arXiv subjects

Oleg Gutik

Publications and source records attributed to Oleg Gutik.

At least 37 records · Page 2Linked to original sources

arXiv2023

On the semigroup of injective monoid endomor\-phisms of the monoid $\boldsymbol{B}_ω^{\mathscr{F}}$ with the two-elements family $\mathscr{F}$ of inductive nonempty subsets of $ω$

We study injective endomorphisms of the semigroup $\boldsymbol{B}_ω^{\mathscr{F}}$ with the two-elements family $\mathscr{F}$ of inductive nonempty subsets of $ω$. We describe the elements of the semigroup $\boldsymbol{End}^1_*(\boldsymbol{B}_ω^{\mathscr{F}})$ of all injective monoid endomorphisms of the monoid $\boldsymbol{B}_ω^{\mathscr{F}}$, and show that Green's relations $\mathscr{R}$, $\mathscr{L}$, $\mathscr{H}$, $\mathscr{D}$, and $\mathscr{J}$ on $\boldsymbol{End}^1_*(\boldsymbol{B}_ω^{\mathscr{F}})$ coincide with the relation of equality.

math.GR↗

arXiv2023

On the semigroup $\boldsymbol{B}_ω^{\mathscr{F}_n}$ which is generated by the family $\mathscr{F}_n$ of finite bounded intervals of $ω$

We study the semigroup $\boldsymbol{B}_ω^{\mathscr{F}}$, which is introduced in the paper [O. Gutik and M. Mykhalenych, \emph{On some generalization of the bicyclic monoid}, Visnyk Lviv. Univ. Ser. Mech.-Mat. \textbf{90} (2020), 5--19 (in Ukrainian)], in the case when the family $\mathscr{F}_n$ generated by the set $\{0,1,\ldots,n\}$. We show that the Green relations $\mathscr{D}$ and $\mathscr{J}$ coincide in $\boldsymbol{B}_ω^{\mathscr{F}_n}$, the semigroup $\boldsymbol{B}_ω^{\mathscr{F}_n}$ is isomorphic to the semigroup $\mathscr{I}_ω^{n+1}(\overrightarrow{\mathrm{conv}})$ of partial convex order isomorphisms of $(ω,\leqslant)$ of the rank $\leqslant n+1$, and $\boldsymbol{B}_ω^{\mathscr{F}_n}$ admits only Rees congruences. Also, we study shift-continuous topologies on the semigroup $\boldsymbol{B}_ω^{\mathscr{F}_n}$. In particular we prove that for any shift-continuous $T_1$-topology $τ$ on the semigroup $\boldsymbol{B}_ω^{\mathscr{F}_n}$ every non-zero element of $\boldsymbol{B}_ω^{\mathscr{F}_n}$ is an isolated point of $(\boldsymbol{B}_ω^{\mathscr{F}_n},τ)$, $\boldsymbol{B}_ω^{\mathscr{F}_n}$ admits the unique compact shift-continuous $T_1$-topology, and every $ω_{\mathfrak{d}}$-compact shift-continuous $T_1$-topology is compact. We describe the closure of the semigroup $\boldsymbol{B}_ω^{\mathscr{F}_n}$ in a Hausdorff semitopological semigroup and prove the criterium when a topological inverse semigroup $\boldsymbol{B}_ω^{\mathscr{F}_n}$ is $H$-closed in the class of Hausdorff topological semigroups.

math.GR↗

arXiv2023

On automorphisms of the semigroup $\boldsymbol{B}_ω^{\mathscr{F}}$ in the case when the family $\mathscr{F}$ consists of nonempty inductive subsets of $ω$

Let $\mathscr{F}$ be a family of nonempty inductive subsets of $ω$. It is proved that an injective endomorphism $\varepsilon$ of the semigroup $\boldsymbol{B}_ω^{\mathscr{F}}$ is the transformation if and only if $\varepsilon$ has three distinct fixed points, which is equivalent to existence non-idempotent element $(i,j,[p))\in\boldsymbol{B}_ω^{\mathscr{F}}$ such that $(i,j,[p))\varepsilon=(i,j,[p))$.

