Advanced Topics in Relation Algebras

Transcription

1 Advanced Topics in Relation Algebras

2 Steven Givant Advanced Topics in Relation Algebras Relation Algebras, Volume 2 123

3 Steven Givant Department of Mathematics Mills College Oakland, CA, USA ISBN ISBN (ebook) DOI / Library of Congress Control Number: Mathematics Subject Classification: 03G15, 03B20, 03C05, 03B35 Springer International Publishing AG 2017 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Printed on acid-free paper This Springer imprint is published by Springer Nature The registered company is Springer International Publishing AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

4 To the memories of Alfred Tarski and Bjarni Jónsson.

5 Preface This is the second volume of a two-volume textbook on relation algebras. The first volume, Introduction to Relation Algebras, begins with the underlying motivation, going back to the calculus of relations of De Morgan, Peirce, and Schröder, and with the basic definitions, axioms, and examples of the subject. There follows a development of the arithmetic of relation algebras in which the most important laws are derived systematically from the axioms, with special emphasis on those laws that apply to, or even characterize, specific types of elements, such as as equivalence elements, functional elements, and ideal elements. The remainder of the first volume is devoted to an exposition of the algebraic side of the subject: subalgebras, homomorphisms, ideals, quotient algebras, simple algebras, direct products, and so on. The purpose of this second volume is to make a systematic, cohesive, and detailed presentation of a selection of more advanced topics of the subject, topics that have been active areas of research over the last few decades, more accessible to readers, with the hope of bringing them to some of the frontiers of research on relation algebras and Boolean algebras with operators. Intended audience This volume is aimed at, but is not limited to, graduate students and professionals in a variety of mathematical disciplines, especially various branches of logic, universal algebra, and theoretical computer science. As regards the background needed to read this volume, it is helpful to have a general understanding of the basic notions and results of the theory of relation algebras, and some familiarity with the basic notions and results of universal algebra. The background provided in the vii

6 viii Preface first volume, Introduction to Relation Algebras, is more than sufficient. Note that this second volume contains numerous, essential references to the first. The reader is strongly encouraged to secure at least electronic access to the first book in order to make use of the second. Any reference in this volume to material in Chapters 1 13 refers to the relevant result in the first volume. Each chapter ends with a historical section and a substantial number of exercises. The exercises vary in difficulty from routine problems that help readers understand the basic definitions and theorems presented in the text, to intermediate problems that extend or enrich the material developed in the text, to difficult problems that often present important results not covered in the text. Hints and solutions to some of the exercises are available for download from the Springer book webpage. The main topics covered in this volume are canonical extensions, completions, representation theorems, varieties and universal classes, and atom structures. Acknowledgements I am very much indebted to Hajnal Andréka, Robert Goldblatt, Ian Hodkinson, Peter Jipsen, Bjarni Jónsson, Richard Kramer, Roger Maddux, Ralph McKenzie, Don Monk, and István Németi for the helpful remarks and suggestions that they provided to me in correspondence during the composition of this work. Some of these remarks are referred to in the historical sections at the end of the chapters. Kexin Liu read the second draft of the entire text, caught numerous typographic errors, and made many suggestions for stylistic improvements. I am very grateful to her. Loretta Bartolini an editor of the mathematical series Graduate Texts in Mathematics, Undergraduate Texts in Mathematics, anduniversitext published by Springer, has served as the editor for these two volumes. She has given me a great deal of advice and guidance during the publication process, and I am very much indebted to her and her entire production team at Springer for pulling out all stops, and doing the best possible job in the fastest possible way, to produce these two companion volumes. Any errors or flaws that remain in the volumes are, of course, my own responsibility. California, USA July 2017 Steven Givant

7 Introduction A mathematical theory of binary relations, together with certain operations on and between these relations, was initiated by De Morgan [26] in It was given a proper foundation by Peirce [116], who over a ten-year period settled on a very natural set of operations and developed a kind of calculus of relations, in analogy with the Boole-Jevons calculus of classes. This calculus was further developed and extended in a very systematic way by Schröder [121]. In 1941, Tarski [132] reformulated the calculus of relations as an abstract, algebraic theory with a finite number of essentially equational axioms later to be called the theory of relation algebras much as Huntington had reformulated the calculus of classes as an abstract, algebraic theory with a finite number of equational axioms, later called the theory of Boolean algebras. Tarski posed several fundamental problems concerning his reformulated theory, and these problems initiated a period of sustained growth and development of the subject and the closely related, but much more general, theory of Boolean algebras with operators. A brief sketch of some of the most important developments during the period is given in the introduction to the first volume, Introduction to Relation Algebras. Since that time, some of the more active areas of research in the subject have been: applications to logic; applications to computer science; canonical extensions and completions of Boolean algebras with operators; connections with the theory of cylindric algebras; representation theory; decision problems; the lattice of varieties of relation algebras; axiomatizability and non-axiomatizability of classes of relation algebras, in particular, the class of representable relation algebras; free relation algebras; and generalized relativizations of relation algebras. ix

