Partial Semigroup Algebras Associated to Partial Actions

TL;DR

This paper introduces algebraic crossed products and partial semigroup algebras associated with inverse semigroup partial actions, establishing conditions for associativity and linking these structures through quotients.

Contribution

It defines algebraic crossed products by partial actions of inverse semigroups and introduces partial semigroup algebras, connecting them via quotients under certain conditions.

Findings

Crossed product is associative under specific conditions.

Partial semigroup algebra K_{Par}(S) is introduced and related to crossed products.

Quotients of K_{Par}(S) serve as a form of crossed product.

Abstract

For a given inverse semigroup S , we introduce the notion of algebraic crossed product by using a given partial action of S, and we will prove that under some condition it is associative. Also we will introduce the concept of partial semigroup algebra K_{Par}(S) , and we show that the suitable quotient of K_{Par}(S) is a sort of crossed product.