Serhii Bardyla

Serhii Bardyla
University of Vienna | UniWien · Kurt Gödel Research Center for Mathematical Logic

PhD in Mathematics

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59
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286
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Publications

Publications (59)
Preprint
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In this paper we establish a connection between categorical closedness and topologizability of semigroups. In particular, for a class $\mathsf T_{\!1}\mathsf S$ of $T_1$ topological semigroups we prove that a countable semigroup $X$ with finite-to-one shifts is injectively $\mathsf T_{\!1}\mathsf S$-closed if and only if $X$ is $\mathsf{T_{\!1}S}$-...
Preprint
We solve the last standing open problem from the seminal paper by J. Gerlits and Zs. Nagy, which was later reposed by A. Miller, T. Orenshtein and B. Tsaban. Namely, we show that under p = c there is a \delta-set, which is not a \gamma-set. Thus we construct a set of reals A such that although Cp(A), the space of all real-valued continuous function...
Article
Let C be a class of T1 topological semigroups which contains all zero-dimensional Hausdorff topological semigroups. A semigroup X is called C-closed if X is closed in each topological semigroup Y∈C containing X as a discrete subsemigroup; X is projectively C-closed if for each congruence ≈ on X the quotient semigroup X/≈ is C-closed. A semigroup X...
Article
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For a Hausdorff topologized semilattice X its Lawson numberΛ¯(X) is the smallest cardinal κ such that for any distinct points x,y∈X there exists a family U of closed neighborhoods of x in X such that |U|≤κ and ⋂U is a subsemilattice of X that does not contain y. It follows that Λ¯(X)≤ψ¯(X), where ψ¯(X) is the smallest cardinal κ such that for any p...
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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 $\mathcal{M}_{\lambda}$ is isomorphic to the direct product $Sym(\lambda){\times}\mathbb{Z}_2$, where $Sym(\lambda)$ is the group of permutations of the...
Preprint
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Let C be a class of Hausdorff topological semigroups which contains all zero-dimensional Hausdorff topological semigroups. A semigroup X is called C-closed if X is closed in each topological semigroup Y ∈ C containing X as a discrete subsemigroup; X is projectively C-closed if for each congruence ≈ on X the quotient semigroup X/≈ is C-closed. A sem...
Article
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A subset A of a semigroup S is called a chain (antichain) if ab∈{a,b} (ab∉{a,b}) for any (distinct) elements a,b∈A. A semigroup S is called periodic if for every element x∈S there exists n∈N such that xn is an idempotent. A semigroup S is called (anti)chain-finite if S contains no infinite (anti)chains. We prove that each antichain-finite semigroup...
Preprint
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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 $\{x \in S : \e...
Article
For each cardinal k we construct an infinite k-bounded (and hence countably compact) regular space R_k such that for any T_1 space Y of pseudocharacter <= k, each continuous function f : R_k--> Y is constant. This result resolves two problems posted by Tzannes [13] and extends results of Ciesielski and Wojciechowski [4] and Herrlich [8].
Article
We construct a metrizable Lawson semitopological semilattice $X$ whose partial order $\le_X\,=\{(x,y)\in X\times X:xy=x\}$ is not closed in $X\times X$. This resolves a problem posed earlier by the authors.
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p>We investigate closed subsets (subsemigroups, resp.) of compact-like topological spaces (semigroups, resp.). We show that each Hausdorff topological space is a closed subspace of some Hausdorff ω-bounded pracompact topological space and describe open dense subspaces of countably pracompact topological spaces. We construct a pseudocompact topologi...
Preprint
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We call a regular topological space $X$ to be $\mathbb R$-rigid if any continuous real-valued function on $X$ is constant. In this paper we construct a number of consistent examples of countably compact $\mathbb R$-rigid spaces with additional properties like separability and first countability. This way we answer several questions of Tzannes, Bana...
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A topologized semilattice X is called complete if each non-empty chain C⊂X\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C\subset X$$\end{document} has infC∈C¯\documen...
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Under Martin's Axiom we construct a Boolean countably compact topological group whose square is not countably pracompact.
Article
Let κ be an infinite cardinal. A topological space X is κ-bounded if the closure of any subset of cardinality ≤κ in X is compact. We discuss the problem of embeddability of topological spaces into Hausdorff (Urysohn, regular) κ-bounded spaces, and present a canonical construction of such an embedding. Also we construct a (consistent) example of a s...
Article
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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 c...
Article
Full-text available
We construct a metrizable semitopological semilattice X whose partial order P = {(x, y) ∈ X × X : xy = x} is a non-closed dense subset of X × 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 convergen...
