Monoids, acts and categories. With applications to wreath products and graphs. A handbook for students and researchers
type: Item
instance of: scholarly article
title: Monoids, acts and categories. With applications to wreath products and graphs. A handbook for students and researchers
zbMATH Open document ID: 0945.20036
author: Mati Kilp
author: Ulrich Knauer
author: A. V. Mikhalev
published in: De Gruyter Expositions in Mathematics
publication date: 2000-03-26T00:00:00Z
review text: 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 f…
reviewed by: Peeter Normak
Mathematics Subject Classification ID: 20M50
Mathematics Subject Classification ID: 18-02
Mathematics Subject Classification ID: 20-02
Mathematics Subject Classification ID: 20M10
Mathematics Subject Classification ID: 20M30
Mathematics Subject Classification ID: 18A30
Mathematics Subject Classification ID: 18A40
Mathematics Subject Classification ID: 18B40
Mathematics Subject Classification ID: 05C25
zbMATH DE Number: 1424797
zbMATH Keywords: monoids
zbMATH Keywords: acts over monoids
zbMATH Keywords: categories of acts
zbMATH Keywords: graphs
zbMATH Keywords: homological classification
zbMATH Keywords: Morita theory
zbMATH Keywords: wreath products
zbMATH Keywords: products
zbMATH Keywords: coproducts
zbMATH Keywords: injectivity
zbMATH Keywords: projectivity
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1424797