论文标题
在半足型半群中$ \ boldsymbol {b}_Ω^{\ mathscr {f}} $当一个家庭$ \ mathscr {f} $由$ω$的归纳非超级子集组成
On a semitopological semigroup $\boldsymbol{B}_ω^{\mathscr{F}}$ when a family $\mathscr{F}$ consists of inductive non-empty subsets of $ω$
论文作者
论文摘要
令$ \ boldsymbol {b}_Ω^{\ mathscr {f}} $为$ω$ clucted $ $ω$ clucted $ $ω$的$ \ mathscr {f} $的biciclic semigroup扩展，$ω$的子集的$ω$ cite {gutik-mykhalenych = 2020}。我们研究了semigroup $ \ boldsymbol {b}_ω^{\ mathscr {f}} $的the the $ \ mathscr {f} $的归纳$ω$ cluctuctive $ω$的子集的$ \ mathscr {f} $。我们将Eberhart-selden和Bertman-West的结果推广到Bicyclic Semigroup \ cite {Bertman-West-1976，Eberhart-Selden = 1969}的拓扑结果，并表明每个Hausdorff Shift-chift-Conlinuul topology in semogroup $ \ boldsymbol {B boldsymbol {b} $ ivcr { hausdorff半序言半群$ s $包含$ \ boldsymbol {b}_Ω^{\ mathscr {f}} $作为适当的密集子群，然后$ s \ setMinus \ setminus \ boldsymbol {b}_Ω另外，我们证明了以下二分法：$ \ boldsymbol {b}_ω^{\ mathscr {f}} $，均具有相邻的零是紧凑或离散的每个Hausdorff均在$ \ boldsymbol {b}_Ω^{\ Mathscr {f}} $上的每个Hausdorff。由于最后的结果，我们得到了$ \ boldsymbol {b}_Ω^{\ Mathscr {f}} $的每个Hausdorff本地紧凑的半群拓扑，均具有离散的零，每个Hausdorff均为Semigroup上的每个Hausdorff semigroup the Semigroup the Semigroup topology the Semigroup the Semigroup topology a $ \ boldsymbol {b}_Ω^{\ mathscr {f}} \ sqcup i $带有相邻的紧凑型理想$ i $是紧凑或理想$ i $是开放的，或者理想的$ i $是开放的，这是许多关于本地紧凑型拓扑的本地紧凑型拓扑的结果$ \ boldsymbol {b}_Ω^{\ mathscr {f}} $。
Let $\boldsymbol{B}_ω^{\mathscr{F}}$ be the bicyclic semigroup extension for the family $\mathscr{F}$ of $ω$-closed subsets of $ω$ which is introduced in \cite{Gutik-Mykhalenych=2020}. We study topologizations of the semigroup $\boldsymbol{B}_ω^{\mathscr{F}}$ for the family $\mathscr{F}$ of inductive $ω$-closed subsets of $ω$. We generalize Eberhart-Selden and Bertman-West results about topologizations of the bicyclic semigroup \cite{Bertman-West-1976, Eberhart-Selden=1969} and show that every Hausdorff shift-continuous topology on the semigroup $\boldsymbol{B}_ω^{\mathscr{F}}$ is discrete and if a Hausdorff semitopological semigroup $S$ contains $\boldsymbol{B}_ω^{\mathscr{F}}$ as a proper dense subsemigroup then $S\setminus\boldsymbol{B}_ω^{\mathscr{F}}$ is an ideal of $S$. Also, we prove the following dichotomy: every Hausdorff locally compact shift-continuous topology on $\boldsymbol{B}_ω^{\mathscr{F}}$ with an adjoined zero is either compact or discrete. As a consequence of the last result we obtain that every Hausdorff locally compact semigroup topology on $\boldsymbol{B}_ω^{\mathscr{F}}$ with an adjoined zero is discrete and every Hausdorff locally compact shift-continuous topology on the semigroup $\boldsymbol{B}_ω^{\mathscr{F}}\sqcup I$ with an adjoined compact ideal $I$ is either compact or the ideal $I$ is open, which extent many results about locally compact topologizations of some classes of semigroups onto extensions of the semigroup $\boldsymbol{B}_ω^{\mathscr{F}}$.
