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COLLOQUIA MATHEMATICA SOCIETATIS JANOS BOLYAI, 54 ALGEBRAIC LOGIC Edited by H. АЫОРЁКА, J. D. MONK, and I. NEMETI NORTH-HOLLAND COLLOQUIA MATHEMATICA SOCIETATIS JANOS BOLYAI, 54. ALGEBRAIC LOGIC Edited by H. ANDREKA, J. D. MONK, and I. NEMETI NORTH-HOLLAND PUBLISHING COMPANY AMSTERDAM - OXFORD - NEW YORK © BOLYAI JANOS MATEMATIKAI TARSULAT Budapest, Hungary, 1991 ISBN North-Hollaad: 0444 88543 9 ISBN Bolyai: 963 8022 57 4 ISSN Bolyai: 0139-3383 Joint edition published by JANOS BOLYAI MATHEMATICAL SOCIETY and ELSEVIER SCIENCE PUBLISHERS B.V. Saraburgerhartstraat 25, P.O. Box 103 1000 AC, Amsterdam, The Netherlands In the U.S.A. and Canada: ELSEVIER SCIENCE PUBLISHING COMPANY INC. 655 Avenue of the Americas New York, NY. 10010 U.S.A. Assistant editors: E. W. KISS and I. SAIN Film transfer by ITEX Laser- and Computingtechnics Ltd. Printed in Hungary Franklin Nyomda Budapest Contents Contents ............................................................................................................(iii) Introduction ................................................................................................. (v) I. H. Anellis and N. Houser: Nineteenth Century Roots of Al gebraic Logic and Universal Algebra........................................................ 1 R. Berghammer, P. Kempf, G. Schmidt, and T. Strohlein: Relation Algebra and Logic of Programs................................................. 37 C. Bergman: Structural Completeness in Algebra and Logic ............. 59 W. J. Blok and D. J. Pigozzi: Local Deduction Theorems in Algebraic Logic ............................................................................................. 75 D. A. Bredikhin: On Relation Algebras with General Superposi tions ....................................................................................................................Ill J. ClRULlS: An Algebraization of First Order Logic with Terms.........125 S. D. Comer: The Representation of Dimension 3 Cylindric Alge bras .................................................................................................................. 147 M. Ferenczi: Measures Defined on Free Products of Formula Al gebras and Analogies with Cylindric Homomorphisms ....................... 173 J. M. Font and V. Verdu: On Some Non-algebraizable Logics ---- 183 S. Givant: Tarski’s Development of Logic and Mathematics based on the Calculus of Relations.........................................................................189 R. Goldblatt: On Closure Under Canonical Embedding Algebras .. 217 G. Hansoul: Modal-axiomatic Classes of Kripke Models ................... 231 P. JlPSEN AND E. LUKACS: Representability of Finite Simple Rela tion Algebras with Many Identity Atoms ............................................... 241 B. JONSSON: The Theory of Binary Relations ........................................ 245 R. L. Kramer: Relativized Relation Algebras...........................................293 J. Lambek: Categorical Versus Algebraic Logic ...................,................ 351 R. D. Maddux: Introductory Course on Relation Algebras, Finite dimensional Cylindric Algebras, and Their Interconnections...............361 V. Manca and A. Salibra: On the Power of Equational Logic: Applications and Extensions ..................................................................... 393 J. D. Monk: Structure Problems for Cylindric Algebras .......................413 I. Nemeti and H. Andreka: On Jonsson’s Clones of Operations on Binary Relations ........................................................................................431 IV CONTENTS E. ORLOWSKA: Relational Interpretation of Modal Logics ................ 445 D. J. Pigozzi: Fregean Algebraic Logic ................................................. 475 В. I. Plotkin: Halmos (polyadic) Algebras in Database Theory.......505 D. Resek and R. J. Thompson: Characterizing Relativized Cylin- dric Algebras ............................................................................................ 519 I. Sain and R. J. Thompson: Strictly Finite Schema Axiomatiza- tion of Quasi-polyadic Algebras ............................................................ 539 A. SALIBRA: A General Theory of Algebras with Quantifiers................573 B. M. SCHEIN: Representation of Subreducts of Tarski Relation Al gebras ......................................................................................................... 621 Gy. SerenY: Neatly Atomic Cylindric Algebras and Representable Isomorphisms ............................................................................................ 