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- W4296412500 abstract "Let $mathcal C$ be a class of topological semigroups. A semigroup $X$ is called (1) $mathcal C$-$closed$ if $X$ is closed in every topological semigroup $Yinmathcal C$ containing $X$ as a discrete subsemigroup, (2) $ideally$ $mathcal C$-$closed$ if for any ideal $I$ in $X$ the quotient semigroup $X/I$ is $mathcal C$-closed; (3) $absolutely$ $mathcal C$-$closed$ if for any homomorphism $h:Xto Y$ to a topological semigroup $Yinmathcal C$, the image $h[X]$ is closed in $Y$, (4) $injectively$ $mathcal C$-$closed$ (resp. $mathcal C$-$discrete$) if for any injective homomorphism $h:Xto Y$ to a topological semigroup $Yinmathcal C$, the image $h[X]$ is closed (resp. discrete) in $Y$. Let $mathsf{T_{!z}S}$ be the class of Tychonoff zero-dimensional topological semigroups. For a semigroup $X$ let $V!E(X)$ be the set of all viable idempotents of $X$, i.e., idempotents $e$ such that the complement $Xsetminusfrac{H_e}e$ of the set $frac{H_e}e={xin X:xe=exin H_e}$ is an ideal in $X$. We prove the following results: (i) for any ideally $mathsf{T_{!z}S}$-closed semigroup $X$ each subgroup of the center $Z(X)={zin X:forall xin X;;(xz=zx)}$ is bounded; (ii) for any $mathsf{T_{!z}S}$-closed semigroup $X$, each subgroup of the ideal center $I!Z(X)={zin Z(X):zXsubseteq Z(X)}$ is bounded; (iii) for any $mathsf{T_{!z}S}$-discrete or injectively $mathsf{T_{!z}S}$-closed semigroup $X$, every subgroup of $Z(X)$ is finite, (iv) for any viable idempotent $e$ in an ideally (and absolutely) $mathsf{T_{!z}S}$-closed semigroup $X$, the maximal subgroup $H_e$ is ideally (and absolutely) $mathsf{T_{!z}S}$-closed and has bounded (and finite) center $Z(H_e)$." @default.
- W4296412500 created "2022-09-20" @default.
- W4296412500 creator A5076106414 @default.
- W4296412500 creator A5088614199 @default.
- W4296412500 date "2022-09-16" @default.
- W4296412500 modified "2023-09-23" @default.
- W4296412500 title "Subgroups of categorically closed semigroups" @default.
- W4296412500 doi "https://doi.org/10.48550/arxiv.2209.08013" @default.
- W4296412500 hasPublicationYear "2022" @default.
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