Algebras of reduced $E$-Fountain semigroups and the generalized ample identity II

TL;DR

This paper explores the structure of reduced $E$-Fountain semigroups satisfying the generalized right ample identity, linking algebraic properties to categorical and Peirce decomposition concepts.

Contribution

It characterizes when such semigroups satisfy the identity via homomorphisms of left actions and interprets their associated categories as Peirce decompositions.

Findings

Semigroups satisfying the identity induce specific homomorphisms.

Associated categories can be viewed as discrete Peirce decompositions.

Provides examples of semigroups fulfilling the generalized right ample identity.

Abstract

We study the generalized right ample identity, introduced by the author in a previous paper. Let $S$ be a reduced $E$-Fountain semigroup which satisfies the congruence condition. We can associate with $S$ a small category $\mathcal{C}(S)$ whose set of objects is identified with the set $E$ of idempotents and its morphisms correspond to elements of $S$. We prove that $S$ satisfies the generalized right ample identity if and only if every element of $S$ induces a homomorphism of left $S$-actions between certain classes of generalized Green's relations. In this case, we interpret the associated category $\mathcal{C}(S)$ as a discrete form of a Peirce decomposition of the semigroup algebra. We also give some natural examples of semigroups satisfying this identity.