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1 - Classical Problems in Mechanics (R. Russo ed.)

2 - Recent Developments in Partial Differential Equations (V. A. Solonnikov ed.)

3 - Recent Progress in Function Spaces (G. Di Maio and L. Hola eds.)

4 - Advances in Fluid Dynamics (P. Maremonti ed.)

5 - Methods of Discrete Mathematics (S. Lowe, F. Mazzocca, N. Melone and U. Ott eds.)

6 - Connections between Model Theory and Algebraic and Analytic Geometry (A. Mac-

intyre ed.)

7 - Homage to Gaetano Fichera (A. Cialdea ed.)

8 - Topics in Infinite Groups (M. Curzio and F. de Giovanni eds.)

9 - Selected Topics in Cauchy-Riemann Geometry (S. Dragomir ed.)

10 - Topics in Mathematical Fluids Mechanics (G.P. Galdi and R. Rannacher eds.)

11 - Model Theory and Applications (L. Belair, Z. Chatzidakis et al. eds.)

12 - Topics in Diagram Geometry (A. Pasini ed.)

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Dispersive nonlinear problems in Mathematical Physics (P. D’Ancona and V. Georgev

eds.)

Kinetic Models of Granular and Reacting Flows (G. Toscani ed.)

Complexity of computations and proofs (J.Krajıcek ed.)

Calculus of Variations: topics from the mathematical heritage of E. De Giorgi (D.

Pallara ed.)

Model Theory and Applications

edited by L. Belair, Z. Chatzidakis, P. D’Aquino, D. Marker, M. Otero, F. Pointand A. Wilkie.

Authors’ addresses:

Serban A. Basarab

Institute of Mathematics of

the Romanian Academy

P.O. Box 1–764,

RO–70700 Bucharest

Romania

e-mail: [email protected]

Oleg Belegradek

Department of Mathematics

Istanbul Bilgi University

80370, Dolapdere–Istanbul,

Turkey

email: [email protected]

Gregory Cherlin

Department of Mathematics,

Rutgers University

Hill Center, Piscataway, NJ 08854,

U.S.A


Paola D’Aquino

Dipartimento di Matematica

Seconda Universita degli Studi di Napoli

via Vivaldi, 43

81100 Caserta

Italy


Deirdre Haskell

Department of Mathematics and Statistics,

McMaster University,

Hamilton, Ontario L8S 4K1,

Canada


Luc Belair

Departement de Mathematiques - UQAM

C.P. 8888, Succ. Centerville

Montreal (Quebec) H3C 3P8

Canada


Zoe Chatzidakis

Universite Paris 7

UFR de Mathematiques, Case 7012

75251 Paris Cedex 05

France


Raf Cluckers


Katholieke Universiteit Leuven

Celestijnenlaan 200B

B-3001 Leuven

Belgium


Lou van den Dries

University of Illinois


1409 W. Green Street

Urbana, IL 61801

U.S.A.


Ehud Hrushovski


Hebrew University

Givat-Ram 91904

Israel


Eric Jaligot

Equipe de Logique Mathematique,

Universite de Paris VII

75251 Paris

France


Dugald Macpherson

Department of Pure Mathematics

University of Leeds

Leeds LS2 9JT

U.K.


Rahim Moosa

Faculty of Mathematics

200 university Ave. W.

Waterloo, Ontario N2L 3G1

Canada


Margarita Otero

Universidad Autonoma de Madrid

Departamento Matematicas

Facultad de Ciencias, C-XV

ctra. de Colmenar Viejo, km. 15

28049 Madrid

Spain


Anand Pillay

University of Illinois


1409 W. Green Street

Urbana, IL 61801

U.S.A.


Angus Macintyre

School of Mathematical Sciences

Queen Mary University of London

Mile End Road

London E1 4NS

U.K.


David Marker


University of Illinois at Chicago

851 S. Morgan St.

Chicago, IL 60607

U.S.A.


Yerulan Mustafin

Institut Girard Desargues

Universite Lyon 1

21 Avenue Claude Bernard

69622 Villeurbanne Cedex

France


Ya’acov Peterzil


University of Haifa

31905 Haifa

Israel


Francoise Point

Universite de Mons-Hainaut

Institut de Mathematiques (Le Pentagone)

6, Avenue du Champ de Mars

7000 Mons

Belgium


Bruno Poizat


Mathematiques, Batiment 101

Universite Claude Bernard - Lyon 1

69622 Villeurbanne cedex

France


Thomas Scanlon

University of California, Berkeley


910 Evans Hall

Berkeley, CA 94720-3840

U.S.A.


Katrin Tent

School of Mathematics

University of Birmingham

Edgbaston

Birmingham B15 2TT

U.K.


Alex Wilkie

24-29 St Giles

Mathematical Institute

University of Oxford

Oxford, OX1 3LB

U. K.


Olivier Roche

Mathematisches Institut

Abteilung fur Mathematische Logik

Albert-Ludwigs-Universitat Freiburg

Eckerstr. 1,

D-79104 Freiburg

Germany


Patrick Speissegger

Department of Mathematics and Statistics,

McMaster University,

Hamilton, Ontario L8S 4K1,

Canada


Frank Wagner


Universite Claude Bernard (Lyon-1)

21 av. Claude Bernard

69622 Villeurbanne Cedex

France


Boris Zilber

24-29 St Giles

Mathematical Institute

University of Oxford

Oxford, OX1 3LB

U. K.


Received May 13, 2003

c© 2002 by Dipartimento di Matematica della Seconda Universita di Napoli

Photocomposed copy prepared from a TEX file.

