Characterizing categorically closed commutative semigroups

Abstract

Let $\mathcal{C}$ be a class of Hausdorff topological semigroups which contains all zero-dimensional Hausdorff topological semigroups. A semigroup $X$ is called $\mathcal{C}$-$closed$ if $X$ is closed in each topological semigroup $Y\in \mathcal{C}$ containing $X$ as a discrete subsemigroup; $X$ is $projectively$ $\mathcal{C}$-$closed$ if for each congruence $\approx$ on $X$ the quotient semigroup $X{/}_{\approx }$ is $\mathcal{C}$-closed. A semigroup $X$ is called $chain$-$finite$ if for any infinite set $I\subseteq X$ there are elements $x,y\in I$ such that $xy\notin \{x,y\}$. We prove that a semigroup $X$ is $\mathcal{C}$-closed if it admits a homomorphism $h:X\to E$ to a chain-finite semilattice $E$ such that for every $e\in E$ the semigroup $h^{-1}(e)$ is $\mathcal{C}$-closed. Applying this theorem, we prove that a commutative semigroup $X$ is $\mathcal{C}$-closed if and only if $X$ is periodic, chain-finite, all subgroups of $X$ are bounded, and for any infinite set $A\subseteq X$ the product $AA$ is not a singleton. A commutative semigroup $X$ is projectively $\mathcal{C}$-closed if and only if $X$ is chain-finite, all subgroups of $X$ are bounded and the union $H(X)$ of all subgroups in $X$ has finite complement $X\setminus H(X)$.

arXiv comment19 pages

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