VARIETIES OF INTERIOR ALGEBRAS

bv

w. J. BLOK

$@@>@

Abstract

Westudy (generalized) Boolean algebras endowedwith an interior operator, called (generalized) interior algebras. Particular attention is paid to the structure of the free (generalized) interior algebra on a finite numberof generators. Free objects in somevarieties of (generalized) interior algebras are determined. Using methods of a universal algebraic nature we investigate the lattice of varieties of interior algebras.

Keywords:(generalized) interior algebra, Heyting algebra, free algebra, *ra1gebra, lattice of varieties, splitting algebra.

AMSMOS70 classification: primary 02 J 05, 06 A 75

secondary 02 C 10, 08 A 15.

Dvuk:HuisdrukkerijUniversiteitvanAmsterdam T _

VARIETIES OF INTERIOR ALGEBRAS

ACADEMISCH PROEFSCHRIFT

TER VERKRIJGING VAN DE GRAAD VAN

DOCTOR IN DE WISKUNDE EN NATUURWETENSCHAPPEN

AAN DE UNIVERSITEIT VAN AMSTERDAM

OP GEZAG VAN DE RECTOR MAGNIFICUS

DR G. DEN BOEF

HOOGLERAAR IN DE FACULTEIT DER WISKUNDE EN NATUURWETENSCHAPPEN

IN HET OPENBAAR TE VERDEDIGEN

IN DE AULA DER UNIVERSITEIT

(TIJDELIJK IN DE LUTHERSE KERK, INGANG SINGEL 4!], HOEK SPUI)

OP WOENSDAG 3 NOVEMBER 1976 DES NAMIDDAGS TE 4 UUR

DOOR

WILLEM JOHANNES BLOK

GEBOREN TE HOORN

Promotor : Prof. Dr. Ph.Dwinger

Coreferent: Prof. Dr. A.S.Troe1stra

Druk: Huisdrukkerii Universiteit van Amsterdam T . I-‘£670/

aan mijn oude/Us

(U171 /LQVLQQ

Acknowledgements

I ammuch indebted to the late prof. J. de Groot, the contact with

whomhas meant a great deal to me.

The origin of this dissertation lies in Chicago, during mystay at

the University of Illinois at ChicagoCircle in the year '73 - '74.

I want to express myfeelings of gratitude to all persons whocontri­

buted to making this stay as pleasant and succesful as I experienced

it, in particular to prof. J. Bermanwhose seminar on "varieties of

lattices" influenced this dissertation in several respects. Prof.

Ph. Dwinger, who introduced me into the subject of closure algebras

and with whomthis research was started (witness Blok and Dwinger [75])

was far more than a supervisor; mathematically as well as personally

he was a constant source of inspiration.

I amgrateful to prof. A.S. Troelstra for his willingness to be

coreferent. The attention he paid to this work has resulted in many

improvements.

Finally I want to thank the Mathematical Institute of the University

of Amsterdamfor providing all facilities which helped realizing this

dissertation. Special thanks are due to Mrs. Y. Cahn and Mrs. L. Molenaar,

whomanaged to decipher my hand-writing in order to produce the present

typewritten paper. Most drawings are by Mrs. Cahn's hand.

CONTENTS

INTRODUCTION

1 Someremarks on the subject and its history

2 Relation to modal logic

3 The subject matter of the paper

CHAPTER 0. PRELIMINARIES

1 Universal algebra

2 Lattices

CHAPTER I.

1 Generalized interior algebras: definitions and

basic properties

(iii) (vii)

GENERAL THEORY OF (GENERALIZED) INTERIOR ALGEBRAS 16

I6

2 Interior algebras: definition, basic properties and

relation with generalized interior algebras

3 Twoinfinite interior algegras generated by one

element

4 Principal ideals in finitely generated free

algebras in 31 and 5;

5 Subalgebras of finitely generated free algebras

in 31 and 3;