637 S. Shelah: On a Problem in Cylindric Algebra......................................645 A. Simon: Finite schema completeness for typeless logic and repre sentable cylindric algebras.........................................................................665 Zs. TuzA: Representations of Relation Algebras and Patterns of Colored Triplets..........................................................................................671 Y. VENEMA: Relational Games...................................................................595 J. D. MONK: Remarks on the Problems in the Books Cylindric Algebras, Part I and Part II and Cylindric Set Algebras .................. 719 J. D. MONK: Corrections for the Books Cylindric Algebras, Part I and Part II and Cylindric Set Algebras ..................................................723 Open problems............................................................................................727 Introduction The Janos Bolyai Mathematical Society held an Algebraic Logic Colloquium between 8-14 August, 1988, in Budapest. The colloquium was co-sponsored by the Association of Symbolic Logic and the IUHPS. This event was one in a series of conferences on Algebraic Logic; to mention a few: Asilomar California 1978, Ames Iowa 1988, Budapest Hungary 1988, Boulder Col orado 1990, Oakland California 1990. The colloquium had 64 participants, 49 from outside Hungary, 20 from overseas, representing 15 countries. An introductory series of lectures on cylindric and relation algebras was given by Roger D. Maddux. There were 10 plenary talks. The present volume is not restricted to papers presented at the confer ence. Instead, it is aimed at providing the reader with a relatively coherent reading on Algebraic Logic (AL), with an emphasis on current research. We could not cover the whole of AL, probably the most important omission being that the category theoretic versions of AL were treated only in their connections with Tarskian (or more traditional) AL. The present volume was prepared in collaboration with the editors of the Proceedings of Ames conference on AL (Springer Lecture Notes in Computer Science Vol425, 1990), and a volume of Studia Logica devoted to AL which was scheduled to go to press in the fall of 1990. Some of the papers originally submitted to the present volume appear in one of the latter. To help the nonspecialist reader, the volume contains an introduction to cylindric and relation algebras by Roger D. Maddux (pp. 361-392). Another paper beginning with introductory sections designed for the nonspecialist is Bjarni Jonsson’s (pp. 245-292). It provides an introduction to Boolean algebras with operators which, besides playing an important role in cylin dric, polyadic, relation algebras, and other algebras whose elements can be conceived as (not necessarily binary) relations, also play an important role in philosophical and nonclassical logics as well as in theoretical computer science. These papers, despite their special introductory role for the volume, are found at their alphabetical places. At the end of the volume, there is an ‘Open problems’ “paper”. Many of the problems listed there were raised at the problem session of the col loquium. This paper is preceded by a report on the status of the problems raised in the monograph by L. Henkin, J. D. Monk, and A. Tarski on AL (despite of the fact that the title of that book is “Cylindric Algebras”, Part VI INTRODUCTION II of the book discusses AL in general) treated together with the problems in the related book L. Henkin, J. D. Monk, A. Tarski, H. Andreka, I. Nemeti: Cylindric Set Algebras. This report was prepared by J. D. Monk. In classifying the papers of the volume, we use, basically, the terminol ogy of the Henkin-Monk Tarski monograph. Common features of cylindric, polyadic, relation, and closely related al gebras are that (i) they can be conceived as expansions of Boolean algebras from algebras whose elements are unary relations to algebras whose elements are relations of higher ranks, and that (ii) they can be considered as al- gebraizations of (some versions of) quantifier logics (e.gof first-order logic). These algebras are discussed in the papers by Berghammer et al; Bredikhin, Clrulis, Comer, Ferenczi, Givant, Jipsen-Lukacs, Jonsson, Kramer, Mad dux, Monk, Nemeti-Andreka, Orlowska, Plotkin, Resek-Thompson, Sain- Thompson, Salibra, Schein, Sereny, Shelah, Simon, Tuza, Venerna. Boolean algebras with operators are treated in the papers by Goldblatt, Hansoul, Jonsson. Universal algebraic logic (general, unifying approaches to algebraic logic, related to the general theory of logics or abstract model theory on the logical side and to universal algebra on the algebraic side) is treated by Bergman, Blok-Pigozzi, Font-Verdu, Pigozzi, Salibra. Applications of cylindric etcalgebras and Boolean algebras with opera tors in computer science, in philosophical logic and in logic in general are treated by Berghammer et al; Blok-Pigozzi, Givant, Goldblatt, Hansoul, Manca-Salibra, Orlowska, Plotkin. Historical aspects and connections with