ISBN 88-7999-411-5

Contents

Introduction

A Representation Theorem for a Class of Arboreal Groups 1Serban A. Basarab

Semi-Bounded Relations in Ordered Abelian Groups 15Oleg Belegradek

Simple Groups of Finite Morley rank, 2002 41Gregory Cherlin

Model Theory of Valued Fields 73Raf Cluckers

O-minimal Preparation Theorems 87Lou van den Dries and Patrick Speissegger

Definable Sets in Valued Fields 117Deirdre Haskell and Dugald Macpherson

Pseudo-Finite Fields and Related Structures 151Ehud Hrushovski

Tame Minimal Simple Groups of Finite Morley Rank and ofOdd Type 213Eric Jaligot

A History of Interactions between Logic and Number Theory 227Angus Macintyre

The Mordell-Lang Conjecture in Positive Characteristic Revisited 273Rahim Moosa and Thomas Scanlon

Sous-Groupes de SL2(K) sur un Corps Superstable 297Yerulan Mustafin

Some Topological and Differentiable Invariants in O-MinimalStructures – A Survey of the Solution to the Torsion Point Problem 307Ya’acov Peterzil

Two Remarks on Differential Fields 325Anand Pillay

Quelques Mauvais Corps de Rang Infini 349Bruno Poizat

Bad Fields with a Torus of Rank 1 367Olivier Roche and Frank Wagner

O-Minimal Expansions of the Real Field 379Patrick Speissegger

Recognizing Groups of Finite Morley Rank via BN-pairs 409Katrin Tent

Model Theory, Geometry and Arithmetic of the UniversalCover of a Semi-Abelian Variety 427Boris Zilber

Open Problems in Model Theory 459

Mathematics subject classification.

A Representation Theorem for a Class of Arboreal Groups: 20E08, 05C25;

Semi-Bounded Relations in Ordered Abelian Groups: 03C64, 03C60, 03C10, 20F60;

Simple Groups of Finite Morley rank, 2002: 20E32, 20D05, 03C45, 20A15,03C60;

Model Theory of Valued Fields: 03C07, 03C98, 03C64, 03C07;

O-minimal Preparation Theorems: 03C64, 03C10, 32B05;

Definable Sets in Valued Fields: 03C60;

Pseudo-Finite Fields and Related Structures: 03C60, 03C45, 12F10;

Tame Minimal Simple Groups of Finite Morley Rank and of Odd Type: 20E32, 03C60;

A History of Interactions between Logic and Number Theory: 03C20;

The Mordell-Lang Conjecture in Positive Characteristic. . . : 11G25, 11G10, 11U09, 14G15;

Sous-Groupes de SL2(K) sur un Corps Superstable: 03C45, 03C60, 20G15;

Some Topological and Differentiable Invariants in O-Minimal Structures. . . : 03C64, 22E30;

Two Remarks on Differential Fields: 12H05, 12L12;

Quelques Mauvais Corps de Rang Infini: 03C45;

Bad Fields with a Torus of Rank 1: 03C45, 12L12;

O-Minimal Expansions of the Real Field: 03C64, 30D60, 58A17;

Recognizing Groups of Finite Morley Rank. . . : 20E42, 03C45, 03C60, 20A15, 20E32;

Model Theory, Geometry and Arithmetic of the Universal Cover. . . : 03C45, 03C60, 14K15.

Introduction

In the last few years model theorists have found deep connections betweentheir subject and other areas of mathematics. Our goal in organizing theEuropean Conference on Model Theory and Applications was to assess thestate of the art, especially for the younger generation.

This conference was also the opportunity to acknowledge, on the occasionof his sixtieth birthday, the influence of Angus Macintyre in the development ofour subject. This has been both immense and diverse and we feel it appropriateto mention a few highlights here.

Angus’ research has links with algebra, geometry, number theory, asymp-totics and even to computational complexity theory and randomized algo-rithms. His early successes included characterizations of uncountably cate-gorical abelian groups and fields as well as results on the elementary theoryof Banach algebras. He also completed the proof of a beautiful equivalence(the “if” direction being due to B H Neumann) between model theoretic andrecursion theoretic notions in combinatorial group theory: a finitely generatedgroup can be embedded in every algebraically closed group if and only if it canbe recursively presented with solvable word problem. Soon after this period,

xiv

while still making deep contributions to the model theory of groups and ringsand, indeed, of arithmetic (both to foundational questions involving PeanoArithmetic and its subtheories, and to the more application orientated “non-standard number theory”), he established his quantifier elimination result forp-adic fields. As we all now know, this was a pivotal moment in applied modeltheory and, through the work of Denef and many others, Angus’ insights atthat time continue to have deep applications in p-adic diophantine geometryvia the theory of Poincare series. It does seem remarkable that the originalwork was done almost thirty years ago. In fact, we list below a chronologicalselection of his papers so that the reader will be able to appreciate more fullythe ramifications of Angus’ ideas on what we now call the model theory of fieldsand his dominant role within it.

Almost all of these papers are extremely familiar to the editors of thisvolume, but to those not brought up under Angus’ influence may we furthersingle out his definitive work on PAC and regularly closed fields, his synthesis ofAx’s methods for finite fields with difficult analytic number-theoretic results toobtain information on primes in nonstandard models of systems of arithmeticand, of course, his work on exponentiation. Many of his well known resultson the decidability and axiomatizability of the real exponential function, andon exponential series, occur in joint papers in the nineties, but it should bepointed out that he had already originated the study of the p-adic analoguemany years before this. Also, he was the first to truly believe, in the earlyeighties, that the real case (Tarski’s problem) was ready for a serious attack:subsequent progress would certainly not have been made were it not for hisearly papers, the input of numerous preprints and his constant encouragement.