Preprint
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Under Martin's Axiom we construct a Boolean countably compact topological group whose square is not countably pracompact.
Article
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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 \({\mathscr {P}}_{\lambda }\) where \(\lambda =|G(E)|\). We show that each graph inverse semigroup G(E) admits the coarsest inverse semi...
Article
In this paper we investigate graph inverse semigroups which are subsemigroups of compact-like topological semigroups. More precisely, we characterize graph inverse semigroups which admit a compact semigroup topology and describe graph inverse semigroups which can be embedded densely into CLP-compact topological semigroups.
Article
A Hausdorff topology τ on the bicyclic monoid with adjoined zero C0 is called weak if it is contained in the coarsest inverse semigroup topology on C0. We show that the lattice W of all weak shift-continuous topologies on C0 is isomorphic to the lattice SIF1×SIF1 where SIF1 is the set of all shift-invariant filters on ω with an attached element 1 e...
Preprint
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In this paper for each cardinal $\kappa$ we construct an infinite $\kappa$-bounded (and hence countably compact) regular space $R_{\kappa}$ such that for any $T_1$ space $Y$ of pseudo-character $\leq\kappa$, each continuous function $f:R_{\kappa}\rightarrow Y$ is constant. This result resolves two problems posted by Tzannes in Open Problems from To...
Preprint
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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 $y$. I...
Preprint
Full-text available
We construct a metrizable Lawson semitopological semilattice $X$ whose partial order $\le_X=\{(x,y)\in X\times X:xy=x\}$ is not closed in $X\times X$. This resolves a problem posed earlier by the authors.
Article
A topologized semilattice X is complete if each non-empty chain C⊂X has inf⁡C∈C¯ and sup⁡C∈C¯. It is proved that for any complete subsemilattice X of a functionally Hausdorff semitopological semilattice Y the partial order ≤X={(x,y)∈X×X:xy=x} of X is closed in Y×Y and hence X is closed in Y. This implies that for any continuous homomorphism h:X→Y f...
Preprint
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A Hausdorff topology $\tau$ on the bicyclic monoid $\mathcal{C}^0$ 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 $\omega$ with an at...
Preprint
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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 $\omega{\times\omega}$-matrix units cannot be embedded into a topological semigroup which is an...
Preprint
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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 t...
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We discuss the problem of embeddibility of a topological space into a Hausdorff $\omega$-bounded space, and present two canonical constructions of such an embedding.
Article
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In the paper we present various characterizations of chain-compact and chain-finite topological semilattices. A topological semilattice $X$ is called chain-compact (resp. chain-finite) if each closed chain in $X$ is compact (finite). In particular, we prove that a (Hausdorff) $T_1$-topological semilattice $X$ is chain-finite (chain-compact) if and...
Preprint
Full-text available
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 c...
Preprint
Full-text available
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...
Article
Full-text available
We study the interplay between three weak topologies on a topological semilattice X: the weak ∘ topology w X∘ (generated by the base consisting of open subsemilattices of X), the weak • topology w X• (generated by the subbase consisting of complements to closed subsemilattices), and the I-weak topology w X (which is the weakest topology in which al...
Preprint
Full-text available
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 compact semigroup topology and describe graph inverse semigroups which can be embeded densely into d-compact topological semigroups.
Preprint
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For each graph inverse semigroup $G(E)$ we describe subsemigroups $D\cup\{0\}$ and $J\cup\{0\}$ of $G(E)$ where $D$ and $J$ are arbitrary $\mathcal{D}$-class and $\mathcal{J}$-class of $G(E)$, respectively. We show that each graph inverse semigroup is a $0$-union of a semilattice $X$ of semigroups $\{S_e\}_{e\in X}$ where each $S_e$ is a Brandt $\l...
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In this paper we discuss the notion of completeness of topologized posets and survey some recent results on closedness properties of complete topologized semilattices.
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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 $Y$...
Preprint
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A topologized semilattice $X$ is called complete if each non-empty chain $C\subset X$ has $\inf C\in\bar C$ and $\sup C\in\bar C$. We prove that for any continuous homomorphism $h:X\to Y$ from a complete topologized semilattice $X$ to a sequential Hausdorff semitopological semilattice $Y$ the image $h(X)$ is closed in $Y$.
Article
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The Comfort topology on a topologized semigroup $X$ is generated by the subbase consisting of the complements to closed subsemigroups of $X$. We prove that the Comfort topology on a Hausdorff semitopological semilattice $X$ is compact if and only if each closed chain in $X$ is compact. Also we prove that the topology of a compact Hausdorff topologi...