categorial logic are treated by Anellis-Houser, Givant, and Lambek. For undefined notion and terminology the reader is referred to pp. 728- 729, 367-376, 245-276, 147-148 of the volume. Namely, several of the pa pers use standard AL notation and terminology without recalling them. All of these can be found in the Henkin-Monk-Tarski monograph mentioned above. To make the volume self-contained, most of these are recalled in §2 of the ‘Open problems’ paper (pp. 728-729) at the end of the volume. We would like to express our thanks for their help in preparing the conference and the present volume to: S. D. Comer, W. Craig, M. Ferenczi, E. W. Kiss, R. D. Maddux, M. Makkai, D. J. Pigozzi, I. Sain, G. Sereny. We also would like to express our thanks to all those authors who helped the technical editors by sending their papers on diskettes too. We thank Dezso Miklos for his technical help in preparing the volume. The Editors COLLOQUIA MATHEMATICA SOCIETATIS JANOS BOLYAI 54. ALGEBRAIC LOGIC, BUDAPEST (HUNGARY), 1988 Nineteenth Century Roots of Algebraic Logic and Universal Algebra IRVING H. ANELLIS and NATHAN R. HOUSER In Memoriam: EVELYN M. NELSON (1943-1987) and in Honor of the Sesquicentennial Celebration of the Birth of CHARLES SANDERS PEIRCE (1839-1914) Historians of mathematical logic frequently tell us that there are two traditions, the algebraic tradition of Boole, Schroder, and Peirce, arising from the algebraization of analysis, and the quantification-theoretical (or logistic) tradition of Peano, Frege, and Russell, arising from the develop ment of the theory of functions. It is said that these two traditions, together with the independent set-theoretical tradition of Cantor, Dedekind, and Zer- melo arising out of the search for a foundation for real analysis in the work of Cauchy, Weierstrass and others, were united by Whitehead and Russell in their Principia mathematica to create mathematical logic. (Grattan- Guinness [1988] agrees that Russell was the founder of mathematical logic, but also holds that mathematical logic arose exclusively from the work of AMS (MOS) 1980 Subject classification (1985 revision): 01A55, 03-03, 03A05, 03G00, 03G05, 06-03, 06E00, 08-03, 08A00 2 IRVING H ANELLIS and NATHAN R. HOUSER Peano and from set-theoretic tradition inspired by the development of real analysis in the work from Cauchy to Weierstrass.) The concern of most historians has been to contrast the algebraic and quantification-theoretic traditions and to show that the algebraic tradition had been the inferior of the two, that it reached a dead-end and was absorbed, along with set theory, into the quantification-theoretic tradition in the Principia. Nev ertheless, algebraic logic and universal algebra remain strong today, and research continues, not only unabated, but making powerful and profound progress. We hold that the distinction between the algebraic and quantification- theoretic traditions is artificial, and that the algebraic logic of the nineteenth century was the mathematical logic of its day. We briefly explore the at titudes of some of those who contributed to its development and suggest reasons, based upon the historiography of logic, for the bifurcation between algebraic logic and quantificational logic. Most contemporary researchers in algebraic logic and universal alge bra have only a very vague conception of their historical roots, and take their primary sources of inspiration from the work of their immediate pre decessors of the 1930s to 1950s, principally Birkhoff, Tarski, and their more prominent contemporaries. For those algebraists who would like to study the fundamental historical roots of their discipline and the ideas of its prin cipal founders, we will sketch the historical and contemporary situation in investigations into the history of algebraic logic and universal algebra and provide a bibliography of readily accessible materials to which the interested reader may turn. Our study is largely historiographical. Working mathematicians are usually interested in the work of bygone colleagues only to the extent that their predecessors have left either in teresting unsolved problems or results that may be profitably employed to obtain some new results. Only infrequently is the work of a bygone math ematician found to be of interest for its own sake, as a contribution to the development of a mathematical area of study. This ahistorical approach is reinforced by the history of mathematics itself, where increased abstraction has allowed for the unification of previously disconnected theories under new, more general, theories. In the case of algebraic logic, the class calculus and the algebra of relations became parts of modern mathematical logic. Similarly, Boolean algebra and the algebra of relations, linear algebra and matrix theory, and the theory of rings and modules, came together to give