You will also see that his more recent research is just as wide-ranging. Itcovers generic automorphisms of algebraically closed fields, the model theoryof analytic functions and its application to sigmoidal neural networks, formalmodels for exponential asymptotics and model-theoretic aspects of the Weilconjectures for finite fields.

But his influence in model theory goes beyond any list of his achievements.It is also due to the excitement he conveys, whether it be in his lectures orjust in everyday conversations at meetings around the world. He is well known

xv

in the community for his accessibility to discuss mathematics, and the circleof his friends and students has always benefitted from his exceptional energy,his enthusiasm and generosity. There is no doubt that through this personalcontact he has helped turn our subject into the thriving area that it now is.

Luc BelairZoe ChatzidakisPaola D’AquinoDave MarkerMargarita OteroFrancoise PointAlex Wilkie

xvii

PH.D. STUDENTS OF ANGUS MACINTYRE

• Yale 1973-1985

– Peter Winkler

– Larry Manevitz

– Emily Grosholz (Philosophy– joint with Korner)

– Lee Davidson (Philosophy– joint with Fitch)

– Steve Garavaglia

– Ken McKenna

– Kay Smith

– Chico Miraglia

– George Loullis

– David Marker

– Zoe Chatzidakis

– Ali Nesin

– Stu Smith

– Luc Belair

– Laura Mayer

• Oxford 1985-1997

– Frank Wagner

– Larry Matthews

– Margarita Otero

– Paola D’Aquino

– Tim Gardener

– Charles Morgan

• Edinburgh 1997-2004

– Ivan Tomasic

– James Gray

– Antongiulio Fornasiero

xix

SELECTED PAPERS OF ANGUS MACINTYRE

1. Weil cohomology and model theory, in Connections between model the-

ory and algebraic and analytic geometry, A. Macintyre ed., Quadernimatematica 6 (2000), 179–199.

2. Logarithmic-exponential power series, J. London Math. Soc. (2) 56(1997), 417–434. With L. van den Dries and D. Marker.

3. Generic automorphism of fields, Annals Pure Applied Logic 88 (1997),165–180.

4. Polynomial bounds for VC dimension of sigmoidal and general Pfaffianneural networks, J. Comput. System Sci. 54 (1997), 169–176. With M.Karpinski.

5. On the decidability of the real exponential field, in Kreiseilania: about

and around Georg Kreisel, P. Odifreddi ed., AK Peters (1996), 441–467.With A. Wilkie.

6. Definable sets over finite fields, J. Reine Angew. Math 427 (1992), 107–135. With Z. Chatzidakis and L. van den Dries.

7. Rationality of Poincare series: Uniformity in p, Annals Pure and Appl.

Logic 49 (1990), 31–74.

8. Degrees of recursively saturated models, Trans. Am. Math. Soc. 282(1984), 539–554. With D. Marker.

9. Constructive logic versus algebraization, in The L.E.J. Brouwer Centen.

Symp. Proc. Conf., North-Holland (1982), 217–260. With G. Kreisel.

10. Decidability and undecidability theorems for PAC-fields, Bull. Am. Math.

Soc., New Ser. 4 (1981), 101–104. With G. Cherlin and L. van den Dries.

11. Nonstandard number theory, in Proc. Int. Congr. Math. Helsinki 1978,

vol. 1 (1980), 253–262.

12. On definable subsets of p-adic fields, J. Symb. Logic 41 (1976), 605–610.

xx

13. Model-completeness for sheaves of structures, Fund. Math. 81 (1973),73–89.

14. The word problem for division rings, J. Symb. Logic 38 (1973), 428–436.

15. On algebraically closed groups, Ann. Math (2) 96 (1972), 53-97.

16. On ω1−categorical theories of fields, Fund. Math. 71 (1971), 1–25.

A Representation Theorem for aClass of Arboreal Groups

Serban A. Basarab

Contents

1. Introduction (3).

2. A representation theorem for ⊥–groups (5).

A Representation Theorem for a Class of Arboreal Groups 3

1. Introduction

In his paper [2], the author introduced a class of groups called median (orarboreal) and used them for the investigation of the partially commutativeArtin-Coxeter groups. Recall that by a median or arboreal group we mean agroup G together with a ternary operation Y on G making it a median set

or generalized tree such that uY (x, y, z) = Y (ux, uy, uz) for all u, x, y, z ∈ G.Notice that, though different, the notion above is similar with the notion ofmedian group introduced by M. Kolibiar in [11]. Setting x ∩ y = Y (x, 1, y),it follows by [2] Proposition 2.2.1. that a median group can be equivalentlydefined as a group G together with a semilattice operation ∩, whose associatedorder ⊂ given by x ⊂ y iff x ∩ y = x satisfies the following conditions: 1 ⊂ x

for all x ∈ G, z−1y ⊂ z−1x whenever x ⊂ y and y ⊂ z, and x−1(x ∩ y) ⊂ x−1y

for all x, y ∈ G.

