Algebras of Reduced E-Fountain Semigroups and the Generalized Ample Identity II

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