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Serhii Bardyla

Publications and source records attributed to Serhii Bardyla.

At least 37 records · Page 2Linked to original sources
arXiv2021

The interplay between weak topologies on topological semilattices

We study the interplay between three weak topologies on a topological semilattice $X$: the weak$^\circ$ topology $\mathcal W^\circ_X$ (generated by the base consiting of open subsemilattices of $X$), the weak$^\bullet$ topology $\mathcal W^\bullet_X$ (generated by the subbase consisting of complements to closed subsemilattices), and the $\mathbb I$-weak topology $\mathcal W_X$ (which is the weakest topology in which all continuous homomorphisms $h:X\to [0,1]$ remain continuous). Also we study the interplay between the weak topologies $\mathcal W^\bullet_X$, $\mathcal W^\circ_X$, $\mathcal W_X$ of a topological semilattice $X$ and the Scott and Lawson topologies $\mathcal S_X$ and $\mathcal L_X$, which are determined by the order structure of the semilattice. We prove that the weak$^\bullet$ topology $\mathcal W^\bullet$ on a Hausdorff semitopological semilattice $X$ is compact if and only if $X$ is chain-compact in the sense that each closed chain in $X$ is compact. This result implies that the Lawson topology $\mathcal L_X$ on a semilattice $X$ is compact if and only if $X$ is a continuous semilattice if and only if $X$ complete in the sense that each non-empty chain $C$ in $X$ has $\inf(C)$ and $\sup(C)$ in $X$. For a chain-compact Hausdorff topological semilattice $X$ with topology $\mathcal T_X$ we prove the inclusions $\mathcal W_X\subset\mathcal L_X\subset\mathcal W^\bullet_X\subset\mathcal T_X$. For a compact topological semilattice $X$ we prove that $\mathcal T_X=\mathcal W^\bullet_X$ if and only if $\mathcal T_X=\mathcal L_X$ if and only if $\mathcal T_X=\mathcal L_X$.

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arXiv2021

On regular separable countably compact $\mathbb{R}$-rigid spaces

A topological space $X$ is said to be {\em $Y$-rigid} if any continuous map $f:X\rightarrow Y$ is constant. In this paper we construct a number of examples of regular countably compact $\mathbb R$-rigid spaces with additional properties like separability and first countability. This way we answer several questions of Tzannes, Banakh, Ravsky, as well as get a consistent example of $\mathbb R$-rigid Nyikos space. Also, we show that it is consistent with ZFC that for every cardinal $κ<\mathfrak c$ there exists a regular separable countably compact space $X$ which is $Y$-rigid with respect to any $T_1$ space $Y$ of pseudocharacter $\leqκ$.

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arXiv2021

On topological McAlister semigroups

In this paper we consider McAlister semigroups over arbitrary cardinals and investigate their algebraic and topological properties. We show that the group of automorphisms of a McAlister semigroup $\mathcal{M}_λ$ is isomorphic to the direct product $Sym(λ){\times}\mathbb{Z}_2$, where $Sym(λ)$ is the group of permutations of the cardinal $λ$. This fact correlates with the result of Mashevitzky, Schein and Zhitomirski which states that the group of automorphisms of the free inverse semigroup over a cardinal $λ$ is isomorphic to the wreath product of $Sym(λ)$ and $\mathbb{Z}_2$. Each McAlister semigroup admits a compact semigroup topology. Consequently, the Green's relations $\mathscr D$ and $\mathscr J$ coincide in McAlister semigroups. The latter fact complements results of Lawson. We showed that each non-zero element of a Hausdorff semitopological McAlister semigroup is isolated. This fact is an analogue of the result of Mesyan, Mitchell, Morayne and Péresse, who proved that each non-zero element of Hausdorff topological polycyclic monoid is isolated. Also, it follows that the free inverse semigroup over a singleton admits only the discrete Hausdorff shift-continuous topology. We proved that a Hausdorff locally compact semitopological semigroup $\mathcal{M}_1$ is either compact or discrete. This fact is similar to the result of Gutik, who showed that a Hausdorff locally compact semitopological polycyclic monoid $\mathcal{P}_1$ is either compact or discrete. However, this dichotomy does not hold for the semigroup $\mathcal{M}_2$. Moreover, $\mathcal{M}_2$ admits continuum many different Hausdorff locally compact inverse semigroup topologies.

