The endomorphism semiring of a commutative inverse semigroup

TL;DR

This paper extends the understanding of the algebraic structure of endomorphism semirings by proving their subdirect irreducibility in the context of commutative inverse semigroups with multiple idempotents, generalizing previous results on semilattices.

Contribution

It establishes that the endomorphism semiring of a commutative inverse semigroup with at least two idempotents is always subdirectly irreducible and characterizes its monolith.

Findings

Endomorphism semiring is subdirectly irreducible for certain semigroups.

Characterization of the monolith in these semirings.

Generalization from semilattices to commutative inverse semigroups.

Abstract

The authors [3] proved that the endomorphism semiring of a nontrivial semilattice is always subdirectly irreducible and described its monolith. Here we prove that the endomorphism semiring of a commutative inverse semigroup with at least two idempotents is always subdirectly irreducible and describe its monolith.