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jonsson b topics universal algebra 1972

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Textbook in PDF format Preface and Acknowledgements. These notes constitute a revised and expanded version of a course given at Vanderbilt University during the academic year 1969-70. The student is assumed to have a working knowledge of the elementary theory of sets. More specifically, he is expected to know the algebra of sets and the most basic facts concerning cardinal and ordinal numbers, and to be able to follov and use arguments based on the axiom of choice, the veil-ordering principle, or the various maximal principles related to the axiom of choice. The approach to set theory will be informal, and no particular system of axioms need be specified. It is important to be aware of the distinction between the concepts of a class and of a set, sets being by definition classes that are members of other classes. Actually this distinction will occasionally be disregarded, but only when this is not likely to lead to the familiar set-theoretic paradoxes, and the abuses could in any case be avoided by reformulating the discussion in a more rigorous but less suggestive language. Preliminaries. The basic symbolism. Families of sets. Relations. Functions. Equivalence relations. Ordering relations. Statural and induced maps. Algebras and relational structures. Basic concepts. Examples of relational structures. Homomorphisms and isomorphisms. Automorphism groups and endomorphism monoids: The concrete representation problem. Automorphism groups and endomorphism monoids: The abstract representation problem. Rigid structures. Endomorphism monoids: More about the abstract representation problem. Substructures. Substructures and subuniverses. Automorphisms of 1-unary algebras. Generating sets. Non-generators. Polynomials and algebraic operations. Intersection structures and closure operators. Multiplicity types and subuniverses. The lattice of subuniverses. Algebras with descending chain condition. Versatile monoids. Congruence relations. Epimorphisms and congruence relations. Algebraic operations and congruence relations. Isomorphisms of quotient structures. Congruence lattices! The concrete representation problem. Partial algebras. Congruence lattices: The abstract representation problem. Congruence lattices! Miscellaneous results and problems. Addenda. Table of notation. Subject index

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jonsson b topics universal algebra 1972

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