  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.
