# Green's relations and stability for subsemigroups

## TL;DR

This paper investigates how Green's relations are inherited by subsemigroups under stability conditions, providing new theoretical results and counterexamples, and reviewing related stability concepts over the past sixty years.

## Contribution

It offers novel results on Green's relations inheritance in stable semigroups and clarifies the relationships among various stability notions in semigroup theory.

## Key findings

- Not all right-stable semigroups are embeddable in left-stable semigroups.
- New inheritance results for Green's relations under stability conditions.
- Counterexamples illustrating limitations of stability inheritance.

## Abstract

We prove new results on inheritance of Green's relations by subsemigroups in the presence of stability of elements. We provide counterexamples in other cases to show in particular that not all right-stable semigroups are embeddable in left-stable semigroups. This is carried out in the context of a survey of the various closely related notions of stability and minimality of Green's classes that have appeared in the literature over the last sixty years, and which have sometimes been presented in different forms.