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arXiv2020

A semigroup is finite if and only if it is chain-finite and antichain-finite

A subset $A$ of a semigroup $S$ is called a $chain$ ($antichain$) if $xy\in\{x,y\}$ ($xy\notin\{x,y\}$) for any (distinct) elements $x,y\in S$. A semigroup $S$ is called ($anti$)$chain$-$finite$ if $S$ contains no infinite (anti)chains. We prove that each antichain-finite semigroup $S$ is periodic and for every idempotent $e$ of $S$ the set $\sqrt[\infty]{e}=\{x\in S:\exists n\in\mathbb N\;\;(x^n=e)\}$ is finite. This property of antichain-finite semigroups is used to prove that a semigroup is finite if and only if it is chain-finite and antichain-finite. Also we present an example of an antichain-finite semilattice that is not a union of finitely many chains.

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arXiv2020

On regular $κ$-bounded spaces admitting only constant continuous mappings into $T_1$ spaces of pseudo-character $\leq κ$

In this paper for each cardinal $κ$ we construct an infinite $κ$-bounded (and hence countably compact) regular space $R_κ$ such that for any $T_1$ space $Y$ of pseudo-character $\leqκ$, each continuous function $f:R_κ\rightarrow Y$ is constant. This result resolves two problems posted by Tzannes in Open Problems from Topology Proceedings and extends results of Ciesielski and Wojciechowski and Herrlich.

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arXiv2019

Embedding of graph inverse semigroups into CLP-compact topological semigroups

In this paper we investigate graph inverse semigroups which are subsemigroups of compact-like topological semigroups. More precisely, we characterise graph inverse semigroups which admit a compact semigroup topology and describe graph inverse semigroups which can be embeded densely into CLP-compact topological semigroups.

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arXiv2019

The Lawson number of a semitopological semilattice

For a Hausdorff topologized semilattice $X$ its $Lawson\;\; number$ $\barΛ(X)$ is the smallest cardinal $κ$ 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 $|\mathcal U|\leκ$ and $\bigcap\mathcal U$ is a subsemilattice of $X$ that does not contain $y$. It follows that $\barΛ(X)\le\barψ(X)$, where $\barψ(X)$ is the smallest cardinal $κ$ such that for any point $x\in X$ there exists a family $\mathcal U$ of closed neighborhoods of $x$ in $X$ such that $|\mathcal U|\leκ$ 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 $\barΛ(X)=1$. Each Hausdorff topological semilattice $X$ has Lawson number $\barΛ(X)\leω$. On the other hand, for any infinite cardinal $λ$ we construct a Hausdorff zero-dimensional semitopological semilattice $X$ such that $|X|=λ$ and $\barΛ(X)=\barψ(X)=cf(λ)$. A topologized semilattice $X$ is called (i) $ω$-$Lawson$ if $\barΛ(X)\leω$; (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 $ω$-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 $ω$-Lawson semitopological semilattice $Y$ the image $h(X)$ is closed in $Y$.

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arXiv2019

On the lattice of weak topologies on the bicyclic monoid with adjoined zero

A Hausdorff topology $τ$ on the bicyclic monoid with adjoined zero $\mathcal{C}^0$ is called {\em weak} if it is contained in the coarsest inverse semigroup topology on $\mathcal{C}^0$. We show that the lattice $\mathcal{W}$ of all weak shift-continuous topologies on $\mathcal{C}^0$ is isomorphic to the lattice of all shift-invariant filters on $ω$ with an attached element $1$ endowed with the following partial order: $\mathcal{F}\leq \mathcal{G}$ iff $\mathcal{G}=1$ or $\mathcal{F}\subset \mathcal{G}$. Also, we investigate cardinal characteristics of the lattice $\mathcal{W}$. In particular, we proved that $\mathcal{W}$ contains an antichain of cardinality $2^{\mathfrak{c}}$ and a well-ordered chain of cardinality $\mathfrak{c}$. Moreover, there exists a well-ordered chain of first-countable weak topologies of order type $\mathfrak{t}$.

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arXiv2019

Closed subsets of compact-like topological spaces

We investigate closed subsets (subsemigroups, resp.) of compact-like topological spaces (semigroups, resp.). We prove that each Hausdorff topological space can be embedded as a closed subspace into an H-closed topological space. However, the semigroup of $ω{\timesω}$-matrix units cannot be embedded into a topological semigroup which is a weakly H-closed topological space. We show that each Hausdorff topological space is a closed subspace of some $ω$-bounded pracompact topological space and describe open dense subspaces of countably pracompact topological spaces. Also, we construct a pseudocompact topological semigroup which contains the bicyclic monoid as a closed subsemigroup, providing a positive solution of a problem posed by Banakh, Dimitrova, and Gutik.