In a median group G, an ordered n-tuple (x1, . . . , xn) ∈ Gn, n ≥ 2, is saidto be reduced (written x1 . . . xn = x1 • x2 • . . . • xn) if x1 . . . xi ⊂ x1 . . . xi+1

for i < n, i.e. x−1i . . . x−1

1 ∩ xi+1 = 1 for i < n. In particular, an orderedpair (x, y) ∈ G2 is reduced iff x ⊂ xy iff x−1 ∩ y = 1. One checks easily that(x1, . . . , xn) is reduced iff (x−1

n , . . . , x−11 ) is reduced, and for all x, y ∈ G, x ∩ y

is the unique element z of G satisfying x = z • (z−1x), y = z • (z−1y) andx−1y = (x−1z) • (z−1y). Note also that for reduced pairs (x, y) and (x, z),xy ∩ xz = x(y ∩ z), in particular, xy ⊂ xz iff y ⊂ z. In other words, x−1y ∩x−1z = x−1(y ∩ z) whenever x ⊂ y and x ⊂ z.

The poset (G,⊂) admits a partially defined least upper bound (l.u.b.) w.r.t.the partial order ⊂ : x∪ y = Y (x, y, z) = z(z−1x∩ z−1y), whenever x ⊂ z andy ⊂ z. Set by convention x ∪ y = ∞ whenever the pair (x, y) is not boundedabove w.r.t. the order ⊂. A median group G is called locally linear, resp.locally boolean, resp. simplicial if its underlying median set is locally linear,resp. locally boolean, resp. simplicial.

∗The author is grateful for the kind hospitality during his visit supported by a grant of

the Royal Society to the Departments of Mathematics of the QMW College of the University

of London and of the University of Edinburgh when this research was partly carried out.

He also acknowledges a partial support from the Grant 47/2002 awarded by the Romanian

Academy.

4 S. A. Basarab

The elements x and y of a median group G are said to be orthogonal (writex ⊥ y) if x ∩ y = 1 and x ∪ y 6= ∞. The median group G is called ⊥–group ifthe following condition is satisfied:

(⊥) x ∪ y = xy = yx whenever x ⊥ y.Given a group G and a set S ⊆ G of generators such that 1 6∈ S and

S ∩ S−1 = S1 := {s ∈ S | s2 = 1}, it turns out according to [2] Theorem 2.4.1.that the canonical order on (G,S), given by x ⊂ y iff l(x) + l(x−1y) = l(y),makes G a ⊥–group if and only if the group G admits the presentation G =<

S; s2 = 1 for s ∈ S1, and sts−1t−1 = 1 for s, t ∈ S, s 6= t subject to the identityst = ts in G >. Such systems (G, S) were called in [2] partially commutative

Artin–Coxeter groups. In particular, for a pair (G,S) satisfying S ∩ S−1 = ∅,the canonical order on (G,S) makes G a ⊥–group iff the group G admits thepresentation

G =< S; sts−1t−1= 1 for s, t ∈ S, s 6= t subject to the identity st = ts in G >.

Such systems (G,S) were introduced by Baudisch in [3, 4] under the nameof semi–free groups, and extensively studied in the last few years under variousnames (right–angled Artin groups, free partially commutative groups, graph

groups) by people working in combinatorial and geometric group theory, asso-ciative algebras, computer science (see for instance the works [8–10]). Theirnice properties were exploited by Bestvina and Brady [5] in their constructionof examples of groups which are of type (FP ) but are not finitely presented, aswell as by Crisp and Paris [7] in their proof of a conjecture of Tits on the sub-group generated by the squares of the generators of an Artin group. Accordingto [8], the finitely generated right–angled Artin groups (moreover, the weaklypartially commutative Artin–Coxeter groups as defined in [2]) are linear andhence equationally noetherian.

On the other hand, any l–group (G, .,≤,∧,∨), not necessarily commutative,has a canonical structure of median group. Indeed, as the underlying lattice ofG is distributive, G has a canonical structure of median set with the mediandefined by Y (x, y, z) = (x ∧ y) ∨ (y ∧ z) ∨ (z ∧ x) = (x ∨ y) ∧ (y ∨ z) ∧ (z ∨ x).Obviously, the median operation is compatible with the multiplication, andhence, in particular, G becomes a median group. Notice that x ⊂ y iff x+ ≤ y+

and x− ≤ y−, (x∩ y)+ = x+ ∧ y+, (x∩ y)− = x− ∧ y−, where x+ = x∨ 1, x− =


(x−1)+ = (x∧ 1)−1. For x, y ∈ G, x ⊥ y iff x and y are orthogonal as elementsof the l–group G, i.e. |x| ∧ |y| = 1, where |x| = x∨x−1 = x+x−. Consequently,G is a ⊥–group by [6] Proposition 3.1.3. According to [2] Theorem 2.4.1.,the ⊥–group G is simplicial iff G is abelian, freely generated by the minimalpositive elements.

It is well known that any l–group is isomorphic to a subdirect product oftransitive groups [6] Theorem 4.1.7., in particular, any commutative l–group isisomorphic to a subdirect product of totally ordered abelian groups [6] Corol-lary 4.1.8. As the structure of ⊥–groups is far more intricate than the structureof the l–groups, we cannot expect to obtain a perfect analogue of the represen-tation theorem above. However we will show in this paper that any ⊥–groupnaturally admits a simple transitive action on a subdirect product of locallylinear median sets.

The paper has one section. Some basic notions of the theory of l–groups aresuitably extended to ⊥–groups in order to prove the main result of the paper(Corollary 2.8.).

2. A representation theorem for ⊥–groups

Lemma 2.1. Assume G is a ⊥–group. Given a reduced decomposition x =x1 • . . . • xn of some element x ∈ G, for any y ∈ G such that y ⊂ x there exist

yi ∈ G, i = 1, . . . , n, such that y = y1 • . . . • yn, and yi ⊂ xi, i = 1, . . . , n.