8 x Introduction It would be impossible to treat all of these topics in depth in this one volume. We focus on a few of them in the main text, and briefly discuss some of the others in the Epilogue. Description and highlights of this volume This volume contain six chapters that deal with more advanced topics. Chapter 14 introduces the notion of the canonical, or perfect, extension of a Boolean algebra with operators A as an analogue of the well-known construction of the Boolean algebra of all subsets of the set of ultrafilters in a given Boolean algebra. The canonical extension of A is a complete and atomic extension of A with certain additional properties, namely the compactness and atom separation properties. It is shown that every Boolean algebra with operators A has a canonical extension, and this extension is unique up to isomorphisms that are the identity function on A. The preservation theorems for canonical extensions say that all positive equations, and certain types of positive implications, are preserved under the passage to canonical extensions. It follows, in particular, that every relation algebra has an essentially unique canonical extension that is also a relation algebra. An interesting consequence of the preservation theorems is that a relation algebra has a total direct decomposition into the product of finitely many simple factors if and only if its canonical extension has a corresponding total direct decomposition into the product of finitely many simple factors. It is also shown that every homomorphism ϕ between Boolean algebras with operators A and B canbeextendedinauniqueway to a complete homomorphism between the canonical extensions of A and B, and this extension is one-to-one or onto if and only if ϕ is one-to-one or onto. This extension theorem implies a number of algebraic preservation theorems for canonical extensions. For example, if A is a subalgebra of B, then the canonical extension of A is a complete subalgebra of the canonical extension of B. A characterization is given of when the canonical extension of a subalgebra of a full set relation algebra A is itself a complete subalgebra of A. Finally, at the end of the chapter it is shown that every homomorphic image of a relation algebra A is isomorphic to a relativization of A to some (closed) ideal element belonging to the canonical extension of A. Thus, every

9 Introduction xi homomorphic image of a relation algebra A is rather close to being a subalgebra of A (up to isomorphisms). A parallel development for the notion of the completion of a Boolean algebra with quasi-complete operators A is given in Chapter 15. This is the analogue of the construction of the completion of a Boolean algebra via the algebra of complete ideals, or equivalently, via Dedekind cuts, and consequently it is also the analogue of the construction of the real numbers from the rational numbers via Dedekind cuts. It is shown that every Boolean algebra with quasi-complete operators A has a completion that is unique up to isomorphisms that are the identity function on A. Just as in the case of canonical extensions, there are preservation theorems saying that positive equations and certain types of positive implications are preserved under the passage to completions. In particular, every relation algebra has an essentially unique completion that is also a relation algebra. An interesting consequence of the preservation theorems is that a relation algebra has a total direct decomposition into a product of any number of simple factors if and only if its completion has a corresponding total direct decomposition into a product of simple factors. It is also shown that every complete homomorphism ϕ from a Boolean algebra with quasi-complete operators A into a complete Boolean algebra with quasi-complete operators B can be extended in a unique way to a complete homomorphism from the completion of A into B. Moreover,ifϕ is one-to-one or if B is the completion of the image of A under ϕ, then the extension of ϕ is one-to-one or onto respectively. As an application of this extension theorem, it is shown that the completion of A can be characterized as the minimal complete, regular extension of A. The extension theorem also implies a number of algebraic preservation theorems for completions. For example, if A is a regular subalgebra of B, then the completion of A is, up to isomorphism, a regular subalgebra of the completion of B. Finally, it is shown that every complete homomorphic image of a relation algebra A is isomorphic to a relativization of A to some ideal element belonging to the completion of A. A representation of a relation algebra A is an isomorphism from A to a set relation algebra. A representation is said to be complete if it preserves all existing sums as unions. Representations are the subject of Chapters 16 and 17. The focus of first of these chapters is on the algebra of representations. A Löwenheim-Skolem-Tarski type theorem is proved for representations: if a relation algebra of cardinality κ is representable as a set relation algebra on an infinite set, then it is rep-

