Cohomology monoids of monoids with coefficients in semimodules I

Abstract

Using the techniques of homological algebra of semimodules developed in our previous papers, we introduce new cohomology monoids of an arbitrary monoid $$M$$ with coeffcients in semimodules over $$M$$ , that is, with coeffcients in abelian monoids on which $$M$$ acts. The construction is similar to the construction of the Eilenberg-Mac Lane cohomology groups of monoids. In particular, we use an $$M$$ -semimodule analog of the classical normalized bar resolution. An explicit computation of these cohomology monoids in the case where $$M$$ is a finite cyclic group is given.