$E$-separated semigroups

TL;DR

This paper characterizes $E$-separated semigroups by their idempotent commutativity properties and extends these results to $ ext{pi}$-regular $E$-semigroups, building on prior work by Putcha and Weissglass.

Contribution

It provides new characterizations of $E$-separated semigroups through idempotent commutativity and extends these to $ ext{pi}$-regular $E$-semigroups, advancing the theoretical understanding.

Findings

Characterization of $E$-separated semigroups via idempotent commutativity

Extension of characterizations to $ ext{pi}$-regular $E$-semigroups

Connection to prior results by Putcha and Weissglass

Abstract

A semigroup is called $E$-$separated$ if for any distinct idempotents $x,y\in X$ there exists a homomorphism $h:X\to Y$ to a semilattice $Y$ such that $h(x)\ne h(y)$. Developing results of Putcha and Weissglass, we characterize $E$-separated semigroups via certain commutativity properties of idempotents of $X$. Also we characterize $E$-separated semigroups in the class of $\pi$-regular $E$-semigroups.