arXiv subjects

Pavlo Khylynskyi

Publications and source records attributed to Pavlo Khylynskyi.

3 recordsLinked to original sources

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↗

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 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↗