24

30

36

50

6 Functional freeness of finitely generated

algebras in gi and Q; 57

7 Someremarks on free products, injectives

and weakly projectives in fii and g; 70

CHAPTER II. ON SOME VARIETIES OF (GENERALIZED) INTERIOR 85

ALGEBRAS

1 Relations between subvarieties of gi and

III: , B. and H-, B. and BI 86-1 - -1 -1

2 The variety generated by all (generalized)

interior *-algebras 95

3 The free algebra on one generator in §;* 104

4 Injectives and projectives in fig and §;* 112

5 Varieties generated by (generalized) interior

algebras whose lattices of open elements are

chains I19

6 Finitely generated free objects in Q; and

M , n 6 N 128-n

7 Free objects in fl- and fl 145

CHAPTER III. THE LATTICE OF SUBVARIETIES OF fii I52

1 General results 153

2 Equations defining subvarieties of fii I57

3 Varieties associated with finite subdirectly

irreducibles 167

A Locally finite and finite varieties 178

5 The lattice of subvarieties of M 189

6 The lattice of subvarieties of (fii: K3) 200 7 The relation between the lattices of

subvarieties of fii and E 209

8 On the cardinality of some sublattices of 52 219

9 Subvarieties of Qi not generated by their finite members 229

REFERENCES 238

SAMENVATTING 246

(1)

INTRODUCTION

1 Someremarks on the subject and its history

In an extensive paper titled "The algebra of topology", J.C.C. McKinsey and A. Tarski [44] started the investigation of a class of algebraic struc­ tures which they termed "closure algebras". The notion of closure algebra developed quite naturally from set theoretic topology. Already in 1922, C. Kuratowski gave a definition of the concept of topological space in terms of a (topological) closure operator defined on the field of all subsets of a set. Bya process of abstraction one arrives from topological spaces defined in this manner at closure algebras, just as one may inves­ tigate fields of sets in the abstract setting of Booleanalgebras. A clo­ sure algebra is thus an algebra (L,(+,.,',C,0,l)) such that (L,(+,.,',0,l)) is a Booleanalgebra, where +,.,' are operations satisfying certain postu­ lates so as to guarantee that they behave as the operations of union, in­ tersection and complementation do on fields of sets and where 0 and l are nullary operations denoting the smallest element and largest element of L respectively. The operation C is a closure operator, that is, C is a unary operation on L satisfying the well-known "Kuratowski axioms"

(i) x s x° (ii) xcc = xc

(iii) (x+y)° = x° + y“ (iv) 0° = o.

The present paper is largely devoted to a further investigation of classes of these algebras. However, in our treatment, not the closure operator C will be taken as the basic operation, but instead the interior operator 0, which relates to C by x0 = x'C' and which satisfies the postulates (i)' x0 s x, (ii)' x°° = x°, (iii)' (xy)° = x°y° and (iv)' 1° = 1, corresponding to (i) - (iv). Accordingly, we shall speak of interior algebras rather than closure algebras. The reason for our favouring the interior operator is the following. An important feature in the structure of an interior algebra is the set of closed elements, or, equivalently, the set of open elements. In a continuation of their work on closure algebras, "On closed elements in closure algebrasfi McKinseyand Tarski showed that the set of closed elements

(ii)

of a closure algebra may be regarded in a natural way as what one would now call a dual Heyting algebra. Hence the set of open elements may be

taken as a Heyting algebra, that is, a relatively pseudo-complemented distributive lattice with 0,], treated as an algebra (L,(+,.,+,O,l)) where + is defined by a + b = max {z I az 5 b}. Therefore, since the theory of Heyting algebras is nowwell-established, it seems pre­ ferable to deal with the open elements and hence with the interior operator such as to makeknownresults more easily applicable to the algebras under consideration.

when they started the study of closure algebras McKinseyand Tarski wanted to create an algebraic apparatus adequate to the treatment of certain portions of topology. Theywere particularly interested in the question as to whether the interior algebras of all subsets of spaces like the Cantor discontinuum or the Euclidean spaces of any number of dimensions .are functionally free, i.e. if they satisfy only those to­ pological equations which hold in any interior algebra. By topological equations we understand those whose terms are expressions involving only the operations of interior algebras. McKinseyand Tarski proved that the answer to this question is in the affirmative: the interior algebra of any separable metric space which is dense in itself is func­ tionally free. Hence, every topological equation which holds in Eucli­ dean space of a given number of dimensions also holds in every other topological space.

However, for a deeper study of topology in an algebraic framework in­ terior algebras prove to be too coarse an instrument. For instance, even a basic notion like the derivative of a set cannot be defined in terms

of the interior operator. A possible approach, which was suggested in McKinseyand Tarski [44] and realized in Pierce [70], would be to consi­

der Boolean algebras endowedwith more operations of a topological nature than just the interior operator. That will not be the course taken here. Weshall stay with the interior algebras, not only because the algebraic theory of these structures is interesting, but also since interior alge­ bras, rather unexpectedly, appear in still another branch of mathematics, namely, i