Monoids, Acts and Categories: With Applications to Wreath Products and Graphs. A Handbook for Students and Researchers (De Gruyter Expositions in Mathematics 29)

"The authors of this book are well-known specialists in the area of homological classification of monoids. The presentation is extremely clear and incisive. Without a doubt, the book could serve as a textbook for a very good graduate course on representation theory of monoids, homological algebra and category theory." __Mathematical Reviews__ "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." __Zentralblatt für Mathematik__

The aim of the series is to present new and important developments in pure and applied mathematics. Well established in the community over two decades, it offers a large library of mathematics including several important classics.

The volumes supply thorough and detailed expositions of the methods and ideas essential to the topics in question. In addition, they convey their relationships to other parts of mathematics. The series is addressed to advanced readers wishing to thoroughly study the topic.

Editorial Board

Lev Birbrair, Universidade Federal do Ceará, Fortaleza, Brasil
Walter D. Neumann, Columbia University, New York, USA
Markus J. Pflaum, University of Colorado, Boulder, USA
Dierk Schleicher, Jacobs University, Bremen, Germany
Katrin Wendland, University of Freiburg, Germany

Honorary Editor

Victor P. Maslov, Russian Academy of Sciences, Moscow, Russia

Titles in planning include

Yuri A. Bahturin, Identical Relations in Lie Algebras (2019)
Yakov G. Berkovich and Z. Janko, Groups of Prime Power Order, Volume 6 (2019)
Yakov G. Berkovich, Lev G. Kazarin, and Emmanuel M. Zhmud', Characters of Finite Groups, Volume 2 (2019)
Jorge Herbert Soares de Lira, Variational Problems for Hypersurfaces in Riemannian Manifolds (2019)
Volker Mayer, Mariusz Urba?ski, and Anna Zdunik, Random and Conformal Dynamical Systems (2021)
Ioannis Diamantis, Boštjan Gabrovšek, Sofia Lambropoulou, and Maciej Mroczkowski, Knot Theory of Lens Spaces (2021)

Foreword Introduction I Elementary properties of monoids, acts and categories 1 Sets and relations 2 Groupoids, semigroups and monoids 3 Some classes of semigroups 4 Acts over monoids (monoid automata) 5 Decompositions and components 6 Categories 7 Functors II Constructions 1 Products and coproducts 2 Pullbacks and pushouts 3 Free objects and generators 4 Cofree objects and cogenerators 5 Tensor products 6 Wreath products of monoids and acts 7 The wreath product of a monoid with a small category III Classes of acts 1 Injective acts 2 Divisible acts 3 Principally weakly injective acts 4 fg-weakly injective acts 5 Weakly injective acts 6 Absolutely pure acts 7 Cogenerators and overview 8 Torsion free acts 9 Flatness of acts and related properties 10 Principally weakly flat acts 11 Weakly flat acts 12 Flat acts 13 Acts satisfying Condition (P) 14 Acts satisfying Condition (E) 15 Equalizer flat acts 16 Pullback flat acts and overview 17 Projective acts 18 Generators 19 Regular acts and overview IV Homological classification of monoids 1 Principal weak injectivity 2 On fg-weak injectivity 3 Weak injectivity 4 Absolute purity 5 Injectivity and overview 6 Torsion freeness and principal weak flatness 7 Weak flatness 8 Flatness 9 Condition (P) 10 Strong flatness 11 Projectivity 12 Projective generators 13 Freeness and overview 14 Regularity of acts V Equivalence and Duality 1 Adjoint functors 2 Categories equivalent to Act — S 3 Morita equivalence of monoids 4 Endomorphism monoids of generators 5 On Morita duality Bibliography Index of symbols Index The aim of the Expositions is to present new and important developments in pure and applied mathematics. Well established in the community over more than two decades, the series offers a large library of mathematical works, including several important classics. The volumes supply thorough and detailed expositions of the methods and ideas essential to the topics in question. In addition, they convey their relationships to other parts of mathematics. The series is addressed to advanced readers interested in a thorough study of the subject. Editorial Board Lev Birbrair, Universidade Federal do Cear, Fortaleza, BrasilWalter D. Neumann, Columbia University, New York, USAMarkus J. Pflaum, University of Colorado, Boulder, USADierk Schleicher, Jacobs University, Bremen, GermanyKatrin Wendland, University of Freiburg, Germany Honorary Editor Victor P. Maslov, Russian Academy of Sciences, Moscow, Russia Titles in planning include Yuri A. Bahturin, Identical Relations in Lie Algebras (2019)Yakov G. Berkovich, Lev G. Kazarin, and Emmanuel M. Zhmud', Characters of Finite Groups, Volume 2 (2019)Jorge Herbert Soares de Lira, Variational Problems for Hypersurfaces in Riemannian Manifolds (2019)Volker Mayer, Mariusz Urbaski, and Anna Zdunik, Random and Conformal Dynamical Systems (2021)Ioannis Diamantis, Bostjan Gabrovsek, Sofia Lambropoulou, and Maciej Mroczkowski, Knot Theory of Lens Spaces (2021) A discussion of monoids, acts and categories with applications to wreath products and graphs. It is divided into sections which address: elementary properties of monoids, acts and categories; constructions; classes of acts; homological classification of monoids; and equivalence and duality. In Section 1.1 we give the necessary terminology and also include some basic vocabulary from graph theory which will be useful to gain examples later on.