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Algebras, lattices, varieties. Volume I. (English) Zbl 0611.08001
The Wadsworth & Brooks/Cole Mathematics Series. Monterey, California: Wadsworth & Brooks/Cole Advanced Books & Software. XII, 361 p.; $ 44.95 (1987).
The book is the first of four volumes on universal algebra. Volume 1 is an introduction to general algebra and lattice theory; besides the fundamental concepts and elementary results, it contains several harder results that will be required in later volumes and a final advanced chapter on unique factorization.
The first three chapters (Basic Concepts; Lattices; Unary and Binary Operations) introduce basic concepts: subalgebras, homomorphisms, direct products, congruences, complete lattices, closure systems, modular and distributive lattices, congruences on lattices, unary algebras, semigroups, groups, quasigroups, categories; most important classes of algebras are introduced, the material is illustrated on numerous examples and can serve as a text for a one-semester undergraduate course in abstract algebra. The deepest result is the proof of the direct join decomposition theorem for modular lattices of finite height; the presented development relies heavily on B. Jónsson’s work.
Chapter 4 (Fundamental Algebraic Results) is a complete presentation of basic universal algebra. The concepts, results, perspectives and intuitions shared by the workers in the field are explained. The sections deal with term operations and clones, isomorphism theorems, the congruence generation theorem, Pálfy-Pudlák’s lemma on the restriction of congruences to the range of an idempotent polynomial, direct and subdirect representations, the subdirect representation theorem (illustrated on the examples of monadic algebras and rings satisfying \(x^ n=x)\), algebraic lattices and lattices of subuniverses, Tarski’s interpolation theorem, permuting congruences, projective geometries, distributive congruence lattices, class operators, varieties, free algebras, the HSP theorem, equivalence and interpretation of varieties and the rudiments of commutator theory. The deepest result is the decomposition of a complemented modular algebraic lattice into a product of projective geometries.
The last chapter 5 (Unique Factorization) is essentially self-contained and it is not a requirement for understanding the material in later volumes. The chapter contains many deep and beautiful results on unique factorization of algebras. Fundamental is B. Jónsson’s theorem on the unique factorization of finite algebras with modular congruence lattice and a one-element subalgebra. The Jónsson-Tarski theorem is proved: Every algebra A with zero (an element 0 such that \(x+0=0+x=x\) for a binary operation \(+\) of A) whose center is finite is uniquely factorable, provided that each of its direct factors is decomposable into a finite product of indecomposable algebras. Results on the refinement property for direct factorizations and on the strict refinement property are formulated and proved; McKenzie’s related Lemma is included. The intermediate refinement property is formulated in terms of decomposition operations and some applications are given. The last section deals with the cancellation property \(A\times C\cong B\times C\to A\cong B\) and the related properties for algebras and relational structures; the famous results of L. Lovász are proved.
There are numerous exercises extending the material.
Reviewer: J.Ježek

MSC:
08-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to general algebraic systems
06-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to ordered structures
08A30 Subalgebras, congruence relations
08B10 Congruence modularity, congruence distributivity
06C05 Modular lattices, Desarguesian lattices
06B05 Structure theory of lattices
06C20 Complemented modular lattices, continuous geometries
08Axx Algebraic structures
08Bxx Varieties