10 xii Introduction resentable as a set relation algebra on sets of every infinite cardinality greaterthanorequaltoκ. Most of these representations are incomplete. Also, it is shown that the property of being representable is preserved under the standard algebraic constructions: subalgebras, relativizations, homomorphic images, direct products, and directed unions of representable relation algebras are all representable. Highlights of the chapter include a proof that the canonical extension of a representable relation algebra is completely representable, and a proof that the completion of a completely representable relation algebra is completely representable. As regards the relationship between representability and complete representability, it is shown that a completely representable relation algebra is necessarily atomic, and if an atomic relation algebra A has the property that the relative product of any two atoms is a finite sum of atoms, then from every representation of A one may construct a complete representation of A. Chapter 17 contains a number of representation theorems for concrete classes of relation algebras. For example, every formula relation algebra is representable, and every representable relation algebra is isomorphic to a relativization of a formula relation algebra. Every Boolean relation algebra is representable, and every atomic Boolean relation algebra is completely representable. Every complex algebra of a group is completely representable via the Cayley representation, and every complete representation of this complex algebra is equivalent to its Cayley representation. Not every complex algebra of a projective geometry is representable; in fact, there are infinitely many complex algebras of finite projective lines that are not representable. The representability of the complex algebra of a projective geometry P may be characterized in terms of P as follows: the complex algebra of P is representable if and only if P is embeddable into a projective geometry of one higher dimension, and if this is the case, then P has a complete representation in terms of an affine geometry of which P is the geometry at infinity. A connection between the collineations of a projective geometry and the complete representations of the complex algebra of the geometry is established, and using this connection, a formula for the number of inequivalent representations of the complex algebras of finite projective lines is given. For example, the complex algebra of a projective line of order nine has 56,700 inequivalent representations. The representability of various small relation algebras is also studied, and an example of a minimal non-representable relation algebra (having four atoms) is given. An analysis of this algebra also

11 Introduction xiii yields a concrete equation that is true in all representable relation algebras but not in all relation algebras. The second part of Chapter 17 begins with a general theorem to the effect that every relation algebra has a quasi-representation, and every atomic relation has a complete quasi-representation. A quasirepresentation is a bijection from a relation algebra to a set relation algebra that maps the identity element to the identity relation and preserves sums as unions, relative products as relational compositions, and converses as relational inverses; but such a mapping need not preserve products as intersections, nor complements as set-theoretic complements. With the help of this theorem, it is shown that every atomic relation algebra with functional atoms is completely representable. Similarly, it is shown that a singleton-dense relation algebra that is to say, a relation algebra in which every non-zero element is above an element that behaves like a singleton relation is representable. Chapter 18 is concerned with varieties of relation algebras and universal classes of simple relation algebras, that is to say, classes of relation algebras axiomatizable by sets of equations and first-order universal sentences respectively. The initial sections contain proofs of the following theorems: the Fundamental Theorem of Ultraproducts, which says that first-order properties are preserved under the passage to ultraproducts; the closely related SP u -Theorem, which says that a class of algebras is axiomatizable by a set of universal sentences if and only if it is closed under ultraproducts and subalgebras; and the wellknown HSP-Theorem, which says that a class of algebras is axiomatizable by a set of equations if and only if it is closed under subalgebras, direct products, and homomorphic images. The Correspondence Theorem for the lattice of varieties says that the correspondence mapping every variety K of relation algebras to the universal class of simple relation algebras in K is a lattice isomorphism from the lattice of varieties of relation algebras to the lattice of universal classes of simple relation algebras. Consequently, one may gain information about the lattice of varieties by studying the lattice of universal classes of simple relation algebras. Some of the highlights of the chapter include the following results. The class of representable relation algebras is a variety, but it is not finitely axiomatizable, and it is not even axiomatizable by an infinite set of equations that uses only finitely many variables. Above the zero element of the lattice of varieties, there are three minimal varieties of relation algebras, namely the varieties M 1, M 2,andM 3 generated

12 xiv Introduction by the minimal set relation algebras on sets of cardinalities one, two, and three respectively. Just above the minimal varieties, there are the varieties that are the joins of two of the minimal varieties. There are no other varieties that are directly above M 1, and exactly one other variety that is immediately above M 2, namely the variety generated by the full set relation algebra on a set of cardinality two. As regards M 3, there are exactly three varieties generated by non-integral relation algebras that are immediately above M 3. There are seventeen known examples of varieties immediately above M 3 that are generated by finite integral relation algebras, and one that is generated by an infinite integral relation algebra. Just below the unit of the lattice of varieties of relation algebras, there are infinitely many maximal varieties. In particular, for each natural number n 1, the variety generated by the class of simple relation algebras that are not isomorphic the full set relation algebra on a set of cardinality n is a maximal element of the lattice. The full set relation algebras on sets of finite cardinalities n generate varieties V n that are distinct from one another for distinct natural numbers n 1. On other hand, the full set relation algebras on infinite sets all have the same equational theory and therefore generate the same variety V ω. The variety of representable relation algebras is the irreducible join of the varieties V n for 1 n ω. The lattice of all subsets of the natural numbers is embeddable into the lattice of varieties of representable relation algebras. In fact, it is embeddable into the interval [M 3, V f ] consisting of all subvarieties of V f that include the variety M 3,whereV f is the join of the varieties V n for positive integers n (thus, V ω is excluded from this join). Consequently, this interval contains continuum many varieties, and it even contains a chain of the same order type as the real numbers. The interval [M 3, V hi ]also contains continuum many varieties, where V hi is the variety generated by the class of hereditarily infinite representable relation algebras, that is to say, the variety generated by the class of infinite, representable relation algebras all of whose non-minimal subalgebras are infinite. Thus, there are continuum many varieties of hereditarily infinite, representable relation algebras. In Chapter 19, the duality between Boolean algebras with operators and relational structures is studied. It is shown that the complex algebra of every relational structure is a complete and atomic Boolean algebra with complete operators; and conversely, every complete and atomic Boolean algebra with complete operators is isomorphic to the complex algebra of some relational structure, namely the atom struc-

