Semigroup Cohomology and Applications

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.I am indebted to Prof. K. Roggenkamp and Prof. M. Ştefanescu for the support of my participation in the Workshop.

Partial projective representations and partial actions

“…Take the abstract semigroup T which is generated by elements g, h, t with the defining relations (20). The maps (17)- (19) determine an action T on G × G. Thus G × G becomes a T -set and Theorem 4 can be restated as follows:…”

“…Corollaries 2 and 3 show how the multiplier of a semigroup is constructed from such components. An appropriate theory of cohomology (0-cohomology of semigroups) was developed in [19] (see also [20]). …”

Recent developments around partial actions

“…The theory of partial projective representations is strongly related to Exel's semigroup S(G). In fact, they can be alternatively defined via projective representations of S(G), so that the theory of projective representations of semigroups and their Schur multipliers, elaborated by Novikov in [238][239][240] (see also [241]), comes into the picture as an essential working tool. The usual cohomology of semigroups does not serve the projective semigroup representations, instead the more general 0-cohomology [240] fits them with its natural partial flavor.…”

0-Cohomology of Semigroups

“…Besides, modifications are 0-cancellative (if ax = bx = 0 or xa = xb = 0 then a = b). Some examples of modification were considered in [38] and [41].…”