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arXiv2019

Embeddings into countably compact Hausdorff spaces

In this paper we consider the problem of characterization of topological spaces that embed into countably compact Hausdorff spaces. We study the separation axioms of subspaces of countably compact Hausdorff spaces and construct an example of a regular separable scattered topological space which cannot be embedded into an Urysohn countably compact topological space but embeds into a Hausdorff countably compact space.

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arXiv2019

An alternative look at the structure of graph inverse semigroups

For any graph inverse semigroup $G(E)$ we describe subsemigroups $D^0=D\cup\{0\}$ and $J^0=J\cup\{0\}$ of $G(E)$ where $D$ and $J$ are arbitrary $\mathcal{D}$-class and $\mathcal{J}$-class of $G(E)$, respectively. In particular, we prove that for each $\mathcal{D}$-class $D$ of a graph inverse semigroup over an acyclic graph the semigroup $D^0$ is isomorphic to a semigroup of matrix units. Also we show that for any elements $a,b$ of a graph inverse semigroup $G(E)$, $J_a\cdot J_b\cup J_b\cdot J_a\subset J_b^0$ if there exists a path $w$ such that $s(w)\in J_a$ and $r(w)\in J_b$.

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arXiv2019

The closedness of complete subsemilattices in functionally Hausdorff semitopological semilattices

A topologized semilattice $X$ is complete if each non-empty chain $C\subset X$ has $\inf C\in\bar C$ and $\sup C\in\bar C$. It is proved that for any complete subsemilattice $X$ of a functionally Hausdorff semitopological semilattice $Y$ the partial order $P=\{(x,y)\in X\times X:xy=x\}$ 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 a functionally Hausdorff semitopological semilattice $Y$ the image $h(X)$ is closed in $Y$. The functional Hausdorffness of $Y$ in these two results can be replaced by the weaker separation axiom $\vec T_{2δ}$, defined in this paper.

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arXiv2019

Positive answers to Koch's problem in special cases

A topological semigroup is monothetic provided it contains a dense cyclic subsemigroup. The Koch problem asks whether every locally compact monothetic monoid is compact. This problem was opened for more than sixty years, till in 2018 Zelenyuk obtained a negative answer. In this paper we obtain a positive answer for Koch's problem for some special classes of topological monoids. Namely, we show that a locally compact monothetic topological monoid is a compact topological group if and only if $S$ is a submonoid of a quasitopological group if and only if $S$ has open shifts if and only if $S$ is non-viscous in the sense of Averbukh. The last condition means that any neighborhood $U$ of the identity $1$ of $S$ and for any element $a\in S$ there exists a neighborhood $V$ of $a$ such that any element $x\in S$ with $(xV\cup Vx)\cap V\ne\emptyset$ belongs to the neighborhood $U$ of 1.

math.GR↗
arXiv2019

A metrizable semitopological semilattice with non-closed partial order

We construct a metrizable semitopological semilattice $X$ whose partial order $P=\{(x,y)\in X\times X:xy=x\}$ is a non-closed dense subset of $X\times X$. As a by-product we find necessary and sufficient conditions for the existence of a (metrizable) Hausdorff topology on a set, act, semigroup or semilattice, having a prescribed countable family of convergent sequences.

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arXiv2018

On universal objects in the class of graph inverse semigroups

In this paper we show that polycyclic monoids are universal objects in the class of graph inverse semigroups. In particular, we prove that a graph inverse semigroup $G(E)$ over a directed graph $E$ embeds into the polycyclic monoid $\mathcal{P}_λ$ where $λ=|G(E)|$. We show that each graph inverse semigroup $G(E)$ admits the coarsest inverse semigroup topology $τ$. Moreover, each injective homomorphism from $(G(E),τ)$ to the $(\mathcal{P}_{|G(E)|},τ)$ is a topological embedding.

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arXiv2018

On locally compact semitopological graph inverse semigroups

In this paper we investigate locally compact semitopological graph inverse semigroups. Our main result is the following: if a directed graph $E$ is strongly connected and contains a finite amount of vertices then a locally compact semitopological graph inverse semigroup $G(E)$ is either compact or discrete. This result generalizes results of Gutik and Bardyla who proved the above dichotomy for locally compact semitopological polycyclic monoids $\mathcal{P}_1$ and $\mathcal{P}_λ$, respectively.

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