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 $\mathcal {H}$ -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 Schützenberger groups have the property.

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.