Universal algebra

Universal Algebra Second Edition

George Grätzer

Universal Algebra Second Edition

George Grätzer Department of Mathematics University of Manitoba Winnipeg, Manitoba R3T 2N2 Canada [email protected]

ISBN: 978-0-387-77486-2 e-ISBN: 978-0-387-77487-9 DOI: 10.1007/978-0-387-77487-9 Library of Congress Control Number: 2008922052 Mathematics Subject Classification (2000): 08-01, 08-02 © 2008, Second Edition with updates, 1979 Second Edition, Springer Science+Business Media, LLC

Originally published in the University Series in Higher Mathematics (D. Van Nostrand Company); edited by M. H. Stone, L. Nirenberg, and S. S. Chern, 1968 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA) and the author, except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed on acid-free paper. 987654321 springer.com

EPILOGUE

I met the young man of about twenty-eight at the Polo Park shopping mall in Winnipeg.1 He walked much faster than I, so I was looking at his back as he passed by. He looked very familiar. Rather thin, with a lot of brown hair, obviously in a hurry. I caught up with him when he paused in front of a shop window. He turned around half-way; he immediately knew who I was. I cannot say that he was happy to see me. “I did not do so badly,” I stammered. “Really,” he responded. “Just compare. When I wrote Universal Algebra, I knew it all. Remember? At Penn State, we spent three weeks in the seminar to decide not to include an article in the book. I knew most everything that was published. Can you say the same?” “No, I cannot,” I replied. “And remember your undertaking: Even though you started on General Lattice Theory after completing Universal Algebra, you resolved to keep your work evenly balanced between the two fields,” he called me to account. “True, but the numbers were against me. Since I finished Universal Algebra in 1966, more than 5,000 papers were published in this field and over 13,000 in lattice theory. I would have had to average two papers a day (including more than a hundred books!), just to keep up,” I replied. The young man was mad at me, and with good reason. For about ten years after I finished Universal Algebra, I concentrated on lattices. There was so much to do. You cannot write a book on lattices without free products and uniquely complemented lattices, and so much else. And so little was known. . . . Indeed, fewer than 20% of my papers after 1966 were written on universal algebraic topics and most of them were written before 1980. *** 1

After F. Karinthy, Atheneum, 1913.

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A lot has happened in Universal Algebra in forty years. I cannot attempt here to survey the 5,000 papers and dozens of books. But I would like to point out that many of the important papers in these forty years are in—or utilize the results of—five main developments. 1. Theory of quasivarieties. The oldest of these five fields is the theory of quasivarieties started by A. I. Maltsev but seriously explored by V. Gorbunov, his students, and colleagues in Siberia. More recently, some East European and some American mathematicians have also made important contributions. A part of this theory was covered in the book V. A. Gorbunov, Algebraic theory of quasivarieties.2 In fact, he considers universal Horn classes, which include quasivarieties and anti-varieties. Thus, the results apply to a wide variety of algebraic systems that may have relational symbols in their language: graphs, ordered structures, some geometrical models, formal languages, and others. Gorbunov’s book contains more than 300 references. The Birkhoff-Maltsev problem has been one of the main driving forces in this field for the last 25 years. It asks for a characterization of lattices that can be represented as the lattice of all subquasivarieties of a quasivariety. The problem is still open even for finite lattices. A very thorough coverage of this field was published in a special double issue of Studia Logica 78 (2004). A survey article by M. E. Adams, K. V. Adaricheva, W. Dziobiak, and A. V. Kravchenko and eighteen research articles in almost 400 pages survey the various aspects of the field, including a listing of open problems. 2. Commutator theory. Originally introduced by J. D. H. Smith for permutable varieties in 1976 and extended to modular varieties by C. Herrmann and J. Hagemann in 1979, commutators were fully developed in the book, Commutator theory for congruence modular varieties by R. Freese and R. N. McKenzie.3 This book shows that a natural commutator operation can be defined on the congruence lattice of an algebra in a variety with modular congruence lattices. As S. Oates-Williams wrote: “It is quite remarkable that anything resembling a commutator can be defined in a general algebra; that it should have such nice properties when restricted to algebras lying in a congruence modular variety is almost too much to expect.” Freese and McKenzie develop the theory and use it to prove deep results. For example, the subdirectly irreducible algebras in a finitely generated congruence modular variety either have a finite bound on their cardinality or no cardinal bound at all. Their concepts and results found many applications in later papers. 2

Translated from the Russian. Siberian School of Algebra and Logic. Consultants Bureau, New York, 1998. 3 London Mathematical Society Lecture Note Series, 125. Cambridge University Press, Cambridge, 1987.

