Semigroup Algebras of Inverse Semigroups

Abstract

In view of the Vagner-Preston theorem, the inverse semigroups can be naturally viewed as semigroups of partial monomorphisms of a given set X. The paper provides an algorithm to describe all indecomposable projective representations of any semigroup of partial monomorphisms of a finite set X that are compatible with a given relation on X contained in a total order on X. The algorithm yields an explicit description of the structure of regular representations of such a given semigroup by exhibiting a complete set of primitive orthogonal idempotents of the respective semigroup algebra and thus presenting the quivers of the respective semigroup algebra. It is at this stage that one can consider questions related to the representation type of a given semigroup and provide in some cases a complete classification of all indecomposable representations. The methods, including those of the graph semigroups, are prepared to describe representation of sets that are unions of finite chains and diamonds.