math.GR↗

arXiv2023

On homomorphisms of bicyclic extensions of archimedean totally ordered groups

Let $\mathscr{B}^+(G)$ be the bicyclic extension of a totally ordered group $G$ which is defined in [O. Gutik, D. Pagon, and K. Pavlyk, Congruences on bicyclic extensions of a linearly ordered group, Acta Comment. Univ. Tartu. Math. 15 (2011), no. 2, 61-80, (arXiv:1111.2401)]. We show that if $G$ and $H$ are archimedean totally ordered groups then every $o$-homomorphism $\widehatφ\colon G\to H$ generates a monoid homomorphim $\widetildeφ\colon \mathscr{B}^+(G)\to \mathscr{B}^+(H)$, and every monoid homomorphism $\widetildeφ\colon \mathscr{B}^+(G)\to \mathscr{B}^+(H)$ generates $o$-homomorphism $\widehatφ\colon G\to H$.

math.GR↗

arXiv2023

On a semitopological semigroup $\boldsymbol{B}_ω^{\mathscr{F}}$ when a family $\mathscr{F}$ consists of inductive non-empty subsets of $ω$

Let $\boldsymbol{B}_ω^{\mathscr{F}}$ be the bicyclic semigroup extension for the family $\mathscr{F}$ of $ω$-closed subsets of $ω$ which is introduced in \cite{Gutik-Mykhalenych=2020}. We study topologizations of the semigroup $\boldsymbol{B}_ω^{\mathscr{F}}$ for the family $\mathscr{F}$ of inductive $ω$-closed subsets of $ω$. We generalize Eberhart-Selden and Bertman-West results about topologizations of the bicyclic semigroup \cite{Bertman-West-1976, Eberhart-Selden=1969} and show that every Hausdorff shift-continuous topology on the semigroup $\boldsymbol{B}_ω^{\mathscr{F}}$ is discrete and if a Hausdorff semitopological semigroup $S$ contains $\boldsymbol{B}_ω^{\mathscr{F}}$ as a proper dense subsemigroup then $S\setminus\boldsymbol{B}_ω^{\mathscr{F}}$ is an ideal of $S$. Also, we prove the following dichotomy: every Hausdorff locally compact shift-continuous topology on $\boldsymbol{B}_ω^{\mathscr{F}}$ with an adjoined zero is either compact or discrete. As a consequence of the last result we obtain that every Hausdorff locally compact semigroup topology on $\boldsymbol{B}_ω^{\mathscr{F}}$ with an adjoined zero is discrete and every Hausdorff locally compact shift-continuous topology on the semigroup $\boldsymbol{B}_ω^{\mathscr{F}}\sqcup I$ with an adjoined compact ideal $I$ is either compact or the ideal $I$ is open, which extent many results about locally compact topologizations of some classes of semigroups onto extensions of the semigroup $\boldsymbol{B}_ω^{\mathscr{F}}$.

math.GR↗

arXiv2023

On the semigroup $\boldsymbol{B}_ω^{\mathscr{F}}$ which is generated by the family $\mathscr{F}$ of atomic subsets of $ω$

We study the semigroup $\boldsymbol{B}_ω^{\mathscr{F}}$, which is introduced in [O. Gutik and M. Mykhalenych, \emph{On some generalization of the bicyclic monoid}, Visnyk Lviv. Univ. Ser. Mech.-Mat. \textbf{90} (2020), 5--19], in the case when the family $\mathscr{F}$ of subsets of cardinality $\leqslant 1$ in $ω$. We show that $\boldsymbol{B}_ω^{\mathscr{F}}$ is isomorphic to the subsemigroup $\mathscr{B}_ω^{\Rsh}(\boldsymbol{F}_{\min})$ of the Brandt $ω$-extension of the semilattice $\boldsymbol{F}_{\min}$ and describe all shift-continuous feebly compact $T_1$-topologies on the semigroup $\mathscr{B}_ω^{\Rsh}(\boldsymbol{F}_{\min})$. In particulary we prove that every shift-continuous feebly compact $T_1$-topology $τ$ on $\mathscr{B}_ω^{\Rsh}(\boldsymbol{F}_{\min})$ is compact and moreover in this case the space $(\mathscr{B}_ω^{\Rsh}(\boldsymbol{F}_{\min}),τ)$ is homeomorphic to the one-point Alexandroff compactification of the discrete countable space $\mathfrak{D}(ω)$. We study the closure of $\boldsymbol{B}_ω^{\mathscr{F}}$ in a semitopological semigroup. In particularly we show that $\boldsymbol{B}_ω^{\mathscr{F}}$ is algebraically complete in the class of Hausdorff semitopological inverse semigroups with continuous inversion, and a Hausdorff topological inverse semigroup $\boldsymbol{B}_ω^{\mathscr{F}}$ is closed in any Hausdorff topological semigroup if and only if the band $E(\boldsymbol{B}_ω^{\mathscr{F}})$ is compact.

