论文标题
明确封闭的半群的亚组
Subgroups of categorically closed semigroups
论文作者
论文摘要
令$ \ Mathcal C $为一类拓扑半群。如果$ x $在\ in \ Mathcal c $中关闭$ x $，则称为（1）$ \ MATHCAL C $ - $关闭$，如果$ x $在\ in \ Mathcal c $ in \ Mathcal c $中，含有$ x $作为离散的subigroup，（2）$ $ $ $ $ $ $ \ $ $ \ $ \ $ \ mathcal c $ - 对于任何理想的$ x $ x $ x $ x $ x $ x $ x $ x $ x $ i \ i i i \ - i i $ i i i i i i i i i i i c. （3）绝对$ $ $ $ $ $ $ $ $ $ $关闭$，如果任何同构$ h：x \ to y $ to y $ to to topology semigroup $ y \ in \ mathcal c $，image $ h [x] $在$ y $中封闭，（4）$ $ $ $ $ $ \ nift $ \ nift $ $ $ $ $ $ $ $ \ nift $ \ nift $ \ nift $ \ nift $ \ nift $ \ nift $ \ nift。 $ h：x \ to y $ to topolication semigroup $ y \ in \ mathcal c $，图像$ h [x] $在$ y $中关闭（分别离散）。令$ \ mathsf {t _ {\！z} s} $为Tychonoff零维拓扑半群的类。对于半群，$ x $ let $ v \！理想的$ x $。我们证明了以下结果：（i）对于任何理想的$ \ mathsf {t _ {\！z} s} $ - 封闭的semigroup $ x $ x $每个中心$ z（x）= \ {z \ in x：\ in x：\ in x：\ in x：\ forall x \ in x \ \; \ \ \; \; \ \;（xz = zx） （ii）对于任何$ \ mathsf {t _ {\！z} s} $ - 封闭的semigroup $ x $，理想中心$ i \！z（x）= \ {z \ in z（x）：zx \ subseteq z（x）（x）\} $的每个子组; （iii）对于任何$ \ mathsf { $ \ mathsf {t _ {\！z} s} $ - 封闭的semigroup $ x $，最大子组$ h_e $是理想情况下（和绝对）$ \ mathsf {t _ {\！z} s} s} $ - 封闭并已封闭，已限制（和finite）中心$ z（h_e）$。
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 $Y\in\mathcal C$ 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 $h:X\to Y$ to a topological semigroup $Y\in\mathcal C$, the image $h[X]$ is closed in $Y$, (4) $injectively$ $\mathcal C$-$closed$ (resp. $\mathcal C$-$discrete$) if for any injective homomorphism $h:X\to Y$ to a topological semigroup $Y\in\mathcal C$, 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 $X\setminus\frac{H_e}e$ of the set $\frac{H_e}e=\{x\in X:xe=ex\in H_e\}$ 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 $Z(X)=\{z\in X:\forall x\in X\;\;(xz=zx)\}$ is bounded; (ii) for any $\mathsf{T_{\!z}S}$-closed semigroup $X$, each subgroup of the ideal center $I\!Z(X)=\{z\in Z(X):zX\subseteq Z(X)\}$ 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 $Z(H_e)$.