Proof – We proceed by induction on n, the case n = 1 being trivial.Assuming n ≥ 2, set x′ = x1 • . . . • xn−1 ⊂ x, y′ = y ∩ x′ ⊂ x′. By inductionhypothesis there exist yi ∈ G, i = 1, . . . , n− 1 such that y′ = y1 • . . . • yn−1 andyi ⊂ xi, i = 1, . . . , n − 1. Setting yn = (y′)−1y, we get yn ∩ (y′)−1x′ = 1 andyn ⊂ (y′)−1x = ((y′)−1x′) • xn, therefore yn ⊥ (y′)−1x′. As G is a ⊥–group, itfollows that yn ⊂ xn and hence y = y′ •yn = y1 • . . .•yn−1 •yn, as required.

Before considering the general case, let us treat the simpler case of the locallyboolean ⊥–groups. Recall that a median set X is locally boolean if any cell C

of X equals its boundary ∂C, i.e. for all x, y ∈ X and for every z ∈ [x, y] (i.e.Y (x, y, z) = z) there exists a (unique) u ∈ X such that [x, y] = [z, u]. A rooted

6 S. A. Basarab

locally boolean median set (X, 0), i.e. a locally boolean median set X togetherwith a fixed element (the root) 0 ∈ X, has a canonical structure of a quasi–

boolean algebra with the partial order x ⊂ y if x ∈ [0, y], the least element 0,the lattice operations x ∩ y = Y (x, 0, y), x ∪ y = the (unique) element z ∈ X

for which [x, y] = [x ∩ y, z], and the relative complement (negation) ¬xy = theunique element u for which [0, x] = [x ∩ y, u].

Proposition 2.1. Given a rooted locally boolean median set (X, 0), there

exists a unique group law + on X with 0 as neutral element such that X

becomes a ⊥–group w.r.t. + and the median operation on X. The group law

+ is commutative, and 2x = 0 for all x ∈ X.

Proof – Define x + y = (¬xy) ∪ (¬yx), the symmetrical difference of theelements x, y ∈ X. One checks easily that X is a 2–elementary ⊥–group w.r.t.+ and the median Y . Conversely, assuming . is a group operation on X with0 as neutral element, such that (X, ., Y ) is a ⊥–group, we have to show thatx . y = x + y for all x, y ∈ X. First notice that, assuming x ⊥ y, i.e. x ∩ y = 0since X is locally boolean, we must have x . y = y . x = x ∪ y = x + y. Forarbitrary x, y ∈ X we get x ∪ y 6= ∞, as X is locally boolean, and hencex ∪ y = x(x ∩ y)−1y = y(x ∩ y)−1x by the property (⊥). Considering the pair(x∩y,¬xy) of orthogonal elements, we get x = (x∩y)(¬xy) = (¬xy)(x∩y) for allx, y ∈ X, whence x(x∩y)−1 = ¬xy = (x∩y)−1x, and hence x∪y = (x∩y)−1xy,i.e. xy = yx = (x ∩ y)(x ∪ y). Thus the group operation . is commutative.Setting y = x−1, it follows that 0 = x . x−1 = (x∩x−1)(x∪x−1). On the otherhand, setting y = x ∩ x−1, we get y−1 = x ∪ x−1. As y ⊂ y−1, it follows thaty−1 = yz, where z = ¬y−1y ⊂ y−1. Consequently, y−1 = yz ⊂ y, thereforex ∩ x−1 = y = y−1 = x ∪ x−1 and hence x = x−1, i.e. x2 = 0 for all x ∈ X.Finally we obtain x + y = (¬xy) ∪ (¬yx) = (¬xy)(¬yx) = x(x ∩ y)y(x ∩ y) =xy(x ∩ y)2 = x . y, as required.

Corollary 2.1. Any locally boolean ⊥–group is isomorphic to a subdirect

product of a power set (ZZ/2ZZ)I , with the canonical group and median opera-

tions.

Proof – According to [1] Theorem 5.2.1., any rooted median set (X, 0) isidentified with a subdirect product of the power set (ZZ/2ZZ)I , where I = {P ∈


Spec(X)| 0 ∈ P, P 6= X}, trough the map x 7→ (xP )P∈I , where

xP =

{0 mod 2 if x ∈ P

1 mod 2 otherwise

Recall that Spec(X) = {P ⊆ X|P and X \ P are convex }. Assuming G is alocally boolean ⊥–group, it remains to observe that any P ∈ Spec(G) satisfying0 ∈ P, P 6= G, is a subgroup of G too, and G/P ∼= ZZ/2ZZ.

As in the case of l–groups, the convex subgroups play a special role in the studyof the ⊥–groups. The next lemma provides a first argument in this sense.

Lemma 2.2. If G is a ⊥–group and H is a convex subgroup of G, then the

quotient set G/H inherits from G a canonical structure of median set on which

G acts from the left.

