Active participants of the seminar will present and discuss interesting open problems from various branches of the topological algebra.

In 1999 Raz demonstrated a partial function that had an efficient quantum two-way communication protocol but no efficient classical two-way protocol and asked, whether there existed a function with an efficient quantum one-way protocol, but still no efficient classical two-way protocol. In 2010 Klartag and Regev demonstrated such a function and asked, whether there existed a function with an efficient quantum simultaneous-messages protocol, but still no efficient classical two-way protocol. In this work we answer the latter question affirmatively and present a partial function, which can be computed by a protocol sending entangled simultaneous messages of poly-logarithmic size, and whose classical two-way complexity is lower bounded by a polynomial.

We investigate shift continuous topologies on the polycyclic monoid $P_1$. Also, an example of an non-second countable topology will be constructed.

We study embedding of the bicyclic monoid $\mathcal{C}^0(p,q)=\mathcal{C}(p,q)\sqcup\{0\}$ with adjoined zero into compact-like topological semigroups. More precisely, we give necessary and sufficient conditions which provide an embedding of the bicyclic monoid with adjoined zero $\mathcal{C}^0(p,q)$ into d-compact topological semigroups.

We discuss on algebraic properties of interassociativities of the $\lambda$-polycyclic monoid. In particular the Green relations on an interassociativity of the $\lambda$-polycyclic monoid and their isomorphisms will be described.

We discuss on algebraic properties of interassociativities of the $\lambda$-polycyclic monoid. In particular the Green relations on an interassociativity of the $\lambda$-polycyclic monoid and their isomorphisms will be described.

We discuss on algebraic properties of interassociativities of the $\lambda$-polycyclic monoid. In particular the Green relations on an interassociativity of the $\lambda$-polycyclic monoid and their isomorphisms will be described.

We discuss on algebraic properties of interassociativities of the $\lambda$-polycyclic monoid. In particular the Green relations on an interassociativity of the $\lambda$-polycyclic monoid and their isomorphisms will be described.

We describe minimal semigroup topologies on $\mathcal{C}(p,q)^{0}$ and investigate properties of the semilattice of shift-continuous topologies on the bicyclic monoid with adjoint zero.

We describe minimal semigroup topologies on $\mathcal{C}(p,q)^{0}$ and investigate properties of the semilattice of shift-continuous topologies on the bicyclic monoid with adjoint zero.

We describe the structure of the semigroup $\mathbf{PStH}_{\mathbb{R}^n}$ of star partial homeomorphisms of a finite deminsional Euclidean space $\mathbb{R}^n.$ The structure of the band of $\mathbf{PStH}_{\mathbb{R}^n}$ and Green's relations on $\mathbf{PStH}_{\mathbb{R}^n}$ are described. We show that $\mathbf{PStH}_{\mathbb{R}^n}$ is a bisimple inverse semigroup and every non-unit congruence on $\mathbf{PStH}_{\mathbb{R}^n}$ is a group congruence.

We describe the structure of the semigroup $\mathbf{PStH}_{\mathbb{R}^n}$ of star partial homeomorphisms of a finite deminsional Euclidean space $\mathbb{R}^n.$ The structure of the band of $\mathbf{PStH}_{\mathbb{R}^n}$ and Green's relations on $\mathbf{PStH}_{\mathbb{R}^n}$ are described. We show that $\mathbf{PStH}_{\mathbb{R}^n}$ is a bisimple inverse semigroup and every non-unit congruence on $\mathbf{PStH}_{\mathbb{R}^n}$ is a group congruence.

