BERNSTEIN SCHUR (number), 3-exerc. 1

segment = in te rva l separable elements, 4-1.2 sequence, ordinal-indexed sequence, o( -sequence, 1 - 2 . 2 w -sequence, 1-2.5 set-extensive subset, set-homogeneous r e l a t i o n , 11-8 s e t of f i l t e r s , 2-exerc. 2 set of i n j ec t ive functions ( S I E R P I N S K I ) , 2-8.2 set-pseudo-homogeneous r e l a t i o n , 11-8 s e t - t r a n s i t i v e group, 11-2.2 SHELAH, see HODGES

S I E R P I N S K I , 2.8.2; 2-exerc. 1; 3-3.1; 5-4.4; 5-5.3; 7-exerc. 4 S ie rp insk i ' s p a r t i a l o rder ing , 7-exerc. 4

SCOTT, 1-5.4

SHEPHERDSON, 2-6.7

S I K O R S K I , 1-9 S I L V E R , 3-6.1 SIMMONS, 10-7.3 s ingu la r aleph o r ca rd ina l , 2-6.1 s l i c e s ( p a r t i t i o n ; BONNET, POUZET), 4-1.5

396 Index

SLOMSON, see BELL smaller sequence, 4-2

sparse s e t (topology), 7-2.3 specification (rel-age) , 10-3.3

SPERNER, 3-exerc. 4 stationary s e t , 1-exerc. 2 s t r a t i f i e d partial ordering, 2-5.2 s t r i c t embedding ( < ) , 5-1 s t r i c t inclusion ( c ) , 1-1 s t r i c t l y r ight and s t r i c t l y l e f t indecomposable chain, 6-3.1 strongly amalgamable s e t , 11-1.5 strongly minimal bad sequence, 4-2.8 5trongly scattered chain, 6-2.3 subpotence, 1-1.3 successive elements ( i n a b a r r i e r ) , 8-1.5 successor aleph a+ , 1-6.3 successor barr ier , 8-3.3 successor cardinal, 2-3.10 successor ordinal a+l , 1-2.4 successor s e t , 1-1 sum along a chain, d. -5um, 2-3.6 sum: ordinal sum (+), 1-3.1 and 2-3.6; cardinal sum (+), 1-5.5

SOLOVAY, 5-6

S P E C K E R , 5-7.3

SUPPES, 1-1.4 supremum ( S u p ) , 2-2; fo r ordinals, 1 -2 .1 ; fo r rea ls , 1-4.6 SUSLIN, 5-6 Suslin chain. 5-6.1: Suslin t ree . 5-6.3 system ( r e l a t i o n a l ) ; 11-2.1 SZPILRAJN, 2-4 .2 Szpilrajn chain, 7-4

table (bivalent) , 9-9 t a i l = f inal interval o f a sequence, 8-5 TARSKI, 1-1.2; 2-3.8 t o 3.10; '2-6.5; 2-exerc. 2 ; 12-5.3

thresholds, 12-4.1 and 4 . 2 ; 12-4.6; threshold pa i r , 12-4.6 T O D O R C E V I C , 7-exerc. 2 topology on se t s of integers, 1-8 topology on i n i t i a l intervals , 7-2.2 to ta l ly ordered s e t (by membership), 1-2 total ordering = chain, 2-2.3 tournament, 5-2.7 t ransf in i te induction, 1-2.11 transformation f " , 1-1.2 t ransi t ive closure, 1-5.1 t ransi t ive group, n- t ransi t ive group, 11-2.2 t ransi t ive s e t , 1-2 transposit ion, 1-1.2 t ree , 4-6 trichotomy for ordinals, 1-2.1; fo r cardinals, 2-2.5 t r iv ia l ul t r a f i l t e r , 1-7.2 T U K E Y , 7-exerc. 1 n-tuple, 1-2.3 type (isomorphism), 2-3

TENNENBAUM, 5-5

Index 397

u l t r a f i 1 t e r , ul t r a f i l t e r axiom, 1 - 7 . 2 unbounded s e t or sequence = without any upper bound universal c l a s s , 12-5.2 universa l ly Szp i l r a jn p a i r of r e l a t ions , 7-4.9 universal theory = age, 10-2 unordered pa i r = p a i r , 1-1 upper bound, 2-2

value (+) o r ( - ) , 2-1 VAUGHT, 12-5.3; see POUZET

WAGNER, see L IV INGSTONE WALKER, 3-1.6 weakening, 2-4.1 weakly inaccess ib le aleph, 2-6.7 well-founded cofinal r e s t r i c t i o n (POUZET), 2-5.1 well-founded pa r t i a l ordering, well-ordering, 2-2.4 well mu l t i r e l a t ion , 12-3 well-ordered s e t ( l ex icograph ica l ly ) , 3-2.1 well-ordering axiom, well-orderable s e t , 2-2.5 well p a r t i a l ordering, 4-3.2 ; theorem on i n i t i a l i n t e rva l s (HIGMAN), 4-4.1 well pa r t i a l ordering of f i n i t e t r e e s (KRUSKAL), 5-2.3 well pa r t i a l ordering of words (HIGMAN), 4-4.4 well quasi-ordering, 4-3.2 WHEELER. see HIRSCHFELD WHITAKER, 1-1.4; 3-1.6 WHITEHEAD, 3-1.6 word = f i n i t e sequence, 1-2.3; pa r t i a l ordering of words, 4-2

YACKEL, 3-1.6; see GRAVER younger r e l a t i o n , 10-1.2 A-younger r e l a t i o n , 10-3.2; W -younger r e l a t i o n , 10-4.5

Z = chain of pos i t i ve and negative in t ege r s , 2-2.3

Z F (axioms), 1-2.4 ZORN, see HAUSDORFF

ZAGUIA, 4-9.3 ZERMELO, 1-1.7; 1-2

[Slfm 118] theory of relations roland fraisse (nh 1986)(t)

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