math.GR↗

arXiv2023

On endomorphisms of the bicyclic semigroup and the extended bicyclic semigroup

It is proved that the semigroups $\mathrm{\mathbf{End}}(\boldsymbol{B}_ω)$ and $\mathrm{\mathbf{End}}(\boldsymbol{B}_{\mathbb{Z}})$ of the endomorphisms of the bicyclic semigroup $\boldsymbol{B}_ω$ and the endomorphisms of the extended bicyclic semigroup $\boldsymbol{B}_{\mathbb{Z}}$ are isomorphic to the semidirect products $(ω,+)\rtimes_φ(ω,*)$ and $\mathbb{Z}(+)\rtimes_φ(ω,*)$, respectively.

math.GR↗

arXiv2022

On the group of automorphisms of the semigroup $\boldsymbol{B}_{\mathbb{Z}}^{\mathscr{F}}$ with the family $\mathscr{F}$ of inductive nonempty subsets of $ω$

We study automorphisms of the semigroup $\boldsymbol{B}_{Z\mathbb{}}^{\mathscr{F}}$ with the family $\mathscr{F}$ of inductive nonempty subsets of $ω$ and prove that the group $\mathbf{Aut}(\boldsymbol{B}_{\mathbb{Z}}^{\mathscr{F}})$ of automorphisms of the semigroup $\boldsymbol{B}_{Z\mathbb{}}^{\mathscr{F}}$ is isomorphic to the additive group integers.

math.GR↗

arXiv2022

On a locally compact monoid of cofinite partial isometries of $\mathbb{N}$ with adjoined zero

Let $\mathscr{C}_\mathbb{N}$ be a monoid which is generated by the partial shift $α\colon n\mapsto n+1$ of the set of positive integers $\mathbb{N}$ and its inverse partial shift $β\colon n+1\mapsto n$. In this paper we prove that if $S$ is a submonoid of the monoid $\mathbf{I}\mathbb{N}_{\infty}$ of all partial cofinite isometries of positive integers which contains $\mathscr{C}_\mathbb{N}$ as a submonoid then every Hausdorff locally compact shift-continuous topology on $S$ with adjoined zero is either compact or discrete. Also we show that the similar statement holds for a locally compact semitopological semigroup $S$ with an adjoined compact ideal.

math.GR↗

arXiv2022

On the semigroup of injective endomorphisms of the semigroup $\boldsymbol{B}_ω^{\mathscr{F}_n}$ which is generated by the family $\mathscr{F}_n$ of initial finite intervals of $ω$

In the paper we describe injective endomorphisms of the inverse semigroup $\boldsymbol{B}_ω^{\mathscr{F}}$, which is introduced in the paper [O. Gutik and M. Mykhalenych, \emph{On some generalization of the bicyclic monoid}, Visnyk Lviv. Univ. Ser. Mech.-Mat. \textbf{90} (2020), 5--19 (in Ukrainian)], in the case when the family $\mathscr{F}_n$ is generated by the set $\{0,1,\ldots,n\}$. In particular we show that the semigroup of injective endomorphisms of the semigroup $\boldsymbol{B}_ω^{\mathscr{F}}$ is isomorphic to $(ω,+)$. Also we describe the structure of the semigroup $\mathfrak{End}(\mathscr{B}_λ)$ of all endomorphisms of the semigroup of $λ{\times}λ$-matrix units $\mathscr{B}_λ$.

math.GR↗

arXiv2022

A note on feebly compact semitopological symmetric inverse semigroups of a bounded finite rank

We study feebly compact shift-continuous $T_1$-topologies on the symmetric inverse semigroup $\mathscr{I}_λ^n$ of finite transformations of the rank $\leqslant n$. It is proved that such $T_1$-topology is sequentially pracompact if and only if it is feebly compact. Also, we show that every shift-continuous feebly $ω$-bounded $T_1$-topology on $\mathscr{I}_λ^n$ is compact.