Proof – First of all notice that xH = (x ∪ y)H and Hx = H(x ∪ y)whenever x ∈ G, y ∈ H and x ∪ y 6= ∞. Indeed, as G is a ⊥–group, we obtainx ∪ y = x(x ∩ y)−1y = y(x ∩ y)−1x ∈ xH ∩ Hx since z := x ∩ y ∈ H, asH is convex and 1, y ∈ H, therefore z−1y, yz−1 ∈ H, as H is a subgroup ofG. To prove the lemma we have to show that Y (x, y, zh)H = Y (x, y, z)H forall x, y, z ∈ G,h ∈ H. We get Y (x, y, zh) = zY (z−1x, z−1y, h) = z((z−1x ∩z−1y) ∪ h′), where h′ = (z−1x ∩ h) ∪ (z−1y ∩ h) ∈ H since 1 ∈ H, h′ ⊂ h ∈ H,and H is a convex subset of G. Thanks to the remark above, it follows thatY (x, y, zh)H = z(z−1x ∩ z−1y)H = Y (x, y, z)H as desired.

Since the intersection of a family of convex subgroups is a convex subgrouptoo, we may speak of the convex subgroup generated by any subset A ⊆ G anddenote it by C(A). A more precise description of C(A) is provided by the nextstatement.

Proposition 2.2. Given a subset A of the ⊥–group G, the convex subgroup

C(A) generated by A is the subgroup generated by [A∪{1}], the convex closure

of the subset A ∪ {1}.Proof – It suffices to show that for any A ⊆ G for which 1 ∈ A and y ∈ A

whenever y ⊂ x and x ∈ A, the subgroup B :=< A > generated by A is convex,i.e. the following two conditions are satisfied:

8 S. A. Basarab

(i) y ⊂ x and x ∈ B imply y ∈ B, and(ii) x ∪ y ∈ B whenever x ∪ y 6= ∞, x ∈ B and y ∈ B.First notice that we have only to verify (i), since assuming (i) and x, y ∈ B

such that x∪ y 6= ∞, we get x∪ y = x(x∩ y)−1y, as G is a ⊥–group, and hencex∪y ∈ B since x∩y ∈ B by (i). Let l denote the length function assigned to thepair (B, A\{1}), i.e. for any x ∈ B, l(x) is the minimal length of any expressionx = a1 . . . ad, with ai ∈ (A ∪ A−1) \ {1}, i = 1, . . . , d. Assuming x ∈ B andy ⊂ x, we show by induction on l(x) = d that y ∈ B. The case d = 0, i.e.x = 1, is trivial, so assume d ≥ 1, x = a1 . . . ad. Setting x′ = a1 . . . ad−1, letu = (x′)−1 ∩ ad, x

′ = x′′ • u−1, ad = u • b, x = x′′ • b. As G is a ⊥–group, andy ⊂ x, we get by Lemma 2.1. y = y′•y′′, with y′ = y∩x′′ ⊂ x′′ ⊂ x′, and y′′ ⊂ b.Since y′ ⊂ x′ and l(x′) = d− 1 < d, it follows by the induction hypothesis thaty′ ∈ B. On the other hand, b−1y′′ ⊂ b−1 ⊂ a−1

d . We distinguish two cases:Case (1): a−1

d ∈ A \ {1}. Then, by the choice of A, we obtain b−1 ∈ A, andb−1y′′ ∈ A, therefore y′′ ∈ B.Case (2): ad ∈ A \ {1}. Then adb

−1 ⊂ adb−1y′′ ⊂ ad, therefore, by the choice

of A, adb−1 ∈ A and (adb

−1)y′′ ∈ A, and hence y′′ ∈ B.Thus in the both cases y′′ ∈ B. Since y′ ∈ B too, it follows that y = y′y′′ ∈

B, as desired.

Corollary 2.2. For all x ∈ G, the convex subgroup C(x) generated by x is

the subgroup of G generated by [1, x].

Corollary 2.3. Let H be a convex subgroup of the ⊥–group G, and let x ∈ G.

Then C(H ∪ {x}) is the subgroup generated by H ∪ [1, x].

Proof – One checks easily that the convex closure [H ∪ {x}] of H ∪ {x}is the union of the cells [h, x], where h ranges over H. Thus any elementy ∈ [H ∪ {x}] has the form y = Y (h, x, y) = (h ∩ y) ∪ (x ∩ y) for some h ∈ H.As H is convex, h∩ y ∈ H, so [H ∪ {x}] = {h∪ y | h ∈ H, y ⊂ x, h∪ y 6= ∞} ⊆⋃{yH | y ⊂ x} by the remark from the beginning of the proof of Lemma 2.2.The conclusion of the statement is now immediate by Proposition 2.2.

Denote by C(G) the poset with respect to inclusion of all convex subgroups ofa ⊥–group G.


Proposition 2.3. C(G) is a complete lattice with∧

Hi =⋂

Hi,∨

Hi =<

⋃Hi >, the submonoid generated by

⋃Hi. Moreover the lattice C(G) is

distributive, and H ∩ (∨

Hi) =∨

(H ∩ Hi) for arbitrary families of convex

subgroups (Hi)i∈I .

Proof – To prove that∨

Hi =<⋃

Hi >, let M denote the subset con-sisting of those x ∈ G for which there exist n ∈ IN and xj ∈

⋃Hi, j = 1, . . . , n

such that x = x1 • x2 . . . • xn. As⋃

Hi ⊆ M ⊆<⋃

Hi >, it suffices to showthat M is a convex subgroup. Since M is closed under inversion and G is a⊥–group, we have to show that y ∈ M whenever y ⊂ x and x ∈ M , and M isclosed under multiplication. Assuming x ∈ M of the form above and y ⊂ x, weget y = y1• . . .•yn with yj ⊂ xj , j = 1, . . . , n by Lemma 2.1.. As the subgroupsHi are convex it follows that y ∈ M . On the other hand, assuming x, y ∈ M ,set z = x−1 ∩ y. As xz ⊂ x ∈ M , and y−1z ⊂ y−1 ∈ M , we get xz, z−1y ∈ M

and hence xy = (xz) • (z−1y) ∈ M , as required. It remains to prove the latterequality of the statement, implying in particular the distributivity of the lat-tice C(G). One inclusion is obvious, so let x ∈ H ∩ (

∨Hi), say x = x1 . . . xn,

with xj ∈⋃

Hi, j = 1, . . . , n. Assuming by the first part of the proof that thedecomposition of x above is reduced, we obtain xj ∈ H ∩ (

⋃Hi), j = 1, . . . , n,

therefore x ∈<⋃

Hi >=∨

Hi, as desired.

