arXiv subjects

Kateryna Maksymyk

Publications and source records attributed to Kateryna Maksymyk.

6 recordsLinked to original sources

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↗

arXiv2020

On a semitopological extended bicyclic semigroup with adjoined zero

In the paper it is shown that every Hausdorff locally compact semigroup topology on the extended bicyclic semigroup with adjoined zero $\mathscr{C}_{\mathbb{Z}}^{\mathbf{0}}$ is discrete, but on $\mathscr{C}_{\mathbb{Z}}^{\mathbf{0}}$ there exist $\mathfrak{c}$ many different Hausdorff locally compact shift-continuous topologies. Also, it is constructed on $\mathscr{C}_{\mathbb{Z}}^{\mathbf{0}}$ the unique minimal shift continuous topology and the unique minimal inverse semigroup topology.

math.GR↗

arXiv2020

On locally compact groups with zero

We study algebraic properties on a group G such that if the discrete group G has these properties then every locally compact shift continuous topology on G with adjoined zero is either compact, or discrete. We introduce electorally flexible and electorally stable groups and establish their properties. In particular, we prove that every group with an infinite cyclic subgroup of an infinite index and every uncountable commutative group are electorally flexible, and show that every countable locally finite group is electorally stable. The main result of the paper is the following: if G is a discrete electorally flexible group then every Hausdorff locally compact shift-continuous topology on G with adjoined zero is either compact, or discrete. Also, we construct a non-discrete non-compact Hausdorff locally compact shift-continuous topology on any discrete virtually cyclic group (and hence on a electorally stable group) G with adjoined zero.

math.GR↗

arXiv2018

On variants of the extended bicyclic semigroup

In the paper we describe the group $\mathbf{Aut}\left(\mathscr{C}_{\mathbb{Z}}\right)$ of automorphisms of the extended bicyclic semigroup $\mathscr{C}_{\mathbb{Z}}$ and study the variants $\mathscr{C}_{\mathbb{Z}}^{m,n}$ of the extended bicycle semigroup $\mathscr{C}_{\mathbb{Z}}$, where $m,n\in\mathbb{Z}$. In particular, we prove that $\mathbf{Aut}\left(\mathscr{C}_{\mathbb{Z}}\right)$ is isomorphic to the additive group of integers, the extended bicyclic semigroup $\mathscr{C}_{\mathbb{Z}}$ and every its variant are not finitely generated, and describe the subset of idempotents $E(\mathscr{C}_{\mathbb{Z}}^{m,n})$ and Green's relations on the semigroup $\mathscr{C}_{\mathbb{Z}}^{m,n}$. Also we show that $E(\mathscr{C}_{\mathbb{Z}}^{m,n})$ is an $ω$-chain and any two variants of the extended bicyclic semigroup $\mathscr{C}_{\mathbb{Z}}$ are isomorphic. At the end we discuss shift-continuous Hausdorff topologies on the variant $\mathscr{C}_{\mathbb{Z}}^{0,0}$. In particular, we prove that if $τ$ is a Hausdorff shift-continuous topology on $\mathscr{C}_{\mathbb{Z}}^{0,0}$ then each of inequalities $a>0$ or $b>0$ implies that $(a,b)$ is an isolated point of $\big(\mathscr{C}_{\mathbb{Z}}^{0,0},τ\big)$ and construct an example of a Hausdorff semigroup topology $τ^*$ on the semigroup $\mathscr{C}_{\mathbb{Z}}^{0,0}$ such that all its points with $ab\leqslant 0$ and $a+b\leqslant 0$ are not isolated in $\big(\mathscr{C}_{\mathbb{Z}}^{0,0},τ^*\big)$.

math.GR↗

arXiv2017

On semitopological bicyclic extensions of linearly ordered groups

For a linearly ordered group $G$ let us define a subset $A\subseteq G$ to be a \emph{shift-set} if for any $x,y,z\in A$ with $y < x$ we get $x\cdot y^{-1}\cdot z\in A$. We describe the natural partial order and solutions of equations on the semigroup $\mathscr{B}(A)$ of shifts of positive cones of $A$. We study topologizations of the semigroup $\mathscr{B}(A)$. In particular, we show that for an arbitrary countable linearly ordered group $G$ and a non-empty shift-set $A$ of $G$ every Baire shift-continuous $T_1$-topology $τ$ on $\mathscr{B}(A)$ is discrete. Also we prove that for an arbitrary linearly non-densely ordered group $G$ and a non-empty shift-set $A$ of $G$, every shift-continuous Hausdorff topology $τ$ on the semigroup $\mathscr{B}(A)$ is discrete, and hence $\left(\mathscr{B}(A),τ\right)$ is a discrete subspace of any Hausdorff semitopological semigroup which contains $\mathscr{B}(A)$ as a subsemigroup.

math.GR↗

arXiv2017

On semitopological interassociates of the bicyclic monoid

Semitopological interassociates $\mathscr{C}_{m,n}$ of the bicyclic semigroup $\mathscr{C}(p,q)$ are studied. In particular, we show that for arbitrary non-negative integers $m$, $n$ and every Hausdorff topology $τ$ on $\mathscr{C}_{m,n}$ such that $\left(\mathscr{C}_{m,n},τ\right)$ is a semitopological semigroup, is discrete. Also, we prove that if an interassociate of the bicyclic monoid $\mathscr{C}_{m,n}$ is a dense subsemigroup of a Hausdorff semitopological semigroup $(S,\cdot)$ and $I=S\setminus\mathscr{C}_{m,n}\neq\varnothing$ then $I$ is a two-sided ideal of the semigroup $S$ and show that for arbitrary non-negative integers $m$, $n$, any Hausdorff locally compact semitopological semigroup $\mathscr{C}_{m,n}^0=\mathscr{C}_{m,n}\sqcup\{0\}$ is either discrete or compact.

math.GR↗