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Gutik Oleg - One of the best experts on this subject based on the ideXlab platform.
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On monoids of injective partial cofinite selfmaps
'Walter de Gruyter GmbH', 2019Co-Authors: Gutik Oleg, Repovš DušanAbstract:We study the semigroup ▫$mathscr{I}^{mathrm{cf}}_lambda$▫ of injective partial cofinite selfmaps of an Infinite Cardinal ▫$lambda$▫. We show that ▫$mathscr{I}^{mathrm{cf}}_lambda$▫ is a bisimple inverse semigroup and each chain of idempotents in ▫$mathscr{I}^{mathrm{cf}}_lambda$▫ is contained in a bicyclic subsemigroup of ▫$mathscr{I}^{mathrm{cf}}_lambda$▫, we describe the Green relations on ▫$mathscr{I}^{mathrm{cf}}_lambda$▫ and we prove that every non-trivial congruence on ▫$mathscr{I}^{mathrm{cf}}_lambda$▫ is a group congruence. Also, we describe the structure of the quotient semigroup ▫$mathscr{I}^{mathrm{cf}}_lambda/sigma$▫, where ▫$sigma$▫ is the least group congruence on ▫$mathscr{I}^{mathrm{cf}}_lambda$▫
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The Lawson number of a semitopological semilattice
2019Co-Authors: Banakh Taras, Bardyla Serhii, Gutik OlegAbstract: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 $\bar\Lambda(X)=\bar\psi(X)=cf(\lambda)$. 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 $Y$.Comment: 10 pages. arXiv admin note: text overlap with arXiv:1806.0286
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Extensions of semigroups by symmetric inverse semigroups of a bounded finite rank
2019Co-Authors: Gutik Oleg, Sobol OleksandraAbstract:We study the semigroup extension $\mathscr{I}_\lambda^n(S)$ of a semigroup $S$ by symmetric inverse semigroups of a bounded finite rank. We describe idempotents and regular elements of the semigroups $\mathscr{I}_\lambda^n(S)$ and $\overline{\mathscr{I}_\lambda^n}(S)$ show that the semigroup $\mathscr{I}_\lambda^n(S)$ ($\overline{\mathscr{I}_\lambda^n}(S)$) is regular, orthodox, inverse or stable if and only if so is $S$. Green's relations are described on the semigroup $\mathscr{I}_\lambda^n(S)$ for an arbitrary monoid $S$. We introduce the conception of a semigroup with strongly tight ideal series, and proved that for any Infinite Cardinal $\lambda$ and any positive integer $n$ the semigroup $\mathscr{I}_\lambda^n(S)$ has a strongly tight ideal series provides so has $S$. At the finish we show that for every compact Hausdorff semitopological monoid $(S,\tau_S)$ there exists a unique its compact topological extension $\left(\mathscr{I}_\lambda^n(S),\tau_{\mathscr{I}}^\mathbf{c}\right)$ in the class of Haudorff semitopological semigroups.Comment: 24 page
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On feebly compact semitopological symmetric inverse semigroups of a bounded finite rank
2018Co-Authors: Gutik OlegAbstract:We study feebly compact shift-continuous $T_1$-topologies on the symmetric inverse semigroup $\mathscr{I}_\lambda^n$ of finite transformations of the rank $\leqslant n$. For any positive integer $n\geqslant2$ and any Infinite Cardinal $\lambda$ a Hausdorff countably pracompact non-compact shift-continuous topology on $\mathscr{I}_\lambda^n$ is constructed. We show that for an arbitrary positive integer $n$ and an arbitrary Infinite Cardinal $\lambda$ for a $T_1$-topology $\tau$ on $\mathscr{I}_\lambda^n$ the following conditions are equivalent: $(i)$ $\tau$ is countably pracompact; $(ii)$ $\tau$ is feebly compact; $(iii)$ $\tau$ is $d$-feebly compact; $(iv)$ $\left(\mathscr{I}_\lambda^n,\tau\right)$ is H-closed; $(v)$ $\left(\mathscr{I}_\lambda^n,\tau\right)$ is $\mathbb{N}_{\mathfrak{d}}$-compact for the discrete countable space $\mathbb{N}_{\mathfrak{d}}$; $(vi)$ $\left(\mathscr{I}_\lambda^n,\tau\right)$ is $\mathbb{R}$-compact; $(vii)$ $\left(\mathscr{I}_\lambda^n,\tau\right)$ is infra H-closed. Also we prove that for an arbitrary positive integer $n$ and an arbitrary Infinite Cardinal $\lambda$ every shift-continuous semiregular feebly compact $T_1$-topology $\tau$ on $\mathscr{I}_\lambda^n$ is compact.Comment: 12 pages. arXiv admin note: text overlap with arXiv:1606.0039
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On a complete topological inverse polycyclic monoid
'Vasyl Stefanyk Precarpathian National University', 2016Co-Authors: Bardyla Serhii, Gutik OlegAbstract:We give sufficient conditions when a topological inverse $\lambda$-polycyclic monoid $P_{\lambda}$ is absolutely $H$-closed in the class of topological inverse semigroups. Also, for every Infinite Cardinal $\lambda$ we construct the coarsest semigroup inverse topology $\tau_{mi}$ on $P_\lambda$ and give an example of a topological inverse monoid S which contains the polycyclic monoid $P_2$ as a dense discrete subsemigroup.Comment: 10 page
Szentmiklóssy Zoltán - One of the best experts on this subject based on the ideXlab platform.
