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Oleg Gutik

Publications and source records attributed to Oleg Gutik.

At least 19 recordsLinked to original sources
arXiv2026

On injective endomorphisms of the semigroup $\boldsymbol{B}_ω^{\mathscr{F}^4}$ with a four-element family $\mathscr{F}^4$ of inductive non-empty subsets of $ω$

We describe injective endomorphisms of the semigroup $\boldsymbol{B}_ω^{\mathscr{F}^4}$ with a four-element family $\mathscr{F}^4$ of inductive non-empty subsets of $ω$. In particular we proved that every injective monoid endomorphism of the semigroup $\boldsymbol{B}_ω^{\mathscr{F}^4}$ is the identity transformation. Also we describe all (not necessary monoid) injective endomorphism of $\boldsymbol{B}_ω^{\mathscr{F}^4}$.

math.GR↗
arXiv2026

On semigroups which admit only discrete left-continuous Hausdorff topology

We give the sufficient condition when every left-continuous (right-continuous) Hausdorff topology on a semigroup $S$ is discrete. We construct a submonoid $\mathscr{C}_{+}(a,b)$ (resp., $\mathscr{C}_{-}(a,b)$) of the bicyclic monoid which contains a family $\{S_α\colon α\in\mathfrak{c}\}$ of continuum many subsemigroups with the following properties: $(i)$ every left-continuous (resp., right-continuous) Hausdorff topology on $S_α$ is discrete; $(ii)$ every semigroup $S_α$ admits a non-discrete right-continuous (resp., left-continuous) Hausdorff topology which is not left-continuous (resp., right-continuous); $(iii)$ every semigroup $S_α$ isomorphically embeds into a Hausdorff compact topological semigroup. Also we construct a submonoid $\mathscr{C}_{\mathbb{Z}}^+$ (resp., $\mathscr{C}_{\mathbb{Z}}^-$) of the extended bicyclic semigroup which contains a family $\{S_α\colon α\in\mathfrak{c}\}$ of continuum many subsemigroups with the above described properties.

math.GR↗
arXiv2026

On topologization of the extended bicyclic semigroup

Non-discrete semigroup $T_1$-topologies on the extended bicyclic semigroup $\mathscr{C}_\mathbb{Z}$ are constructed. Also, we present topological conditions, when a semigroup (shift-continuous) $T_1$-topology on $\mathscr{C}_\mathbb{Z}$ is discrete.

math.GR↗
arXiv2026

On non-topologizable semigroups

We find anti-isomorphic submonoids $\mathscr{C}_{+}(a,b)$ and $\mathscr{C}_{-}(a,b)$ of the bicyclic monoid $\mathscr{C}(a,b)$ with the following properties: every Hausdorff left-continuous (right-continuous) topology on $\mathscr{C}_{+}(a,b)$ ($\mathscr{C}_{-}(a,b)$) is discrete and there exists a compact Hausdorff topological monoid $S$ which contains $\mathscr{C}_{+}(a,b)$ ($\mathscr{C}_{-}(a,b)$) as a submonoid. Also, we construct a non-discrete right-continuous (left-continuous) topology $τ_p^+$ ($τ_p^-$) on the semigroup $\mathscr{C}_{+}(a,b)$ ($\mathscr{C}_{-}(a,b)$) which is not left-continuous (right-continuous).

math.GR↗
arXiv2026

On topologization of subsemigroups of the bicyclic monoid

We show that if a subsemigroup $S$ of the bicyclic monoid ${\mathscr{C}}(p,q)$ contains infinitely many idempotents then $S$ admits only the discrete Hausdorff shift-continuous topology. Also we proof that every right-continuous (left-continuous\emph) Hausdorff Baire topology on the semigroup $\mathscr{C}_+(a,b)$ $(\mathscr{C}_-(a,b))$ is discrete and the same statement holds for the bicyclic monoid.

math.GR↗
arXiv2025

On injective endomorphisms of the semigroup $\boldsymbol{B}_ω^{\mathscr{F}^3}$ with a three-element family $\mathscr{F}^3$ of inductive non-empty subsets of $ω$

We describe injective endomorphisms of the semigroup $\boldsymbol{B}_ω^{\mathscr{F}^3}$ with a three-element family $\mathscr{F}^3$ of inductive non-empty subsets of $ω$. In particular we find endomorphisms $\varpi_3$ and $λ$ of $\boldsymbol{B}_ω^{\mathscr{F}^3}$ such that for every injective endomorphism $\varepsilon$ of the semigroup $\boldsymbol{B}_ω^{\mathscr{F}^3}$ there exists an injective endomorphism $ι\in\left\langleλ,\varpi_3\right\rangle$ such that $\varepsilon=α_{[k]}\circι$ for some positive integer $k$, where $α_{[k]}$ is an injective monoid endomorphism of $\boldsymbol{B}_ω^{\mathscr{F}^3}$.

math.GR↗
arXiv2025

On injective endomorphisms of the semigroup $\boldsymbol{B}_{\mathbb{Z}}^{\mathscr{F}^2}$ with the two-element family $\mathscr{F}^2$ of inductive nonempty subsets of $ω$