13 Introduction xv ture of the algebra, that is to say, the relational structure consisting of the atoms in the algebra under certain natural relations that are induced on the atoms by the operations of the algebra. This duality implies the following representation theorem for Boolean algebras with operators: every Boolean algebra with operators is embeddable into the complex algebra of some relational structure. In particular, every relation algebra is embeddable into the complex algebra of an appropriate relational structure. In the case of relation algebras, the class of these appropriate relational structures may be axiomatized by a very simple set of first-order formulas. More generally, if V is any variety of Boolean algebras with complete operators, then the class of atom structures of the atomic algebras in V is axiomatizable by a set of first-order formulas. In general, this axiomatization is infinite and complicated, but if the equations axiomatizing V have a certain simple form, then the class of atom structures of atomic algebras in V has a correspondingly simple form. One of the highlights of Chapter 19 is a proof of the deep theorem that for a class of relational structures closed under ultraproducts, and in particular for a class of relational structures axiomatizable by a set of first-order formulas, the variety generated by the class L of complex algebras of these relational structures is just the class SP(L) ofisomorphic copies of subalgebras of direct products of algebras in L in other words, SP(L) is automatically closed under homomorphic images and furthermore, SP(L) is also closed under canonical extensions in the sense that the canonical extension of every algebra in this variety also belongs to the variety. Several applications of this theorem to classes of relation algebras are given. For example, if L is the class of all complex algebras of groups, or the class of all complex algebras of projective geometries, or the class of all complex algebras of modular lattices with zero, then SP(L) is the variety generated by L, andthis variety is closed under canonical extensions. As another application, an especially transparent proof is given for the theorem that the class of representable relation algebras is a variety With one exception, the chapters in volume 2 are written to be largely independent of one another. Occasionally, a definition or theorem from one chapter is used in a later one, but in this case it suffices to read the statement of the definition or theorem. Chapter 14 (canonical extensions) and Chapter 15 (completions) are developed in parallel, but independently of one another. Chapter 17 (representation theorems) is dependent on much of the material in Chapter 16 (repre-

14 xvi Introduction sentations). Chapter 18 (varieties) and Chapter 19 (atom structures) are largely independent of one another and of the earlier chapters.

15 Contents Preface... Introduction... vii ix 14 Canonical extensions CanonicalextensionsofBooleanalgebras Canonical extensions of Boolean algebras with operators Analternativeapproachtoexistence FirstPreservationTheorem SecondPreservationTheorem Applicationstorelationalgebras Canonicalextensionsofhomomorphisms Applicationstoalgebraicconstructions Canonicalextensionsofsetrelationalgebras Acharacterizationofhomomorphicimages Historicalremarks Exercises Completions CompleteBooleanideals CompletionsofBooleanalgebras CompletionsofBooleanalgebraswithoperators Thepreservationtheorems Applicationstorelationalgebras Completionsofhomomorphisms Minimality Applicationstoalgebraicconstructions xvii

16 xviii Contents 15.9 A characterization of complete homomorphic images Historicalremarks Exercises Representations Alternativeviewsofrepresentations Equivalentrepresentations Completerepresentations Subalgebras and relativizations Simplealgebras Products Canonicalextensions Completions Homomorphicimagesanddirectedunions Historicalremarks Exercises Representation theorems Formularelationalgebras Booleanrelationalgebras Groupcomplexalgebras Geometriccomplexalgebras Smallrelationalgebras Quasi-representations Atomic relation algebras with functional atoms Singletondenserelationalgebras Historicalremarks Exercises Varieties of relation algebras Theoriesandclassesofrelationalgebras Ultraproducts Universalclasses Thelatticeofvarieties Thevarietyofrepresentablerelationalgebras Thestructureofthelattices Minimal and quasi-minimal universal classes and varieties Maximaluniversalclassesandvarieties Universal classes of representable relation algebras

17 Contents xix Historicalremarks Exercises Atom structures Atomstructuresandcomplexalgebras Atomstructuresofrelationalgebras Cycles Axiomatizingclassesofatomstructures Duality Ultraproductsofstructures Universal classes closed under canonical extensions Varieties closed under canonical extensions Applicationstorelationalgebras Polyalgebras Historicalremarks Exercises Epilogue References Index