Let $\mathcal{C}_\mathbb{N}$ be a monoid which is generated by the partial shift $\alpha\colon n\mapsto n+1$ of the set of positive integers $\mathbb{N}$ and its inverse partial shift $\beta\colon n+1\mapsto n$. 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 $\mathcal{C}_\mathbb{N}$ 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.

This is a joint work with Pavlo Khylynskyi.

We describe minimal topologies in some class of semigroup topologies on an infinite semigroup of matrix units.

This is a joint work with Pavlo Khylynskyi.

We shall discuss algebraic and topological properties of McAlister semigroups. In particular, we describe inner structure, group of automorphisms and topologization of McAlister semigroups. Our results imply that a free inverse semigroup over a singleton admits only the discrete Hausdorff semigroup topology. Also, we show that a locally compact topological free inverse semigroup over a singleton with adjoined zero is either compact or discrete.

Given families $\{X_{i}: i\in I\}$, $\{Y_{i}: i\in I\}$ of subspaces of topological spaces $X$ and $Y$ respectively, we say that $(X, \{X_{i}: i\in I\})$ is $M$-equivalent to $(Y, \{Y_{i}: i\in I\})$ if there exists $h\colon F(X)\to F(Y)$ of the free topological groups, such that $h(G(X_{i}))=G(Y_{i})$ for all $i\in I$. We consider a methods for constructing $M$-equivalent families, $M$-invariant properties and full $M$-classification for some special cases of the bundles.

We shall discuss algebraic and topological properties of McAlister semigroups. In particular, we describe inner structure, group of automorphisms and topologization of McAlister semigroups. Our results imply that a free inverse semigroup over a singleton admits only the discrete Hausdorff semigroup topology. Also, we show that a locally compact topological free inverse semigroup over a singleton with adjoined zero is either compact or discrete.

We shall discuss algebraic and topological properties of McAlister semigroups. In particular, we describe inner structure, group of automorphisms and topologization of McAlister semigroups. Our results imply that a free inverse semigroup over a singleton admits only the discrete Hausdorff semigroup topology. Also, we show that a locally compact topological free inverse semigroup over a singleton with adjoined zero is either compact or discrete.

We study shift-continuous topologies on the semigroup $\boldsymbol{B}_{\omega}^{\mathcal{F}_n}$. In particular we prove that for any shift-continuous $T_1$-topology $\tau$ on the semigroup $\boldsymbol{B}_{\omega}^{\mathcal{F}_n}$ every non-zero element of $\boldsymbol{B}_{\omega}^{\mathcal{F}_n}$ is an isolated point of $(\boldsymbol{B}_{\omega}^{\mathcal{F}_n},\tau)$, $\boldsymbol{B}_{\omega}^{\mathcal{F}_n}$ admits the unique compact shift-continuous $T_1$-topology, and every $\omega_{\mathfrak{d}}$-compact shift-continuous $T_1$-topology is compact. We describe the closure of the semigroup $\boldsymbol{B}_{\omega}^{\mathcal{F}_n}$ in a Hausdorff semitopological semigroup and prove the criterium when a topological inverse semigroup $\boldsymbol{B}_{\omega}^{\mathcal{F}_n}$ is $H$-closed in the class of Hausdorff topological semigroups.
This is a joint work with Olha Popadiuk.

We suggest a topologization of a semigroup using so called remote bases, which will allow us to characterize injectively $T_1S$-closed commutative semigroups as bounded nonsingular semigroups with finite Clifford part.
More details can be found in the preprint https://arxiv.org/abs/2208.00074.