Remark 2.1 - In general, the operation ∨ is not distributive w.r.t. an infiniteintersection. For instance, let G = ZZIN with the arboreal structure induced bythe canonical l–group structure on G, H = ZZ(IN),Hi = {x = (xj) ∈ G | xi =0}. Then H ∨ (

⋂i∈IN Hi) = H 6= G =

⋂i∈IN(H ∨Hi).

The following statement provides natural examples of convex subgroups.

Proposition 2.4. For any subset A of a ⊥–group G, the set A⊥ :={y ∈ G |y ⊥ x for all x ∈ A} is a convex subgroup of G.

Proof – Obviously y ∈ A⊥ whenever y ⊂ x and x ∈ A⊥. Notice alsothat A⊥ is closed under inversion by [2] Lemma 2.2.4., so it remains to showthat A⊥ is closed under multiplication. Let x, y ∈ A⊥. Writing xy = x′ • y′,where x′ = x(x−1 ∩ y) ⊂ x ∈ A⊥, (y′)−1 = y−1(x−1 ∩ y) ⊂ y−1 ∈ A⊥, weget x′, y′ ∈ A⊥, so we may assume from the beginning that xy = x • y. For

10 S. A. Basarab

any a ∈ A, we have to show that xy ⊥ a. Setting b = xy ∩ a ⊂ x • y, we getb ⊥ x since b ⊂ a and a ⊥ x. Consequently, b ⊂ a ∩ y = 1, i.e. b = 1, asrequired. Similarly we get a−1 ∩ xy = 1, therefore axy = a • x • y = x • y • a,i.e. xy ∪ a = axy 6= ∞ and hence xy ⊥ a, as desired.

Corollary 2.4. For A ⊆ G,A⊥ = C(A)⊥.

Proof – The inclusion C(A)⊥ ⊆ A⊥ is obvious. Conversely, let b ∈ A⊥,i.e. A ⊆ b⊥. As b⊥ ∈ C(G) by Proposition 2.4., we obtain C(A) ⊆ b⊥, i.e.b ∈ C(A)⊥, and hence A⊥ ⊆ C(A)⊥, as required.

Corollary 2.5. For A ⊆ G,A ⊆ A⊥⊥, and A⊥ = A⊥⊥⊥.

Corollary 2.6. A = B⊥ for some B ⊆ G iff A = A⊥⊥.

Corollary 2.7. Assuming G is a ⊥–group, let x, y ∈ G be such that x∪y 6= ∞.

Then C(x) ∩ C(y) = C(x ∩ y) and C(x) ∨ C(y) = C(x ∪ y). In particular,

C(x) ∩ C(y) = {1} and C(x) ∨ C(y) = C(xy) = C(yx) whenever x ⊥ y.

Proof – The equality C(x) ∨C(y) = C(x ∪ y) is immediate since x ∪ y =x(x∩y)−1y. To prove the equality C(x)∩C(y) = C(x∩y), setting z := x∩y, x =z • x′, y = z • y′, we get x′ ⊥ y′ as x ∪ y 6= ∞. Thus C(x) = C(z) ∨ C(x′)and C(y) = C(z) ∨ C(y′), and hence C(x) ∩ C(y) = C(z) ∨ (C(x′) ∩ C(y′)) byProposition 2.3., so it remains to show that C(x)∩C(y) = {1} whenever x ⊥ y.As x ∈ x⊥⊥, it follows by Proposition 2.4. that C(x) ⊆ x⊥⊥, and similarlyC(y) ⊆ y⊥⊥, so it suffices to check that x⊥⊥ ∩ y⊥⊥ = {1} whenever x ⊥ y.Now, if u ∈ x⊥⊥ ∩ y⊥⊥, it follows that u ∈ y⊥ since u ∈ x⊥⊥ and y ∈ x⊥.Consequently, u ⊥ u, i.e. u = 1, as u ∈ y⊥⊥.

The next lemma is immediate.

Lemma 2.3. The following are equivalent for a convex subgroup H of a ⊥–

group G :

(i) For x, y ∈ G such that x ∪ y 6= ∞, x ∩ y ∈ H implies either x ∈ H or

y ∈ H.

(ii) For x, y ∈ G, x ⊥ y implies either x ∈ H or y ∈ H.

(iii) For x, y, z ∈ G either (x ∩ y)H = (y ∩ z)H or (y ∩ z)H = (z ∩ x)H or

(z ∩ x)H = (x ∩ y)H.


(iv) G/H is a locally linear median set.

Proof – (i) −→ (ii) is obvious.(ii) −→ (iii): As x∩ y, y ∩ z, z ∩x ⊂ Y (x, y, z), it follows that u−1(x∩ y) ⊥

u−1(y ∩ z), u−1(y ∩ z) ⊥ u−1(z ∩ x) and u−1(z ∩ x) ⊥ u−1(x ∩ y), whereu = x ∩ y ∩ z. The implication (ii) −→ (iii) is now immediate.

