arXiv subjects
Publications and source records attributed to Serhii Bardyla.
In this paper we characterise graph inverse semigroups which admit only discrete locally compact semigroup topology. This characterization provides a complete answer on the question of Z. Mesyan, J. D. Mitchell, M. Morayne and Y. H. Péresse posed in their paper "Topological graph inverse semigroups".
A topology $τ$ on a monoid $S$ is called {\em shift-continuous} if for every $a,b\in S$ the two-sided shift $S\to S$, $x\mapsto axb$, is continuous. For every ordinal $α\le ω$, we describe all shift-continuous locally compact Hausdorff topologies on the $α$-bicyclic monoid $\mathcal{B}_α$. More precisely, we prove that the lattice of shift-continuous locally compact Hausdorff topologies on $\mathcal{B}_α$ is anti-isomorphic to the segment of $[1,α]$ of ordinals, endowed with the natural well-order. Also we prove that for each ordinal $α$ the $α+1$-bicyclic monoid $\mathcal{B}_{α+1}$ is isomorphic to the Bruck extension of the $α$-bicyclic monoid $\mathcal{B}_α$.
We find (completeness type) conditions on topological semilattices $X,Y$ guaranteeing that each continuous homomorphism $h:X\to Y$ has closed image $h(X)$ in $Y$.
We present a complete classification of Hausdorff locally compact polycyclic monoids up to a topological isomorphism. A {\em polycyclic monoid} is an inverse monoid with zero, generated by a subset $Λ$ such that $xx^{-1}=1$ for any $x\inΛ$ and $xy^{-1}=0$ for any distinct $x,y\inΛ$. We prove that any non-discrete Hausdorff locally compact topology with continuous shifts on a polycyclic monoid $M$ coincides with the topology of one-point compactification of the discrete space $M\setminus\{0\}$.
In this paper we consider a semitopological $α$-bicyclic monoid $\mathcal{B}_α$ and prove that it is algebraically isomorphic to a semigroup of all order isomorphisms between the principal upper sets of the ordinal $ω^α$. We prove that for every ordinal $α$ for every $(a,b)\in \mathcal{B_α}$ if either $a$ or $b$ is a non-limit ordinal then $(a,b)$ is an isolated point in $\mathcal{B}_α$. We show that for every ordinal $α<ω+1$ every locally compact semigroup topology on $\mathcal{B}_α$ is discrete. However, we construct an example of a non-discrete locally compact topology $τ_{lc}$ on $\mathcal{B}_{ω+1}$ such that $(\mathcal{B}_{ω+1},τ_{lc})$ is a topological inverse semigroup. This example shows that there is a gap in \cite[Theorem~2.9]{Hogan-1984}, where is stated that for every ordinal $α$ there is only discrete locally compact inverse semigroup topology on $\mathcal{B_α}$.
We give sufficient conditions when a topological inverse $λ$-polycyclic monoid $P_λ$ is absolutely $H$-closed in the class of topological inverse semigroups. Also, for every infinite cardinal $λ$ we construct the coarsest semigroup inverse topology $τ_{mi}$ on $P_λ$ and give an example of a topological inverse monoid S which contains the polycyclic monoid $P_2$ as a dense discrete subsemigroup.
We study algebraic structure of the $λ$-polycyclic monoid $P_λ$ and its topologizations. We show that the $λ$-polycyclic monoid for an infinite cardinal $λ\geqslant 2$ has similar algebraic properties so has the polycyclic monoid $P_n$ with finitely many $n\geqslant 2$ generators. In particular we prove that for every infinite cardinal $λ$ the polycyclic monoid $P_λ$ is a congruence-free combinatorial $0$-bisimple $0$-$E$-unitary inverse semigroup. Also we show that every non-zero element $x$ is an isolated point in $(P_λ,τ)$ for every Hausdorff topology $τ$ on $P_λ$, such that $(P_λ,τ)$ is a semitopological semigroup, and every locally compact Hausdorff semigroup topology on $P_λ$ is discrete. The last statement extends results of the paper [33] obtaining for topological inverse graph semigroups. We describe all feebly compact topologies $τ$ on $P_λ$ such that $\left(P_λ,τ\right)$ is a semitopological semigroup and its Bohr compactification as a topological semigroup. We prove that for every cardinal $λ\geqslant 2$ any continuous homomorphism from a topological semigroup $P_λ$ into an arbitrary countably compact topological semigroup is annihilating and there exists no a Hausdorff feebly compact topological semigroup which contains $P_λ$ as a dense subsemigroup.