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Interior Algebras 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. Bermanwhoseseminar 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 whichhelped 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 (iii) 3 The subject matter of the paper (vii) CHAPTER 0. PRELIMINARIES 1 Universal algebra 2 Lattices CHAPTER I. GENERAL THEORY OF (GENERALIZED) INTERIOR ALGEBRAS 16 1 Generalized interior algebras: definitions and basic properties I6 2 Interior algebras: definition, basic properties and relation with generalized interior algebras 24 3 Twoinfinite interior algegras generated by one element 30 4 Principal ideals in finitely generated free algebras in 31 and 5; 36 5 Subalgebras of finitely generated free algebras in 31 and 3; 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: ,-1 B. and -H-,-1 B. and-1 BI 86 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 , n 6 N 128 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. Animportant 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, in the study of certain non-classical, so-called modal logics. (iii) Algebraic structures arising from logic have received a great deal of attention in the past. As early as in the 19th century George Boole initiated the study of the relationship betweenalgebra and classical pro­ positional logic, which resulted in the development of what we nowknow as the theory of Boolean algebras, a subject which has been studied very thoroughly. In the twenties and thirties several new systems of proposi­ tional logic were introduced, notably the intuitionistic logic, created by Brouwer and Heyting [30], various systems of modal logic, introduced by Lewis (see Lewis and Langford [32]), and many-valued logics, proposed by Post [21] and Lukasiewicz. The birth of these non-classical logics sti­ mulated investigations into the relationships between these logics and the corresponding classes of algebras as well as into the structural properties of the algebras associated with these logics. The algebras turn out to be interesting not only from a logical point of view, but also in a purely algebraic sense, and structures like Heyting algebras, Brouwerianalgebras, distributive pseudo-complementedlattices, Post algebras and Lukasiewicz algebras have been studied intensively.
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