On separability finiteness conditions in semigroups

TL;DR

This paper explores various finiteness conditions related to residual finiteness in semigroups, establishing their equivalence in finitely generated commutative cases and examining inheritance by Schützenberger groups.

Contribution

It characterizes when three related separability properties coincide in finitely generated commutative semigroups and investigates their inheritance by Schützenberger groups.

Findings

In finitely generated commutative semigroups, the three properties are equivalent and relate to finiteness of H-classes.

Examples show these properties can differ in general semigroups.

For semigroups with finitely many H-classes, properties are linked to properties of Schützenberger groups.

Abstract

Taking residual finiteness as a starting point, we consider three related finiteness properties: weak subsemigroup separability, strong subsemigroup separability and complete separability. We investigate whether each of these properties is inherited by Sch\"utzenberger groups. The main result of this paper states that for a finitely generated commutative semigroup $S$, these three separability conditions coincide and are equivalent to every $\mathcal{H}$-class of $S$ being finite. We also provide examples to show that these properties in general differ for commutative semigroups and finitely generated semigroups. For a semigroup with finitely many $\mathcal{H}$-classes, we investigate whether it has one of these properties if and only if all its Sch\"utzenberger groups have the property.