Topics in Universal Algebra
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250 Bjarni J6nsson Vanderbilt University, Nashville, TN/USA
Topics in Universal Algebra
Springer-Verlag Berlin' Heidelberg' NewYork 1972
Lecture Notes in Mathematics A collection of informal reports and seminars Edited by A. Dold, Heidelberg and B. Eckmann, ZUrich
250 Bjarni J6nsson Vanderbilt University, Nashville, TN/USA
Topics in Universal Algebra
Springer-Verlag Berlin' Heidelberg' NewYork 1972
AMS Subject Classifications (1970): 08 A 25
ISBN 3-540-05722-6 Springer-Verlag Berlin' Heidelberg· New York ISBN 0-387-05722-6 Springer-Verlag New York· Heidelberg· Berlin This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher.
© by Springer-Verlag Berlin' Heidelberg 1972.Library of Congress Catalog Card Number 70-186526. Printed in Germany. Offsetdruck:Julius Beltz, HemsbachlBergstr.
Preface and These notes constitute a revised and expanded version 01' a course given at Vanderbilt University during the e.c:&d.emic year 1969-70. Thanks are due to Mr. Chi Lin Yen, who read portions 01' the manuscript, and helped with the preparation of the bibliography. During the period when the notes were baing prepared, the author's work was supported in part bY' NSF Grants GP-ll804 and the VanderbiU University Research Council.
and by e. grant from
Contents Chapter 1. 1.1.
Preliminaries. . . . . . . . . . . . . . • • . . . . . • . .
Introduction • • • • •
1. 2.
The basic symbolism •
1.3.
Families of sets.
1.4. 1.5. 1.6.
Functions • Equivalence relations
1.7.
Ordering relations.
10
1.6.
Natural and induced maps.
13
Relations
Chapter 2.
Algebras and relational structures . • . . • . . . . • • . . concepts. • • • • • • • . •
2.1.
3 3 6 6
16 16
2.2.
Examples of relational structures •
16
2.3.
Homomorphisms and isomorphisms ••
24
2.4.
Automorphism groups and endomorphism monoids: The concrete represen tation problem • • • • • • • • • • • • • • • .
33
2.5.
Automorphism groups and endomorphism monoids: The abstract representation problem •
43
2.6. 2.7.
Rigid structures . • • • • • • . • • • • • • •
46
Endomorphism monoids: More about the abstract representation problem • • • . • • • • • • • • • • • • • •
54
Chapter 3. 3.1.
Substructures and subuniverses • • •
66
3.2.
Automorphisms of l-unary algebras.
71
VI 3.3. 3.4. 3.6.
Generating sets •• Non-generators • • • • • • • • Polynomials and algebraic operations Intersection structures and closure operators.
3.7. 3.8. 3.9. 3.10.
Multiplicity types and subuniverses • • • The lattice of subuniverses • • • • • • • Algebras with descending chain condition. Versatile monoids. • • • • •••••
3.5.
Chapter 4. 4.1.
4.2. 4.3.
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