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Coloring Cantor sets and resolvability of pseudocompact spaces
'American Mathematical Society (AMS)', 2018Co-Authors: Juhász István, Soukup Lajos, Szentmiklóssy ZoltánAbstract:summary:Let us denote by $\Phi(\lambda,\mu)$ the statement that $\mathbb{B}(\lambda) = D(\lambda)^\omega$, i.e. the Baire space of weight $\lambda$, has a coloring with $\mu$ colors such that every homeomorphic copy of the Cantor set $\mathbb{C}$ in $\mathbb{B}(\lambda)$ picks up all the $\mu$ colors. We call a space $X$ $\pi$-regular if it is Hausdorff and for every nonempty open set $U$ in $X$ there is a nonempty open set $V$ such that $\overline{V} \subset U$. We recall that a space $X$ is called feebly compact if every locally finite collection of open sets in $X$ is finite. A Tychonov space is pseudocompact if and only if it is feebly compact. The main result of this paper is the following: Let $X$ be a crowded feebly compact $\pi$-regular space and $\mu$ be a fixed (finite or Infinite) Cardinal. If $\Phi(\lambda,\mu)$ holds for all $\lambda < \hat{c}(X)$ then $X$ is $\mu$-resolvable, i.e. $X$ contains $\mu$ pairwise disjoint dense subsets. (Here $\hat{c}(X)$ is the smallest Cardinal $\kappa$ such that $X$ does not contain $\kappa$ many pairwise disjoint open sets.) This significantly improves earlier results of [van Mill J., {Every crowded pseudocompact ccc space is resolvable}, Topology Appl. 213 (2016), 127--134], or [Ortiz-Castillo Y. F., Tomita A. H., {Crowded pseudocompact Tychonoff spaces of cellularity at most the continuum are resolvable}, Conf. talk at Toposym 2016]
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Coloring Cantor sets and resolvability of pseudocompact spaces
2017Co-Authors: Juhász István, Soukup Lajos, Szentmiklóssy ZoltánAbstract:Let us denote by $\Phi(\lambda,\mu)$ the statement that $\mathbb{B}(\lambda) = D(\lambda)^\omega$, i.e. the Baire space of weight $\lambda$, has a coloring with $\mu$ colors such that every homeomorphic copy of the Cantor set $\mathbb{C}$ in $\mathbb{B}(\lambda)$ picks up all the $\mu$ colors. We call a space $X\,$ {\em $\pi$-regular} if it is Hausdorff and for every non-empty open set $U$ in $X$ there is a non-empty open set $V$ such that $\overline{V} \subset U$. We recall that a space $X$ is called {\em feebly compact} if every locally finite collection of open sets in $X$ is finite. A Tychonov space is pseudocompact iff it is feebly compact. The main result of this paper is the following. Theorem. Let $X$ be a crowded feebly compact $\pi$-regular space and $\mu$ be a fixed (finite or Infinite) Cardinal. If $\Phi(\lambda,\mu)$ holds for all $\lambda < \widehat{c}(X)$ then $X$ is $\mu$-resolvable, i.e. contains $\mu$ pairwise disjoint dense subsets. (Here $\widehat{c}(X)$ is the smallest Cardinal $\kappa$ such that $X$ does not contain $\kappa$ many pairwise disjoint open sets.) This significantly improves earlier results of van Mill , resp. Ortiz-Castillo and Tomita.Comment: 8 page
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Anti-Urysohn spaces
2015Co-Authors: Juhász István, Soukup Lajos, Szentmiklóssy ZoltánAbstract:All spaces are assumed to be Infinite Hausdorff spaces. We call a space "anti-Urysohn" $($AU in short$)$ iff any two non-emty regular closed sets in it intersect. We prove that $\bullet$ for every Infinite Cardinal ${\kappa}$ there is a space of size ${\kappa}$ in which fewer than $cf({\kappa})$ many non-empty regular closed sets always intersect; $\bullet$ there is a locally countable AU space of size $\kappa$ iff $\omega \le \kappa \le 2^{\mathfrak c}$. A space with at least two non-isolated points is called "strongly anti-Urysohn" $($SAU in short$)$ iff any two Infinite closed sets in it intersect. We prove that $\bullet$ if $X$ is any SAU space then $ \mathfrak s\le |X|\le 2^{2^{\mathfrak c}}$; $\bullet$ if $\mathfrak