Homological Classification of Hypermonoids

Abstract

One of the very useful notions in many branches of Mathematics as well as in Computer Science is the action of a semigroup or a monoid on a set. In 1922, Suschkewitsch in his dissertation "The Theory of Action as Generalized Group Theory" introduced the notion of semigroup action [61]. A representation of semigroup S by transformation of a set defines an Sact just as representation of a ring R by endomorphisms of an Abelian group defines an Rmodule. More precisely, a right act Xs over the monoid S with identity e is a set X for which a "product" XSE X for XE X and SE S is defined such that for all S 1, S 2 E S, xE X. Probably first, in this form, the definition of S-act appeared in two papers by Hoehnke [38-39] with different names in connection with the consideration of radicals of semigroups. For an S-act the following names are also common: S-sets, S-operands, S-systems, transition system, Sautomata. Acts over semigroups appeared and were used in a variety of applications like algebraic automata theory, mathematical linguistics etc