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We study the generalized right ample identity, introduced by the author in aprevious paper. Let $S$ be a reduced $E$-Fountain semigroup which satisfies thecongruence condition. We can associate with $S$ a small category$\mathcal{C}(S)$ whose set of objects is identified with the set $E$ ofidempotents 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 ofgeneralized Green's relations. In this case, we interpret the associatedcategory $\mathcal{C}(S)$ as a discrete form of a Peirce decomposition of thesemigroup algebra. We also give some natural examples of semigroups satisfyingthis identity.

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