We describe injective endomorphisms of the semigroup $\boldsymbol{B}_{Z\mathbb{}}^{\mathscr{F}^2}$ with the two-element family $\mathscr{F}^2$ of inductive nonempty subsets of $ω$. In particular we show that every injective endomorphism $\mathfrak{e}$ of $\boldsymbol{B}_{Z\mathbb{}}^{\mathscr{F}^2}$ is presented in the form $\mathfrak{e}=\mathfrak{e}_0\mathfrak{a}$, where $\mathfrak{e}_0$ is an injective $(0,0,[0))$-endomorphism of $\boldsymbol{B}_{\mathbb{Z}}^{\mathscr{F}^2}$ and $\mathfrak{a}$ is an automorphism $\mathfrak{a}$ of $\boldsymbol{B}_{\mathbb{Z}}^{\mathscr{F}^2}$. Also we describe all injective $(0,0,[0))$-endomorphisms $\mathfrak{e}_0$ of $\boldsymbol{B}_{\mathbb{Z}}^{\mathscr{F}^2}$, i.e., such that $(0,0,[0))\mathfrak{e}_0=(0,0,[0))$.

math.GR↗
arXiv2025

On the semigroup of endomorphisms of the semigroup $\boldsymbol{B}_ω^{\mathscr{F}^2}$ with the two-element family $\mathscr{F}^2$ of inductive nonempty subsets of $ω$

We study the semigroup $\overline{\boldsymbol{End}}(\boldsymbol{B}_ω^{\mathscr{F}^2})$ of all endomorphisms of the bicyclic extension $\boldsymbol{B}_ω^{\mathscr{F}^2}$ with the two-element family $\mathscr{F}^2$ of inductive nonempty subsets of $ω$. The submonoid $\left\langle\varpi\right\rangle^1$ of $\overline{\boldsymbol{End}}(\boldsymbol{B}_ω^{\mathscr{F}^2})$ with the property that every element of the semigroup $\overline{\boldsymbol{End}}(\boldsymbol{B}_ω^{\mathscr{F}^2})$ has the unique representation as the product of the monoid endomorphism of $\boldsymbol{B}_ω^{\mathscr{F}^2}$ and the element of $\left\langle\varpi\right\rangle^1$ is constructed.

math.GR↗
arXiv2025

On the semigroup of monoid endomorphisms of the semigroup $\mathscr{C}_{+}(a,b)$

Let $\mathscr{C}_{+}(a,b)$ be the submonoid of the bicyclic monoid which is studied in \cite{Makanjuola-Umar=1997}. We describe monoid endomorphisms of the semigroup $\mathscr{C}_{+}(a,b)$ which are generated by the family of all congruences of the bicyclic monoid and all injective monoid endomorphisms of $\mathscr{C}_{+}(a,b)$.

math.GR↗
arXiv2024

On compact topologies on the semigroup of finite partial order isomorphisms of a bounded rank of an infinite linear ordered set

We study topologization of the semigroup $\mathscr{O\!\!I}\!_n(L)$ of finite partial order isomorphisms of a bounded rank of an infinite linear ordered set $(L,\leqslant)$. In particular we show that every $T_1$ left-topological (right-topological) semigroup $\mathscr{O\!\!I}\!_n(L)$ is a completely Hausdorff, Urysohn, totally separated, scattered space. We prove that on the semigroup $\mathscr{O\!\!I}\!_n(L)$ admits a unique Hausdorff countably compact (pseudocompact) shift-continuous topology which is compact, and the Bohr compactification of a Hausdorff topological semigroup $\mathscr{O\!\!I}\!_n(L)$ is the trivial semigroup.

math.GR↗
arXiv2024

The semigroup of finite partial order isomorphisms of a bounded rank of an infinite linear ordered set

We study algebraic properties of the semigroup $\mathscr{O\!\!I\!}_n(L)$ of finite partial order isomorphisms of the rank $\leq n$ of an infinite linearly ordered set $(L,\leqslant)$. In particular we describe its idempotents, the natural partial order and Green's relations on $\mathscr{O\!\!I\!}_n(L)$. It is proved that the semigroup $\mathscr{O\!\!I\!}_n(L)$ is stable and it contains tight ideal series. Moreover, we show that the semigroup $\mathscr{O\!\!I\!}_n(L)$ admits only Rees' congruences and every its homomorphic image is a semigroup with tight ideal series.

math.GR↗
arXiv2024

On semitopological simple inverse $ω$-semigroups with compact maximal subgroups

We describe the structure of ($0$-)simple inverse Hausdorff semitopological $ω$-semigroups with compact maximal subgroups. In particular, we show that if $S$ is a simple inverse Hausdorff semitopological $ω$-semigroup with compact maximal subgroups, then $S$ is topologically isomorphic to the Bruck--Reilly extension $\left(\textbf{BR}(T,θ),τ_{\textbf{BR}}^{\oplus}\right)$ of a finite semilattice $T=\left[E;G_α,φ_{α,β}\right]$ of compact groups $G_α$ in the class of topological inverse semigroups, where $τ_{\textbf{BR}}^{\oplus}$ is the sum direct topology on $\textbf{BR}(T,θ)$. Also we prove that every Hausdorff locally compact shift-continuous topology on the simple inverse Hausdorff semitopological $ω$-semigroups with compact maximal subgroups with adjoined zero is either compact or the zero is an isolated point.

