Semigroup cohomology and applications

TL;DR

This survey reviews various cohomology theories of semigroups, including classic, 0-cohomology, and partial cohomology, highlighting their applications and computational methods.

Contribution

It introduces and discusses generalizations of Eilenberg-MacLane cohomology for semigroups, including 0-cohomology and partial cohomology, with applications.

Findings

Analysis of semigroup cohomology with cohomological dimension 1

Introduction of 0-cohomology and its applications in projective representations

Use of partial cohomology to compute classic cohomology for specific semigroups

Abstract

This article is a survey of the author's research. It consists of three sections concerned three kinds of cohomologies of semigroups. Section 1 considers `classic' cohomology as it was introduced by Eilenberg and MacLane. Here the attention is concentrated mainly on semigroups having cohomological dimension 1. In Section 2 a generalization of the Eilenberg-MacLane cohomology is introduced, the so-called 0-cohomology, which appears in applied topics (projective representations of semigroups, Brauer monoids). At last Section 3 is devoted to further generalizing: partial cohomology defined and discussed in it are used then for calculation of the classic cohomology for some semigroups.