We extend results of the papers [Carl Eberhart and John Selden, On the Closure of the Bicyclic Semigroup, Transactions of the American Mathematical Society, Vol. 144 (Oct., 1969), pp. 115-126 ] and [M. O. Bertman and T. T. West, Conditionally Compact Bicyclic Semitopological Semigroups, Proceedings of the Royal Irish Academy. Section A: Mathematical and Physical Sciences Vol. 76 (1976), pp. 219-226] for the monoid $\mathbf{I}\mathbb{N}_{\infty}^{\textbf{g}[j]}$ of cofinite partial isometries of $\mathbb{N}$ with a bounded finite noise for any positive integer $j$. In particular we show that for any positive integer $j$ every Hausdorff shift-continuous topology $\tau$ on $\mathbf{I}\mathbb{N}_{\infty}^{\textbf{g}[j]}$ is discrete and and if $\mathbf{I}\mathbb{N}_{\infty}^{\textbf{\emph{g}}[j]}$ be a proper dense subsemigroup of a Hausdorff semitopological semigroup $S$, then $S\setminus \mathbf{I}\mathbb{N}_{\infty}^{\textbf{\emph{g}}[j]}$ is a closed ideal of $S$, and moreover if $S$ is a topological inverse semigroup then $S\setminus \mathbf{I}\mathbb{N}_{\infty}^{\textbf{\emph{g}}[j]}$ is a topological group.
This is a joint work with Pavlo Khylynskyi.

We extend results of the papers [Carl Eberhart and John Selden, On the Closure of the Bicyclic Semigroup, Transactions of the American Mathematical Society, Vol. 144 (Oct., 1969), pp. 115-126 ] and [M. O. Bertman and T. T. West, Conditionally Compact Bicyclic Semitopological Semigroups, Proceedings of the Royal Irish Academy. Section A: Mathematical and Physical Sciences Vol. 76 (1976), pp. 219-226] for the monoid $\mathbf{I}\mathbb{N}_{\infty}^{\textbf{g}[j]}$ of cofinite partial isometries of $\mathbb{N}$ with a bounded finite noise for any positive integer $j$. We describe the algebraic and topological structure of the closure of the monoid $\mathbf{I}\mathbb{N}_{\infty}^{\textbf{\emph{g}}[j]}$ in a locally compact topological inverse semigroup.
This is a joint work with Pavlo Khylynskyi.

We will discuss different kinds of filters on the bicyclic monoid $\mathcal C$ and the corresponding topologies on $\mathcal C^0$. Our results provide a powerful method of constructing inverse semigroup topologies on $\mathcal C^0$. We describe the finest non-discrete inverse semigroup topology on $\mathcal C^0$. Also, we show that there exists a well-ordered chain of inverse semigroup topologies on $\mathcal C^0$ of length $\mathfrak b$.

За матеріалами кандидатської дисертації.

За матеріалами кандидатської дисертації.

За матеріалами кандидатської дисертації.

Let $K$ be any cardinality and $\mathcal{IP\!F}(\sigma(\mathbb{N}^K))$ be the semigroup of all order isomorphisms between principal filters of the set $\sigma(\mathbb{N}^K)$ with the product order. We study properties of the semigroup $\mathcal{IP\!F}(\sigma(\mathbb{N}^K))$.

We provide a survey of recent results of Banakh and Bardyla concerning different completeness properties in topological semigroups, semilattices and pospaces.

The reporter discusses on the topic of her PhD thesis. The PhD thesis is devoted to lattice-valued monotonic predicates on continuous semilattices, which are natural generalizations of non-additive real-valued and lattice-valued measures.

We disccus on topologizations of polycyclic extensions of monoids.
This is a joint work with Pavlo Khylynskyi.

We disccus on topologizations of polycyclic extensions of monoids.
This is a joint work with Pavlo Khylynskyi.

We disccus on topologizations of polycyclic extensions of monoids.
This is a joint work with Pavlo Khylynskyi.

We disccus on topologizations of polycyclic extensions of monoids.
This is a joint work with Pavlo Khylynskyi.

We study the Gutik-Mykhalenych semigroup Ɓ_ω^ℱ₁ in the case when the family ℱ₁ consists of the empty set and all singleton subsets of ω. We show that Ɓ_ω^ℱ₁ is isomorphic to the subsemigroup of the Brandt ω-extension of the semilattice (ω,min) and describe all shift-continuous feebly compact T₁-topologies on the semigroup Ɓ_ω^ℱ₁.

We study the semigroup $\mathbf{B}_{\omega}^{\mathcal{F}}$ (which is introduced in the paper [Oleg Gutik, Mykola Mykhalenych, On some generalization of the bicyclic monoid, Visnyk of the Lviv University. Series Mechanics and Mathematics 90 (2020), 5-19]) in the case when the family $\mathcal{F}$ is generated by the family of singletons $\mathcal{F}$ of $\omega$. We describe the algebraic structure of the semigroup $\mathbf{B}_{\omega}^{\mathcal{F}}$, describe all shift-continuous feebly compact $T_1$-topologies on the semigroup $\mathbf{B}_{\omega}^{\mathcal{F}_1}$ and closure of $\mathbf{B}_{\omega}^{\mathcal{F}}$ in a (semi)topological semigroup.
This is a joint work with Oleksandra Lysetska (Sobol)