On separability finiteness conditions in semigroups

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 Schützenberger 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 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 H-classes, we investigate whether it has one of these properties if and only if all its Schützenberger groups have the property.

Residual Finiteness and Related Properties in Monounary Algebras and their Direct Products

“…While these properties have been studied for some time, such as in [5] and [3], the names used for them have not been consistent. The names we use are from [7], as these designed to be more descriptive of the property. We provide characterisations for these properties, and show how they interact with direct products in a similar fashion to how we dealt with residual finiteness.…”