A CHARACTERISATION OF SEMIGROUPS WITH ONLY COUNTABLY MANY SUBDIRECT PRODUCTS WITH Z

Let Z be the additive (semi)group of integers. We prove that for a finite semigroup S the direct product Z× S contains only countably many subdirect products (up to isomorphism) if and only if S is regular. As a corollary we show that Z× S has only countably many subsemigroups (up to isomorphism) if and only if S is completely regular.