math.GR↗
arXiv2024

On the semigroup of monoid endomorphisms of the semigroup $\boldsymbol{B}_ω^{\mathscr{F}}$ with a two-element family $\mathscr{F}$ of inductive nonempty subsets of $ω$

We study the semigroup of non-injective monoid endomorphisms of the semigroup $\boldsymbol{B}_ω^{\mathscr{F}}$ with a two-elements family $\mathscr{F}$ of inductive nonempty subsets of $ω$. We describe the structure of elements of the semigroup $\boldsymbol{End}^*_0(\boldsymbol{B}_ω^{\mathscr{F}})$ of non-injective monoid endomorphisms of the semigroup $\boldsymbol{B}_ω^{\mathscr{F}}$. In particular we show that its subsemigroup $\boldsymbol{End}^*(\boldsymbol{B}_ω^{\mathscr{F}})$ of non-injective non-annihilating monoid endomorphisms of the semigroup $\boldsymbol{B}_ω^{\mathscr{F}}$ is isomorphic to the direct product of the two-element left-zero semigroup and the multiplicative semigroup of positive integers and describe Green's relations on $\boldsymbol{End}^*(\boldsymbol{B}_ω^{\mathscr{F}})$.

math.GR↗
arXiv2024

On the semigroup of injective monoid endomorphisms of the monoid $\boldsymbol{B}_ω^{\mathscr{F}^3}$ with a three element family $\mathscr{F}^3$ of inductive nonempty subsets of $ω$

We describe injective monoid endomorphisms of the semigroup $\boldsymbol{B}_ω^{\mathscr{F}^3}$ with a three element family $\mathscr{F}^3$ of inductive nonempty subsets of $ω$. Also, we show that the monoid $\boldsymbol{End}_*^1(\boldsymbol{B}_ω^{\mathscr{F}})$ of all injective endomorphisms of the semigroup $\boldsymbol{B}_ω^{\mathscr{F}^3}$ is isomorphic to the multiplicative semigroup of positive integers.

math.GR↗
arXiv2024

On topologization of the bicyclic monoid

We construct two non-discrete inverse semigroup $T_1$-topologies and a compact inverse shift-continuous $T_1$-topology on the bicyclic monoid ${\mathscr{C}}(p,q)$. Also we give conditions on a $T_1$-topology $τ$ on ${\mathscr{C}}(p,q)$ to be discrete. In particular, we show that if $τ$ is an inverse semigroup $T_1$-topology on ${\mathscr{C}}(p,q)$ which satisfies one of the following conditions: $τ$ is Baire, $τ$ is quasi-regular or $τ$ is semiregular, then $τ$ is discrete.

math.GR↗
arXiv2024

On locally compact shift continuous topologies on the semigroup $\boldsymbol{B}_{[0,\infty)}$ with an adjoined compact ideal

Let $\boldsymbol{B}_{[0,\infty)}$ be the semigroup which is defined in the Ahre paper \cite{Ahre=1981}. The semigroup $\boldsymbol{B}_{[0,\infty)}$ with the induced usual topology $τ_u$ from $\mathbb{R}^2$, with the topology $τ_L$ which is generated by the natural partial order on $\boldsymbol{B}_{[0,\infty)}$, and the discrete topology are denoted by $\boldsymbol{B}^1_{[0,\infty)}$, $\boldsymbol{B}^2_{[0,\infty)}$, and $\boldsymbol{B}^{\mathfrak{d}}_{[0,\infty)}$, respectively. We show that if $S_1^I$ ($S_2^I$) is a Hausdorff locally compact semitopological semigroup $\boldsymbol{B}^1_{[0,\infty)}$ ($\boldsymbol{B}^2_{[0,\infty)}$) with an adjoined compact ideal $I$ then either $I$ is an open subset of $S_1^I$ ($S_2^I$) or the semigroup $S_1^I$ ($S_2^I$) is compact. Also, we proved that if $S_{\mathfrak{d}}^I$ is a Hausdorff locally compact semitopological semigroup $\boldsymbol{B}^{\mathfrak{d}}_{[0,\infty)}$ with an adjoined compact ideal $I$ then $I$ is an open subset of $S_{\mathfrak{d}}^I$.

math.GR↗
arXiv2024

On some generalization of the bicyclic semigroup: the topological version

We show that every Hausdorff Baire topology $τ$ on $\mathcal{C}=\langle a,b\mid a^2b=a, ab^2=b\rangle$ such that $(\mathcal{C},τ)$ is a semitopological semigroup is discrete and we construct a nondiscrete Hausdorff semigroup topology on $\mathcal{C}$. We also discuss the closure of a semigroup $\mathcal{C}$ in a semitopological semigroup and prove that $\mathcal{C}$ does not embed into a topological semigroup with the countably compact square.

math.GR↗