math.GN↗

arXiv2021

On some generalization of the bicyclic monoid

We introduce an algebraic extension $\boldsymbol{B}_ω^{\mathscr{F}}$ of the bicyclic monoid for an arbitrary $ω$-closed family $\mathscr{F}$ subsets of $ω$ which generalizes the bicyclic monoid, the countable semigroup of matrix units and some other combinatorial inverse semigroups. It is proved that $\boldsymbol{B}_ω^{\mathscr{F}}$ is a combinatorial inverse semigroup and Green's relations, the natural partial order on $\boldsymbol{B}_ω^{\mathscr{F}}$, and its set of idempotents are described. We provide criteria of simplicity, $0$-simplicity, bisimplicity, $0$-bisimplicity of the semigroup $\boldsymbol{B}_ω^{\mathscr{F}}$ and when $\boldsymbol{B}_ω^{\mathscr{F}}$ has the identity, is isomorphic to the bicyclic semigroup or the countable semigroup of matrix units.

math.GR↗

arXiv2021

Polycyclic extensions of semigroups

In the paper we introduce a notion of the Bruck-Reilly $λ$-polycyclic extension of a monoid $S$ with a homomorphism $θ$ which is an analogue of the Bruck-Reilly extension of a monoid $S$. We describe idempotens of the semigroup $\left(\mathscr{P}_λ(θ,S),*\right)$ and Green's relations on $\left(\mathscr{P}_λ(θ,S),*\right)$. It is proved that $\left(\mathscr{P}_λ(θ,S),*\right)$ is a $0$-simple semigroup for any semigroup $S$. We find necessary and sufficient conditions on a monoid $S$ and a homomorphism $θ$ under which the semigroup $\left(\mathscr{P}_λ(θ,S),*\right)$ is regular, inverse, $0$-bisimple, combinatorial, congruence free, or inverse 0-E-unitary. Also we study topologizations of the semigroup $\left(\mathscr{P}_λ(θ,S),*\right)$.

math.GR↗

arXiv2021

On the semigroup generating by extended bicyclic semigroup and a $ω$-closed family

The algebraic extension $\boldsymbol{B}_{\mathbb{Z}}^{\mathscr{F}}$ of the extended bicyclic semigroup for an arbitrary $ω$-closed family $\mathscr{F}$ subsets of $ω$ is introduced. It is proven that $\boldsymbol{B}_{\mathbb{Z}}^{\mathscr{F}}$ is a combinatorial inverse semigroup. Green's relations, the natural partial order on the semigroup $\boldsymbol{B}_{\mathbb{Z}}^{\mathscr{F}}$ and its set of idempotents are described. The criteria of simplicity, $0$-simplicity, bisimplicity, $0$-bisimplicity of the semigroup $\boldsymbol{B}_{\mathbb{Z}}^{\mathscr{F}}$ and the criterion for $\boldsymbol{B}_{\mathbb{Z}}^{\mathscr{F}}$ to be isomorphic to the extended bicyclic semigroup or the countable semigroup of matrix units are derived. It is proved that in the case when the family $\mathscr{F}$ consists of all singletons of $ω$ and the empty set, the semigroup $\boldsymbol{B}_{\mathbb{Z}}^{\mathscr{F}}$ is isomorphic to the Brandt $λ$-extension of the semilattice $(ω,\min)$.

math.GR↗

arXiv2021

On the monoid of cofinite partial isometries of $\mathbb{N}$ with a bounded finite noise

