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The Lawson number of a semitopological semilatticeBanakh, Taras, 1968- ; Bardyla, Serhii ; Gutik, OlegFor 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 ▫$\bar\Lambda(X)=\bar\psi(X)=cf(\lambda)$▫. A topologized semilattice ▫$X$▫ is called (i) ▫$\omega$▫-▫$Lawson$▫ if ▫$\bar\Lambda(X) \le \omega$▫; (ii) ▫$complete$▫ if each non-empty chain ▫$C\subset X$▫ 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 ▫$h:X\to Y$▫ from a compete topologized semilattice ▫$X$▫ to an ▫$\omega$▫-Lawson semitopological semilattice ▫$Y$▫ the image ▫$h(X)$▫ is closed in ▫$Y$▫.Source: Semigroup forum. - ISSN 0037-1912 (Vol. 103, iss. 1, Aug. 2021, str. 24-37)Type of material - article, component part ; adult, seriousPublish date - 2021Language - englishCOBISS.SI-ID - 64074755
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Banakh, Taras, 1968- | 31193 |
Bardyla, Serhii | |
Gutik, Oleg |
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