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Abstract 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.
Clayton et al. (Fri,) studied this question.
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