On separability properties in direct products of semigroups

TL;DR

This paper explores how various residual finiteness properties of semigroups, such as separability conditions, are preserved or inherited in direct product constructions, providing necessary and sufficient criteria for finite semigroups.

Contribution

It offers new necessary and sufficient conditions for the preservation of strong and monogenic subsemigroup separability in direct products of finite semigroups.

Findings

Conditions for preservation of residual finiteness properties in direct products

Inheritance of properties by factors in direct products

Characterization of finite semigroups maintaining separability properties

Abstract

We investigate four finiteness conditions related to residual finiteness: complete separability, strong subsemigroup separability, weak subsemigroup separability and monogenic subsemigroup separability. For each of these properties we examine under which conditions the property is preserved under direct products. We also consider if any of the properties are inherited by the factors in a direct product. We give necessary and sufficient conditions for finite semigroups to preserve the properties of strong subsemigroup separability and monogenic subsemigroup separability in a direct product.