arXiv subjects

Mykola Mykhalenych

Publications and source records attributed to Mykola Mykhalenych.

4 recordsLinked to original sources

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 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↗

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

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↗