A note on the relation $\mathcal{J}$ in $le$-semigroups

TL;DR

This paper investigates the structure of le-semigroups with commuting left ideal elements, showing that certain J-classes form subsemigroups and characterizing semisimple le-semigroups as semilattices of simpler components.

Contribution

It establishes conditions under which J-classes are subsemigroups and characterizes semisimple le-semigroups as semilattices of specific simple components.

Findings

J-classes satisfying the Green condition are subsemigroups.

Semisimple le-semigroups with commuting left ideals decompose into semilattices of simple components.

Characterization of semisimple le-semigroups satisfying condition mbda.

Abstract

We prove that if $S$ is a $le$-semigroup in which left ideal elements commute (condition which is called $\mathbf{\Lambda}$), then any $\mathcal{J}$-class satisfying the Green condition is a subsemigroup of $S$. As a corollary of this we show that semisimple $le$-semigroups satisfying $\mathbf{\Lambda}$ are precisely those that decompose as a semilattice of left simple $\vee e$-semigroups which are in addition semisimple, intra-regular and satisfy $\mathbf{\Lambda}$.