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Register a new userCharacterizing categorically closed commutative semigroups
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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 $Yin mathcal C$ 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 $Isubseteq X$ there are elements $x,yin I$ such that $xy otin{x,y}$. We prove that a semigroup $X$ is $mathcal C$-closed if it admits a homomorphism $h:Xto E$ to a chain-finite semilattice $E$ such that for every $ein E$ 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 $Asubseteq X$ 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 $Xsetminus H(X)$.
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