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A CHARACTERISATION OF SEMIGROUPS WITH ONLY COUNTABLY MANY SUBDIRECT PRODUCTS WITH $\mathbb {Z}$
Abstract Let $\mathbb {Z}$ be the additive (semi)group of integers. We prove that for a finite semigroup S the direct product $\mathbb {Z}\times S$ contains only countably many subdirect products (up to isomorphism) if and only if S is regular. As a corollary we show that $\mathbb {Z}\times S$ has …