On minimal conditions in semigroups and biacts

TL;DR

This paper investigates minimal conditions related to Green's relations and stability in semigroups and biacts, analyzing how these properties are preserved or inherited through quotients, substructures, and extensions.

Contribution

It provides new results on the preservation and inheritance of minimal Green's class conditions and stability notions in semigroups and biacts, including counterexamples and characterizations.

Findings

Conditions $M_L$, $M_R$, and $M_J$ are preserved under quotients.

$M_L$, $M_R$, and stability are inherited by subsemigroups of finite Green index.

The equivalence of $M_L$ (or $M_R$) with ideals and Rees quotients is established.

Abstract

We systematically study the minimal conditions on $\mathcal{L}$-, $\mathcal{R}$- and $\mathcal{J}$-classes, denoted by $M_L$, $M_R$ and $M_J$, as well as the related notions of left/right/two-sided stability, in semigroups and biacts. In particular, we investigate the behaviour of these conditions with respect to quotients, substructures and extensions. Among other results, the following are proved. The conditions $M_L$, $M_R$ and $M_J$ are preserved under quotients (for both semigroups and biacts), but this is not the case for one- and two-sided stability. For semigroups, the conditions $M_L$, $M_R$ and one- and two-sided stability are inherited by subsemigroups of finite Green index and also by bi-ideals (and hence by one- and two-sided ideals). Moreover, a semigroup satisfies $M_L$ (or $M_R$) if and only if both an ideal and the associated Rees quotient do, but the analogue of this result fails in both directions for the condition $M_J$, and in the reverse direction for one- and two-sided stability.