In the paper we study algebraic properties of the monoid $\mathbf{I}\mathbb{N}_{\infty}^{\boldsymbol{g}[j]}$ of cofinite partial isometries of the set of positive integers $\mathbb{N}$ with the bounded finite noise $j$. For the monoids $\mathbf{I}\mathbb{N}_{\infty}^{\boldsymbol{g}[j]}$ we prove counterparts of some classical results of Eberhart and Selden describing the closure of the bicyclic semigroup in a locally compact topological inverse semigroup. In particular we show that for any positive integer $j$ every Hausdorff shift-continuous topology $τ$ on $\mathbf{I}\mathbb{N}_{\infty}^{\boldsymbol{g}[j]}$ is discrete and if $\mathbf{I}\mathbb{N}_{\infty}^{\boldsymbol{g}[j]}$ is a proper dense subsemigroup of a Hausdorff semitopological semigroup $S$, then $S\setminus \mathbf{I}\mathbb{N}_{\infty}^{\boldsymbol{g}[j]}$ is a closed ideal of $S$, and moreover if $S$ is a topological inverse semigroup then $S\setminus \mathbf{I}\mathbb{N}_{\infty}^{\boldsymbol{g}[j]}$ is a topological group. Also we describe the algebraic and topological structure of the closure of the monoid $\mathbf{I}\mathbb{N}_{\infty}^{\boldsymbol{g}[j]}$ in a locally compact topological inverse semigroup.

math.GN↗

arXiv2021

On group congruences on the semigroup $\boldsymbol{B}_ω^{\mathscr{F}}$ and its homomorphic retracts in the case when a family $\mathscr{F}$ consists of inductive non-empty subsets of $ω$

We study group congruences on the semigroup $\boldsymbol{B}_ω^{\mathscr{F}}$ and its homomorphic retracts in the case when an $ω$-closed family $\mathscr{F}$ which consists of inductive non-empty subsets of $ω$. It is proven that a congruence $\mathfrak{C}$ on $\boldsymbol{B}_ω^{\mathscr{F}}$ is a group congruence if and only if its restriction on a subsemigroup of $\boldsymbol{B}_ω^{\mathscr{F}}$, which is isomorphic to the bicyclic semigroup, is not the identity relation. Also, all non-trivial homomorphic retracts and isomorphisms of the semigroup $\boldsymbol{B}_ω^{\mathscr{F}}$ are described.

math.GR↗

arXiv2021

On the monoid of cofinite partial isometries of $\mathbb{N}$ with the usual metric

In the paper we show that the monoid $\mathbf{I}\mathbb{N}_{\infty}$ of all partial cofinite isometries of positive integers does not embed isomorphically into the monoid $\mathbf{ID}_{\infty}$ of all partial cofinite isometries of integers. Moreover, every non-annihilating homomorphism $\mathfrak{h}\colon \mathbf{I}\mathbb{N}_{\infty}\to\mathbf{ID}_{\infty}$ has the following property: the image $(\mathbf{I}\mathbb{N}_{\infty})\mathfrak{h}$ is isomorphic either to the two-element cyclic group $\mathbb{Z}_2$ or to the additive group of integers $\mathbb{Z}(+)$. Also we prove that the monoid $\mathbf{I}\mathbb{N}_{\infty}$ is not finitely generated, and, moreover, monoid $\mathbf{I}\mathbb{N}_{\infty}$ does not contain a minimal generating set.

math.GR↗

arXiv2020

The monoid of monotone injective partial selfmaps of the poset $(\mathbb{N}^{3},\leqslant)$ with cofinite domains and images

Let $n$ be a positive integer $\geqslant 2$ and $\mathbb{N}^n_{\leqslant}$ be the $n$-th power of positive integers with the product order of the usual order on $\mathbb{N}$. In the paper we study the semigroup of injective partial monotone selfmaps of $\mathbb{N}^n_{\leqslant}$ with cofinite domains and images. We show that the group of units $H(\mathbb{I})$ of the semigroup $\mathscr{P\!O}\!_{\infty}(\mathbb{N}^n_{\leqslant})$ is isomorphic to the group $\mathscr{S}_n$ of permutations of an $n$-element set, and describe the subsemigroup of idempotents of $\mathscr{P\!O}\!_{\infty}(\mathbb{N}^n_{\leqslant})$. Also in the case $n=3$ we describe the property of elements of the semigroup $\mathscr{P\!O}\!_{\infty}(\mathbb{N}^3_{\leqslant})$ as partial bijections of the poset $\mathbb{N}^3_{\leqslant}$ and Green's relations on the semigroup $\mathscr{P\!O}\!_{\infty}(\mathbb{N}^3_{\leqslant})$. In particular we show that $\mathscr{D}=\mathscr{J}$ in $\mathscr{P\!O}\!_{\infty}(\mathbb{N}^3_{\leqslant})$.

math.GR↗