r=\mathfrak c$ then there is a separable, crowded, locally countable, SAU space of Cardinality $\mathfrak c$; \item if $\lambda > \omega$ Cohen reals are added to any ground model then in the extension there are SAU spaces of size $\kappa$ for all $\kappa \in [\omega_1,\lambda]$; $\bullet$ if GCH holds and $\kappa \le\lambda$ are uncountable regular Cardinals then in some CCC generic extension we have $\mathfrak s={\kappa}$, $\,\mathfrak c={\lambda}$, and for every Cardinal ${\mu}\in [\mathfrak s, \mathfrak c]$ there is an SAU space of Cardinality ${\mu}$. The questions if SAU spaces exist in ZFC or if SAU spaces of Cardinality $> \mathfrak c$ can exist remain open
Juhász István - One of the best experts on this subject based on the ideXlab platform.
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Coloring Cantor sets and resolvability of pseudocompact spaces
'American Mathematical Society (AMS)', 2018Co-Authors: Juhász István, Soukup Lajos, Szentmiklóssy ZoltánAbstract:summary:Let us denote by $\Phi(\lambda,\mu)$ the statement that $\mathbb{B}(\lambda) = D(\lambda)^\omega$, i.e. the Baire space of weight $\lambda$, has a coloring with $\mu$ colors such that every homeomorphic copy of the Cantor set $\mathbb{C}$ in $\mathbb{B}(\lambda)$ picks up all the $\mu$ colors. We call a space $X$ $\pi$-regular if it is Hausdorff and for every nonempty open set $U$ in $X$ there is a nonempty open set $V$ such that $\overline{V} \subset U$. We recall that a space $X$ is called feebly compact if every locally finite collection of open sets in $X$ is finite. A Tychonov space is pseudocompact if and only if it is feebly compact. The main result of this paper is the following: Let $X$ be a crowded feebly compact $\pi$-regular space and $\mu$ be a fixed (finite or Infinite) Cardinal. If $\Phi(\lambda,\mu)$ holds for all $\lambda < \hat{c}(X)$ then $X$ is $\mu$-resolvable, i.e. $X$ contains $\mu$ pairwise disjoint dense subsets. (Here $\hat{c}(X)$ is the smallest Cardinal $\kappa$ such that $X$ does not contain $\kappa$ many pairwise disjoint open sets.) This significantly improves earlier results of [van Mill J., {Every crowded pseudocompact ccc space is resolvable}, Topology Appl. 213 (2016), 127--134], or [Ortiz-Castillo Y. F., Tomita A. H., {Crowded pseudocompact Tychonoff spaces of cellularity at most the continuum are resolvable}, Conf. talk at Toposym 2016]
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Coloring Cantor sets and resolvability of pseudocompact spaces
2017Co-Authors: Juhász István, Soukup Lajos, Szentmiklóssy ZoltánAbstract:Let us denote by $\Phi(\lambda,\mu)$ the statement that $\mathbb{B}(\lambda) = D(\lambda)^\omega$, i.e. the Baire space of weight $\lambda$, has a coloring with $\mu$ colors such that every homeomorphic copy of the Cantor set $\mathbb{C}$ in $\mathbb{B}(\lambda)$ picks up all the $\mu$ colors. We call a space $X\,$ {\em $\pi$-regular} if it is Hausdorff and for every non-empty open set $U$ in $X$ there is a non-empty open set $V$ such that $\overline{V} \subset U$. We recall that a space $X$ is called {\em feebly compact} if every locally finite collection of open sets in $X$ is finite. A Tychonov space is pseudocompact iff it is feebly compact. The main result of this paper is the following. Theorem. Let $X$ be a crowded feebly compact $\pi$-regular space and $\mu$ be a fixed (finite or Infinite) Cardinal. If $\Phi(\lambda,\mu)$ holds for all $\lambda < \widehat{c}(X)$ then $X$ is $\mu$-resolvable, i.e. contains $\mu$ pairwise disjoint dense subsets. (Here $\widehat{c}(X)$ is the smallest Cardinal $\kappa$ such that $X$ does not contain $\kappa$ many pairwise disjoint open sets.) This significantly improves earlier results of van Mill , resp. Ortiz-Castillo and Tomita.Comment: 8 page
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Anti-Urysohn spaces