(iii) −→ (iv): Let x, y, z ∈ G be such that xH ⊂ zH, i.e. xH = (x ∩ z)H,and yH ⊂ zH, i.e. yH = (y ∩ z)H. By assumption it follows that either(x ∩ y)H = (y ∩ z)H = yH or yH = (y ∩ z)H = (z ∩ x)H = xH or xH =(z ∩x)H = (x∩ y)H, and hence either xH ⊂ yH or yH ⊂ xH, i.e. the medianset G/H is locally linear.

(iv) −→ (i): Assuming that x∪ y 6= ∞ and x∩ y ∈ H, it follows that xH ∈[H, zH], yH ∈ [H, zH], where z = x∪ y, and hence either xH = (x∩ y)H = H,i.e. x ∈ H, or yH = (x ∩ y)H = H, i.e. y ∈ H, as required.

Call prime any convex subgroup H of a ⊥–group G satisfying the equivalentconditions from Lemma 2.3. Notice that a prime convex subgroup H is notnecessarily prime as a convex subset of the median set G, i.e. G \ H is notnecessarily a convex subset. Since

⋂i∈I Hi is obviously prime whenever (Hi)i∈I

is a chain of prime convex subgroups, it follows by Zorn’s lemma that anyprime convex subgroup contains a minimal prime. Notice also that any convexsubgroup H ′ lying over a prime convex subgroup H is obviously prime. Asin the case of l–groups, call regular any convex subgroup H of G such thatH =

⋂i∈I Hi, where Hi ∈ C(G), i ∈ I, implies H = Hi for some i ∈ I.

Lemma 2.4. Any regular convex subgroup H of a ⊥–group G is prime.

Proof – Let x, y ∈ G be such that x ⊥ y, and set H1 = H ∨ C(x), H2 =H ∨ C(y). By Proposition 2.3. and Corollary 2.7. we obtain H1 ∩ H2 =H ∨ (C(x) ∩ C(y)) = H, therefore, by the regularity of H, either H1 = H, i.e.x ∈ H, or H2 = H, i.e. y ∈ H.

For any element a ∈ G\{1}, let M(a) denote the set of those convex subgroupsH of G which are maximal with the property that a 6∈ H. The next lemmaprovides an equivalent description for regular convex subgroups.

12 S. A. Basarab

Lemma 2.5. The necessary and sufficient condition for a convex subgroup

H 6= G to be regular is that there exists a ∈ G \ {1} such that H ∈M(a).

Proof – Assuming H is regular, set H∗ =⋂{U ∈ C(G) | H ⊆ U,U 6=

H}. As H is regular, H 6= H∗, so H ∈ M(a) for all a ∈ H∗ \H. Conversely,assume H ∈ M(a) for some a ∈ G \ {1}, and H =

⋂i∈I Hi,Hi ∈ C(G). Let

i ∈ I be such that a 6∈ Hi. Since H is maximal amongst the convex subgroupswhich do not contain a, it follows that H = Hi, and hence H is regular.

The next statement is now immediate by Zorn’s lemma.

Proposition 2.5. For any convex subgroup H of a ⊥–group G and for any

a ∈ G \ H, there exists M ∈ M(a) such that H ⊆ M . In particular, any

convex subgroup H is an intersection of regular, resp. of minimal prime convex

subgroups.

Corollary 2.8. Given a ⊥–group, there exists a canonical simple transitive

action of G on a subdirect product of locally linear median sets.

Proof – Let Pmin(G) denote the set of the minimal prime convex sub-groups of G. By Proposition 2.5., G is naturally embedded as median setinto the product

∏H∈Pmin(G)

G/H of locally linear median sets, on which G acts

canonically from the left.

References

[1] Basarab, S. A.: The dual of the category of generalized trees, An. St.

Univ. Ovidius Constanta 9 (1) (2001), 1–20.

[2] Basarab, S. A.: Partially commutative Artin–Coxeter groups and theirarboreal structure, J. Pure Appl. Algebra 176 (1) (2002), 1–25.

[3] Baudisch, A.: Kommutationsgleichungen in Semifreien Gruppen, Acta

Math. Acad. Sci. Hungaricae (3–4) 29 (1977), 235–249.


[4] Baudisch, A.: Subgroups of semifree groups, Acta Math. Acad. Sci.

Hungaricae (1–4) 81 (1981), 19–28.

[5] Bestvina, M. and Brady, N.: Morse theory and finiteness propertiesof groups, Invent. Math. 129 (1997), 470–495.

[6] Bigard, A., Keimel, K. and Wolfenstein, S.: Groupes et AnneauxReticules, Lecture Notes Math. 608, Springer, Heidelberg (1977).

[7] Crisp, J. and Paris, L.: The solution to a conjecture of Tits on thesubgroup generated by the squares of the generators of an Artin group,Invent. Math. 145 (2001), 19–36.

[8] Davis, M. and Januszkiewicz, T.: Right–angled Artin groups arecommensurable with right–angled Coxeter groups, J. Pure Appl. Algebra

153 (2001), 229–235.

[9] Duchamp, G. and Krob, D.: The free partially commutative Lie alge-bra: bases and ranks, Adv. Math. (1) 95 (1992), 92–126.

[10] Duchamp, G. and Krob, D.: Free partially commutative structures, J.

Algebra (2) 126 (1993), 318–361.

[11] Kolibiar, M.: Median groups, Arch. Math., Brno (1/2) 25 (1989),73–82.

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