de Gruyter Expositions in Mathematics. 29. Berlin: Walter de Gruyter. xvii, 529 p. DM 298.00 (2000).
This is the first monograph on representation theory of monoids and on non-additive homological algebra which discusses the relations between monoids and their categories of acts. The book consists of five chapters and of a detailed list of references.
The first chapter is devoted to the basic definitions and results about different classes of semigroups and acts over monoids, congruences, categories and functors. In functor and category theory, special attention has been paid to the categories of acts.
The second chapter starts with “universal” constructions in general categories: products and coproducts, pullbacks and pushouts, free and cofree objects, generators and cogenerators. Additionally, these notions and some others (for example, amalgamated coproducts, coamalgamated products, subdirect irreducibility, finite approximation and partitioning subobjects) are discussed in $S$-Act, in the category of all acts over a monoid $S$. The rest of the chapter deals with tensor products of acts and wreath products of monoids and acts as well as with the wreath product of a monoid with a small category. The latter construction is used for representing endomorphism monoids of different algebraic structures.
The third chapter discusses different classes of acts. The main properties considered are: injectivity and its generalizations (weak injectivity, fg-weak injectivity, principal weak injectivity, divisibility, absolute purity, absolute 1-purity), projectivity and its generalizations (pullback flatness, equalizer flatness, conditions (P) or (E), flatness, weak flatness, torsion freeness) and regularity. The main aim of the chapter is to describe acts having properties listed above. For doing this a number of additional important concepts have been discussed (for example, essential extensions, injective envelopes, faithfulness, perfectness). The theory developed leads to two graphs demonstrating relations between properties around injectivity and projectivity.
The fourth chapter concentrates on the homological classification of monoids trying to answer questions of the type “for which monoids $S$, all $S$-acts from a certain class of $S$-acts having property $P$ have property $Q$ as well” or of the type “for which monoids $S$, all $S$-acts from a certain class of $S$-acts have property $P$”, where $P$ and $Q$ are properties discussed in the previous chapter. A big part of the results is summarised in informative tables. The classes for which separate tables are presented are: 1) all right $S$-acts and all (or all finitely generated or all principal) right ideals of $S$ for classification of monoids by properties related to injectivity, 2) all right $S$-acts, all cyclic right $S$-acts, all monocyclic right $S$-acts, all Rees factor acts for properties related to projectivity.
In the fifth chapter, Morita theory is developed, i.e., it is determined to what extent a monoid is characterized by a certain act category (called Morita subcategory). To be self-contained, the chapter contains some more category theory (adjoint functors, complete description of the category of all $S$-acts etc.). Among other results, conditions for Morita equivalence of two monoids are given, Morita induced isomorphisms of endomorphism monoids of generators are described and Morita dual monoids are fully characterized.
Several open problems together with many exercises are presented. At many places, comments with historical remarks and about relations to other directions are inserted. The material of the book is well organised and suitable for a broad audience interested in monoids, acts, non-abelian categories as well as in formal languages, automata theory and other applications of semigroups. The book is comprehensive and self-contained and can be used both as study material for courses on representation theory on monoids, homological algebra and category theory, and as a handbook for students and researchers in the area. This is why the book should not be missing in any library of the faculty of mathematics of any university.