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3. Tame congruence theory. Apart from a paper of P. P. P´alfy, this theory— like Athena, born fully grown from the forehead of Zeus—burst onto the scene fully developed in D. Hobby and R. McKenzie, The structure of finite algebras.4 An excellent write up of this book, is J. Berman’s review in the Mathematical Reviews. Three important papers by McKenzie motivated by tame congruence theory to obtain deep results are also jointly reviewed by J. Berman.5 They include the solution, in the negative, of A. Tarski’s famous problem from the early 1960’s: Does there exist an algorithm which, when given an effective description of a finite algebra A, determines whether or not A has a finite basis for its equational theory? Tame congruence theory has been effectively applied in a large number of papers to solve various problems related to finite algebras and locally finite varieties. 4. The shape of congruence lattices. The forthcoming book by K. A. Kearnes and E. W. Kiss presents a beautiful theory. Here is one example: The congruence lattices of a variety V satisfy SD∨ iff they satisfy SD∧ and V satisfies a nontrivial congruence identity. In particular, SD∧ and SD∨ are equivalent for varieties satisfying a nontrivial congruence identity. This was known for locally finite varieties (using tame congruence theory) but that is far more restrictive than the full result. 5. Natural duality theory. The foundations of this theory were laid in 1980 by B. A. Davey and H. Werner. They showed that there is a common universal-algebraic framework for various classical topological dualities, including the dualities for abelian groups (L. S. Pontryagin, 1934), Boolean algebras (M. H. Stone, 1936), and distributive lattices (H. A. Priestley, 1970). They developed methods for finding many topological dualities, in addition to the classical ones. For example, if a finite algebra has a near-unanimity term—and in particular, if it is lattice-based—then there is a natural duality for the quasivariety it generates. The book by D. M. Clark and B. A. Davey, Natural dualities for the working algebraist6 covers the first eighteen years of the theory of natural dualities. More recent results, including work on full dualities and strong dualities, is covered in the book by J. G. Pitkethly and B. A. Davey, Dualisability: unary algebras and beyond.7 *** And what has happened with the problems I proposed? In Appendix 2, I report on (partial) solutions to a large number of problems. I would say that roughly half of the problems remain unresolved. For instance, the problem: 4

Contemporary Mathematics. 76. American Mathematical Society, Providence, RI, 1988. MR1371732-4 (97e:08002a-c) 6Studies in Advanced Mathematics. 57. Cambridge University Press, Cambridge, 1998. 7Advances in Mathematics. 9. Springer, New York, 2005. 5

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Is every finite lattice the congruence lattice of a finite algebra? now seems very difficult, because a positive solution to the “much easier problem” to embed every finite lattice into a finite partition lattice turned out to be discouragingly hard. Some problems shifted in focus. In 1979, it seemed likely that every distributive algebraic lattice could be represented as the congruence lattice of a lattice. Now that F. Wehrung has provided a counterexample, we ask: 1. Can every distributive algebraic lattice can be represented as the congruence lattice of an algebra in a congruence distributive variety? or even stronger: 2. Is there a congruence distributive variety V such that every distributive algebraic lattice can be represented as the congruence lattice of an algebra in V? *** So what does the excited young man say in this book that is still relevant after all these profound developments? A lot, I think. A vast superstructure has been built up, but the foundation is still basically the same. Despite the numerous books on specialized topics—such as the few mentioned above— it may still be the best introduction to Universal Algebra to learn the basic concepts as presented here and to work out some of the 750 exercises. Then one could proceed to the specialized books. This is a good way to get started. And remember, these specialized topics are all interconnected. Even if you cannot become a researcher in them all, you must have a passing knowledge of all five fields to become successful. Maybe, I am prejudiced. But if I were a young man, this is how I would proceed. Acknowledgement: I would like to thank K. Adaricheva, J. Berman, G. M. Bergman, B. Davey, R. Freese, K. A. Kearnes, R. N. McKenzie, J. B. Nation, W. Taylor, and F. Wehrung for their help and encouragement.