Article
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A topology τ on a monoid S is called shift-continuous if for every a, b ∈ S the two-sided shift S → S, x ↦ axb, is continuous. For every ordinal α ≤ ω, we describe all shift-continuous locally compact Hausdorff topologies on the α-bicyclic monoid Bα. More precisely, we prove that the lattice of shift-continuous locally compact Hausdorff topologies...
Preprint
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In this paper we show that the λ-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 λ-polycyclic monoid P_λ where λ = |G(E)|. We show that each graph inverse semigroup G(E) admits the coarsest inverse semigroup topolog...
Article
Full-text available
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...
Article
An $H$-closed quasitopological group is a Hausdorff quasitopological group which is contained in each Hausdorff quasitopological group as a closed subspace. We obtained a sufficient condition for a quasitopological group to be $H$-closed, which allowed us to solve a problem by Arhangel'skii and Choban and to show that a topological group $G$ is $H$...
Article
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In this paper we prove that a graph inverse semigroup G(E) over a directed graph E embeds into the λ-polycyclic monoid P λ where λ = |G(E)|. Moreover, each countable graph inverse semigroup embeds into the polycyclic monoid P 2. We shall follow the terminology of [11] and [20]. By |A| we denote the cardinality of a set A and by ω we denote the firs...
Article
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In this paper for every ordinal $\alpha<\omega+1$ we describe all locally compact Hausdorff topologies which make $\alpha$-bicyclic monoid $\mathcal{B}_{\alpha}$ a semitopological semigroup. In particular, we prove that there exist exactly $k$ distinct locally compact Hausdorff topologies which make $\mathcal{B}_{k}$ a semitopological semigroup and...
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In this paper we characterise graph inverse semigroups which admit only discrete locally compact semigroup topology. This characterization provide a complete answer on the question of Z. Mesyan, J. D. Mitchell, M. Morayne and Y. H. P\'eresse posed in in their 2016 paper "Topological graph inverse semigroups".
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We find (completeness type) conditions on topological semilattices $X,Y$ guaranteeing that each continuous homomorphism $h:X\to Y$ has closed image $h(X)$ in $Y$.
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Classifying locally compact semitopological polycyclic monoids, Math. Bull. Shevchenko Sci. Soc. 13 (2016) 1–9. We present a complete classification of Hausdorff locally compact polycyclic monoids up to a topological isomorphism. A polycyclic monoid is an inverse monoid with zero, generated by a subset Λ such that xx −1 = 1 for any x ∈ Λ and xy −1...
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In this paper for each non-zero cardinal $\lambda$ we describe all locally compact topologies $\tau$ on the $\lambda$-polycyclic monoid $\mathcal{P}_{\lambda}$ such that $(\mathcal{P}_{\lambda},\tau)$ is a semitopological semigroup. In particular we prove that a locally compact semitopological $\lambda$-Polycyclic monoid is either compact or discre...
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In this paper we consider a semitopological α-bicyclic monoid B α and prove that it is algebraically isomorphic to a semigroup of all order isomorphisms between the principal upper sets of ordinal ω α. We prove that for every ordinal α for every (a, b) ∈ B α if a or b is a non-limit ordinal then (a, b) is an isolated point in B α. We show that for...
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We study algebraic structure of the λ-polycyclic monoid Pλ and its topologizations. We show that the λ-polycyclic monoid for an infinite cardinal λ⩾2 has similar algebraic properties so has the polycyclic monoid Pn with finitely many n⩾2 generators. In particular we prove that for every infinite cardinal λ the polycyclic monoid Pλ is a congruence-f...
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In this paper we consider the $\alpha$-bicyclic monoid $\mathcal{B}_{\alpha}$ as a semitopological semigroup and prove that it is algebraically isomorphic to the semigroup of all order isomorphisms between the principal upper sets of ordinal $\omega^{\alpha}$. Also we prove that for every ordinal $\alpha$ for every $(a,b)\in \mathcal{B_{\alpha}}$ i...
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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...
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A b s t r ac t. We study algebraic structure of the λ-polycyclic monoid P λ and its topologizations. We show that the λ-polycyclic monoid for an infinite cardinal λ 2 has similar algebraic properties so has the polycyclic monoid P n with finitely many n 2 generators. In particular we prove that for every infinite cardinal λ the polycyclic monoid P...
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In the paper we describe the structure of $\mathscr{AH}$-completions and $\mathscr{H}$-completions of the discrete semilattices $(\mathbb{N},\min)$ and $(\mathbb{N},\max)$. We give an example of an $\mathscr{H}$-complete topological semilattice which is not $\mathscr{AH}$-complete. Also we construct an $\mathscr{H}$-complete topological semilattice...