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Oleksandra Sobol

Publications and source records attributed to Oleksandra Sobol.

4 recordsLinked to original sources
arXiv2019

Extensions of semigroups by symmetric inverse semigroups of a bounded finite rank

We study the semigroup extension $\mathscr{I}_λ^n(S)$ of a semigroup $S$ by symmetric inverse semigroups of a bounded finite rank. We describe idempotents and regular elements of the semigroups $\mathscr{I}_λ^n(S)$ and $\overline{\mathscr{I}_λ^n}(S)$ show that the semigroup $\mathscr{I}_λ^n(S)$ ($\overline{\mathscr{I}_λ^n}(S)$) is regular, orthodox, inverse or stable if and only if so is $S$. Green's relations are described on the semigroup $\mathscr{I}_λ^n(S)$ for an arbitrary monoid $S$. We introduce the conception of a semigroup with strongly tight ideal series, and proved that for any infinite cardinal $λ$ and any positive integer $n$ the semigroup $\mathscr{I}_λ^n(S)$ has a strongly tight ideal series provides so has $S$. At the finish we show that for every compact Hausdorff semitopological monoid $(S,τ_S)$ there exists a unique its compact topological extension $\left(\mathscr{I}_λ^n(S),τ_{\mathscr{I}}^\mathbf{c}\right)$ in the class of Haudorff semitopological semigroups.

math.GR↗
arXiv2017

On feebly compact shift-continuous topologies on the semilattice $\exp_nλ$

We study feebly compact topologies $τ$ on the semilattice $\left(\exp_nλ,\cap\right)$ such that $\left(\exp_nλ,τ\right)$ is a semitopological semilattice and prove that for any shift-continuous $T_1$-topology $τ$ on $\exp_nλ$ the following conditions are equivalent: $(i)$~$τ$ is countably pracompact; $(ii)$ $τ$ is feebly compact; $(iii)$ $τ$ is $d$-feebly compact; $(iv)$ $\left(\exp_nλ,τ\right)$ is an $H$-closed space.

math.GN↗
arXiv2017

On feebly compact topologies on the semilattice $\exp_nλ$

We study feebly compact topologies $τ$ on the semilattice $\left(\exp_nλ,\cap\right)$ such that $\left(\exp_nλ,τ\right)$ is a semitopological semilattice. All compact semilattice $T_1$-topologies on $\exp_nλ$ are described. Also we prove that for an arbitrary positive integer $n$ and an arbitrary infinite cardinal $λ$ for a $T_1$-topology $τ$ on $\exp_nλ$ the following conditions are equivalent: $(i)$ $\left(\exp_nλ,τ\right)$ is a compact topological semilattice; $(ii)$ $\left(\exp_nλ,τ\right)$ is a countably compact topological semilattice; $(iii)$ $\left(\exp_nλ,τ\right)$ is a feebly compact topological semilattice; $(iv)$ $\left(\exp_nλ,τ\right)$ is a compact semitopological semilattice; $(v)$ $\left(\exp_nλ,τ\right)$ is a countably compact semitopological semilattice. We construct a countably pracompact $H$-closed quasiregular non-semiregular topology $τ_{\operatorname{\textsf{fc}}}^2$ such that $\left(\exp_2λ,τ_{\operatorname{\textsf{fc}}}^2\right)$ is a semitopological semilattice with discontinuous semilattice operation and prove that for an arbitrary positive integer $n$ and an arbitrary infinite cardinal $λ$ every $T_1$-semiregular feebly compact semitopological semilattice $\exp_nλ$ is a compact topological semilattice.

math.GN↗