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Congruence-modular varieties of algebras. (Kongruehnts-modulyarnye mnogoobraziya algebr.) (Russian) Zbl 0714.08003

Irkutsk: Izdatel’stvo Irkutskogo Universiteta. 132 p. R. 0.85 (1986).
The book, giving a survey of results on congruence modular varieties, is divided into eleven chapters. Chapter 1 is introductory. Chapter 2 is a collection of the most fundamental theorems of Mal’tsev type. The exposition of commutator theory in the next chapter follows the approach taken by Gumm, Herrmann, Hageman, Freese and McKenzie. In the chapter on residually small varieties, the classical work of Taylor is followed by Quackenbush’s result on residual finiteness of a locally finite variety with finitely many finite subdirectly irreducible algebras; and the characterization, by Freese and McKenzie, of finitely generated congruence modular residually small varieties. Chapter 5 is devoted to the proof of McKenzie’s characterization of direct representability for finitely generated congruence modular varieties. Chapter 6 is an exposition of the theory of primal and related algebras (mainly the results of Foster and Pixley). Chapter 7 on spectra and skeleta includes results of Grätzer, Baldwin, McKenzie, Quackenbush, Taylor and a result of the author on the embeddability of quasiordered sets into the epimorphism skeleton of a congruence distributive variety. Results of McKenzie, Baldwin, Berman, Köhler and Pigozzi on definability of principal congruences are contained in chapter 8. The next chapter on finite basedness contains two results: a finitely generated congruence distributive variety is finitely based (Baker) and a finitely based congruence distributive and permutable variety is one-based (McKenzie, Padmanabhan, Quackenbush). Chapter 10 deals with the structure of decidable finitely generated congruence modular varieties (the theory developed by Burris and McKenzie). The last chapter is concerned with congruence varieties and includes results by Day, Freese, Czédli, Nation and Jónsson.
Reviewer: J.Ježek

MSC:

08B10 Congruence modularity, congruence distributivity
08-02 Research exposition (monographs, survey articles) pertaining to general algebraic systems