2015Co-Authors: Juhász István, Soukup Lajos, Szentmiklóssy ZoltánAbstract:All spaces are assumed to be Infinite Hausdorff spaces. We call a space "anti-Urysohn" $($AU in short$)$ iff any two non-emty regular closed sets in it intersect. We prove that $\bullet$ for every Infinite Cardinal ${\kappa}$ there is a space of size ${\kappa}$ in which fewer than $cf({\kappa})$ many non-empty regular closed sets always intersect; $\bullet$ there is a locally countable AU space of size $\kappa$ iff $\omega \le \kappa \le 2^{\mathfrak c}$. A space with at least two non-isolated points is called "strongly anti-Urysohn" $($SAU in short$)$ iff any two Infinite closed sets in it intersect. We prove that $\bullet$ if $X$ is any SAU space then $ \mathfrak s\le |X|\le 2^{2^{\mathfrak c}}$; $\bullet$ if $\mathfrak r=\mathfrak c$ then there is a separable, crowded, locally countable, SAU space of Cardinality $\mathfrak c$; \item if $\lambda > \omega$ Cohen reals are added to any ground model then in the extension there are SAU spaces of size $\kappa$ for all $\kappa \in [\omega_1,\lambda]$; $\bullet$ if GCH holds and $\kappa \le\lambda$ are uncountable regular Cardinals then in some CCC generic extension we have $\mathfrak s={\kappa}$, $\,\mathfrak c={\lambda}$, and for every Cardinal ${\mu}\in [\mathfrak s, \mathfrak c]$ there is an SAU space of Cardinality ${\mu}$. The questions if SAU spaces exist in ZFC or if SAU spaces of Cardinality $> \mathfrak c$ can exist remain open
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Covering compacta by discrete subspaces
Elsevier B.V., 2007Co-Authors: Juhász István, Van Mill JanAbstract:AbstractFor any space X, denote by dis(X) the smallest (Infinite) Cardinal κ such that κ many discrete subspaces are needed to cover X. It is easy to see that if X is any crowded (i.e. dense-in-itself) compactum then dis(X)⩾m, where m denotes the additivity of the meager ideal on the reals. It is a natural, and apparently quite difficult, question whether in this inequality m could be replaced by c. Here we show that this can be done if X is also hereditarily normal.Moreover, we prove the following mapping theorem that involves the Cardinal function dis(X). If f:X→Y is a continuous surjection of a countably compact T2 space X onto a perfect T3 space Y then |{y∈Y:f−1y is countable}|⩽dis(X)
Samuel Christian - One of the best experts on this subject based on the ideXlab platform.
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Copies of c 0 (τ ) spaces in projective tensor products
'American Mathematical Society (AMS)', 2020Co-Authors: Cortes Vinícius, Galego Medina, Samuel ChristianAbstract:International audienceThis paper is dedicated to the memory of our friend Eve Oja. Abstract. Let X and Y be Banach spaces and consider the projective tensor product X ⊗ π Y. Suppose that τ is an Infinite Cardinal, X has the bounded approximation property and the density character of X is strictly smaller than the cofinality of τ. We prove the following c 0 (τ) generalizations of classical c 0 results due to Oja (1991) and Kwapień (1974) respectively: (i) If c 0 (τ) is isomorphic to a complemented subspace of X ⊗ π Y , then c 0 (τ) is isomorphic to a complemented subspace of Y. (ii) If c 0 (τ) is isomorphic to a subspace of X ⊗ π Y , then c 0 (τ) is isomorphic to a subspace of Y. We also show that the result (i) is optimal for regular Cardinals τ and Banach spaces X without copies of c 0 (τ). In order to do so, we provide a c 0 (τ) extension of a classical c 0 result due to Emmanuele (1988) concerning the c 0 (τ) complemented subspaces of Lp(D τ , Y) spaces, 1 ≤ p ≤ ∞, where D τ is the Cantor cube. Finally, as a consequence of (i) we conclude that under the continuum hypothesis, the space c 0 (ℵα), with α > 1, is not isomorphic to a complemented subspace of l∞ ⊗ π l∞(ℵα)
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When is c 0 (τ ) complemented in tensor products of l p (I) and X?
