Separability conditions in acts over monoids

TL;DR

This paper explores various separability conditions for acts over monoids, establishing which monoids ensure all their acts satisfy these properties, with specific results for finite, commutative, idempotent, and Clifford monoids.

Contribution

It characterizes monoids based on the separability properties of their acts, providing new results for classes like finite, commutative, idempotent, and Clifford monoids.

Findings

All acts over a finite monoid are completely separable.

Finitely generated acts over finitely generated commutative monoids are residually finite.

Acts over a Clifford monoid are strongly subact separable.

Abstract

We discuss residual finiteness and several related separability conditions for the class of monoid acts, namely weak subact separability, strong subact separability and complete separability. For each of these four separability conditions, we investigate which monoids have the property that all their (finitely generated) acts satisfy the condition. In particular, we prove that: all acts over a finite monoid are completely separable (and hence satisfy the other three separability conditions); all finitely generated acts over a finitely generated commutative monoid are residually finite and strongly subact separable (and hence weakly subact separable); all acts over a commutative idempotent monoid are residually finite and strongly subact separable; and all acts over a Clifford monoid are strongly subact separable.