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The Lawson number of a semitopological semilattice
Semigroup Forum ( IF 0.7 ) Pub Date : 2021-05-13 , DOI: 10.1007/s00233-021-10184-z
Taras Banakh , Serhii Bardyla , Oleg Gutik

For a Hausdorff topologized semilattice X its Lawson number \(\bar{\Lambda }(X)\) is the smallest cardinal \(\kappa \) such that for any distinct points \(x,y\in X\) there exists a family \(\mathcal U\) of closed neighborhoods of x in X such that \(|\mathcal U|\le \kappa \) and \(\bigcap \mathcal U\) is a subsemilattice of X that does not contain y. It follows that \(\bar{\Lambda }(X)\le \bar{\psi }(X)\), where \(\bar{\psi }(X)\) is the smallest cardinal \(\kappa \) such that for any point \(x\in X\) there exists a family \(\mathcal U\) of closed neighborhoods of x in X such that \(|\mathcal U|\le \kappa \) and \(\bigcap \mathcal U=\{x\}\). We prove that a compact Hausdorff semitopological semilattice X is Lawson (i.e., has a base of the topology consisting of subsemilattices) if and only if \(\bar{\Lambda }(X)=1\). Each Hausdorff topological semilattice X has Lawson number \(\bar{\Lambda }(X)\le \omega \). On the other hand, for any infinite cardinal \(\lambda \) we construct a Hausdorff zero-dimensional semitopological semilattice X such that \(|X|=\lambda \) and \(\bar{\Lambda }(X)=\bar{\psi }(X)=\mathrm {cf}(\lambda )\). A topologized semilattice X is called (i) \(\omega \)-Lawson if \(\bar{\Lambda }(X)\le \omega \); (ii) complete if each non-empty chain \(C\subseteq X\) has \(\inf C\in {\overline{C}}\) and \(\sup C\in {\overline{C}}\). We prove that for any complete subsemilattice X of an \(\omega \)-Lawson semitopological semilattice Y, the partial order \(\le _X=\{(x,y)\in X\times X:xy=x\}\) of X is closed in \(Y\times Y\) and hence X is closed in Y. This implies that for any continuous homomorphism \(h:X\rightarrow Y\) from a complete topologized semilattice X to an \(\omega \)-Lawson semitopological semilattice Y the image h(X) is closed in Y.



中文翻译:

半拓扑半格的Lawson数

对于Hausdorff拓扑半格X,劳森数 \(\ bar {\ Lambda}(X)\)是最小基数\(\ kappa \),因此对于任何不同点\(x,y \ X \)都存在家庭\(\ mathcal U \)的封闭社区的XX,使得\(| \ mathcal U | \文件\卡帕\)\(\ bigcap \ mathcal U \)中的subsemilattice X不包含ÿ。因此,\(\ bar {\ Lambda}(X)\ le \ bar {\ psi}(X)\),其中\(\ bar {\ psi}(X)\)是最小基数\(\ kappa \)这样,对于任何点\(X \在X \)存在一个家庭\(\ mathcal U \)的封闭社区的XX,使得\(| \ mathcal U | \文件\卡帕\)\(\ bigcap \ mathcal U = \ {x \} \)。我们证明了当且仅当\(\ bar {\ Lambda}(X)= 1 \)时,紧Hausdorff半拓扑半格X才是Lawson(即,具有由半子半词组成的拓扑的基础。每个Hausdorff拓扑半晶格X的Lawson数为\(\ bar {\ Lambda}(X)\ le \ omega \)。另一方面,对于任何无限基数\(\ lambda \)我们构造一个Hausdorff零维半拓扑半格X,使得\(| X | = \ lambda \)\(\ bar {\ Lambda}(X)= \ bar {\ psi}(X)= \ mathrm {cf} (\ lambda)\)。被拓扑化的半晶格X称为(i)\(\ omega \) - Lawson if \(\ bar {\ Lambda}(X)\ le \ omega \) ; (ⅱ)完全如果每个非空链\(C \ subseteq X \)具有\(\ INFÇ\在{\划线{C}} \)\(\ SUPÇ\在{\划线{C}} \)。我们证明了对任何完整subsemilattice X的的\(\欧米茄\) -Lawson semitopological半格ÿ中,偏序\(\文件_X = \ {(X,Y)\在X \倍X:XY = X \} \)X中被关闭\(Y \倍Y \) ,因此X在关闭ÿ。这意味着对于任何连续同态\(H:X \ RIGHTARROW Y \)从一个完整的拓扑结构化半格X\(\欧米加\) -Lawson semitopological半格ý图像ħX)中被关闭ÿ

更新日期:2021-05-14
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