'Wiley', 2019Co-Authors: Cortes Vinícius, Galego Medina, Samuel ChristianAbstract:International audienceLet X be a Banach space, I an Infinite set, τ an Infinite Cardinal and p ∈ [1, ∞). In contrast to a classical c 0 result due independently to Cembranos and Freniche, we prove that if the cofinality of τ is greater than the Cardinality of I, then the injective tensor product p(I) ⊗ ε X contains a complemented copy of c 0 (τ) if and only if X does. This result is optimal for every regular Cardinal τ. On the other hand, we provide a generalization of a c 0 result of Oya by proving that if τ is an Infinite Cardinal, then the projective tensor product p(I) ⊗ π X contains a complemented copy of c 0 (τ) if and only if X does. These results are obtained via useful descriptions of tensor products as convenient generalized sequence spaces
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Complemented copies of c_0(tau) in tensor products of L_p[0,1]
'Mathematical Sciences Publishers', 2019Co-Authors: Cortes Vinícius, Galego Medina, Samuel ChristianAbstract:International audienceLet X be a Banach space and τ an Infinite Cardinal. We show that if τ has uncountable cofinality, p ∈ [1, ∞) and either the Lebesgue-Bochner space L p ([0, 1], X) or the injective tensor product L_p [0, 1] ⊗_ ε X contains a complemented copy of c_0 (τ), then so does X. We show also that if p ∈ (1, ∞) and the projective tensor product L_p [0, 1] ⊗_π X contains a complemented copy of c_0 (τ), then so does X
Oleg Gutik - One of the best experts on this subject based on the ideXlab platform.
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the lawson number of a semitopological semilattice
Semigroup Forum, 2021Co-Authors: Serhii Bardyla, Taras Banakh, Oleg GutikAbstract: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.
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the lawson number of a semitopological semilattice
arXiv: General Topology, 2019Co-Authors: Serhii Bardyla, Taras Banakh, Oleg GutikAbstract: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 $\bar\Lambda(X)=\bar\psi(X)=cf(\lambda)$. 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 $Y$.
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on feebly compact topologies on the semilattice exp_n lambda
Matematychni Studii, 2016Co-Authors: Oleg Gutik, Oleksandra SobolAbstract:We study feebly compact topologies $\tau$ on the semilattice $\left(\exp_n\lambda,\cap\right)$ such that $\left(\exp_n\lambda,\tau\right)$ is a semitopological semilattice. All compact semilattice $T_1$-topologies on $\exp_n\lambda$ are described. Also we prove that for an arbitrary positive integer $n$ and an arbitrary Infinite Cardinal $\lambda$ for a $T_1$-topology $\tau$ on $\exp_n\lambda$ the following conditions are equivalent: $(i)$ $\left(\exp_n\lambda,\tau\right)$ is a compact topological semilattice; $(ii)$ $\left(\exp_n\lambda,\tau\right)$ is a countably compact topological semilattice; $(iii)$ $\left(\exp_n\lambda,\tau\right)$ is a feebly compact topological semilattice; $(iv)$ $\left(\exp_n\lambda,\tau\right)$ is a compact semitopological semilattice; $(v)$ $\left(\exp_n\lambda,\tau\right)$ is a countably compact semitopological semilattice. We construct a countably pracompact $H$-closed quasiregular non-semiregular topology $\tau_{\operatorname{\textsf{fc}}}^2$ such that $\left(\exp_2\lambda,\tau_{\operatorname{\textsf{fc}}}^2\right)$ is a semitopological semilattice with discontinuous semilattice operation and prove that for an arbitrary positive integer $n$ and an arbitrary Infinite Cardinal $\lambda$ every $T_1$-semiregular feebly compact semitopological semilattice $\exp_n\lambda$ is a compact topological semilattice.