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Given a countable monoid $S$, is the set of all (isomorphic representatives of) $S$-acts a small set?

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I assume that $S$-act means a set with an $S$-action that can't be decomposed into two nonempty subsets both invariant under $S$. Then the answer to the question is no. Let $S$ be the 2-element monoid that isn't a group, i.e., the multiplicative monoid $\{0,1\}$, and let $\kappa$ be any non-zero cardinal number. Then there is an $S$-act of cardinality $\kappa$ in which 0 acts as a constant map. For different cardinals $\kappa$, you get non-isomorphic $S$-acts.

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