We discuss on the monoid $\mathscr{I}_{\infty}^{\,\Rsh\!\!\!\nearrow}(\mathbb{N})$ of almost monotone injective co-finite partial selfmaps of positive integers $\mathbb{N}$. We show that every automorphism of the submonoid $\mathbf{I}\mathbb{N}_{\infty}\subseteq \mathscr{I}_{\infty}^{\,\Rsh\!\!\!\nearrow}(\mathbb{N})$ of all partial cofinite isometries of $\mathbb{N}$ is trivial. We construct a subsemigroup $\mathbf{I}\mathbb{N}_{\infty}^{[\underline{1}]}$ of $\mathscr{I}_{\infty}^{\,\Rsh\!\!\!\nearrow}(\mathbb{N})$ with the following amazing property: if $S$ be an inverse subsemigroup of $\mathscr{I}_{\infty}^{\,\Rsh\!\!\!\nearrow}(\mathbb{N})$ such that $S$ contains $\mathbf{I}\mathbb{N}_{\infty}^{[\underline{1}]}$ as a subsemigroup, then every non-identity congruence $\mathfrak{C}$ on $S$ is a group congruence. Let $\mathscr{C}_{\mathbb{N}}$ be a subsemigroup $\mathscr{I}_{\infty}^{\,\Rsh\!\!\!\nearrow}(\mathbb{N})$ which is generated by the partial shift $n\mapsto n+1$ and its inverse of $\mathbb{N}$. We show that if $S$ is an inverse subsemigroup of $\mathscr{I}_{\infty}^{\,\Rsh\!\!\!\nearrow}(\mathbb{N})$ such that $S$ contains $\mathscr{C}_{\mathbb{N}}$ as a subsemigroup, then $S$ is simple and the quotient semigroup $S/\mathfrak{C}_{\mathbf{mg}}$, where $\mathfrak{C}_{\mathbf{mg}}$ is minimum group congruence on $S$, is isomorphic to the additive group of integers $\mathbb{Z}(+)$. Also, we discuss topologizations of inverse subsemigroups of $\mathscr{I}_{\infty}^{\,\Rsh\!\!\!\nearrow}(\mathbb{N})$ such that $S$ contains $\mathscr{C}_{\mathbb{N}}$ and embeddings of such semigroups into compact-like topological semigroups.

This talk about our joint research with Taras Banakh is inspired by two questions which remain unanswered from 2010. The first was posed by Gjergji Zaimi at MathOverflow and the second was posed by Aryabhata at Mathematics StackExchange.
Let $S$ be a subset of an abelian group. A set $S$ is called \emph{decomposable} provided $S+S\supset S$. Let $z(S)$ be the smallest size of a non-empty subset $T$ of $S$ such that $\sum T=0$, and $z(S)=\infty$, otherwise. Given a natural number $n$ put $z(n)=\sup\{z(S): S\subset S+S\subset\mathbb R, |S|=n\},$ that is $z(n)$ is the smallest number $m$ such that any decomposable set of $n$ real numbers has a non-empty subset $T$ of size $m$ such that $\sum T=0$. Our main conjecture is that for any $n\ge 2$, $z(n)=\left\lfloor\tfrac n2\right\rfloor$ and our main result is the proof of the lower bound for $z(n)$ claimed by the conjecture.

Let $n$ be any positive integer and $\mathcal{IPF}(\mathbb{N}^n)$ be the semigroup of all order isomorphisms between principal filters of the $n$-th power of the set of positive integers $\mathbb{N}$ with the product order. We show that every Hausdorff locally compact shift-continuous toplogy on $\mathcal{IPF}(\mathbb{N}^n)^0=\mathcal{IPF}(\mathbb{N}^n)\sqcup\{0\}$ is either compact or discrete.

Let $n$ be any positive integer and $\mathcal{IPF}(\mathbb{N}^n)$ be the semigroup of all order isomorphisms between principal filters of the $n$-th power of the set of positive integers $\mathbb{N}$ with the product order. We show that every Hausdorff locally compact shift-continuous toplogy on $\mathcal{IPF}(\mathbb{N}^n)^0=\mathcal{IPF}(\mathbb{N}^n)\sqcup\{0\}$ is either compact or discrete.

A Hausdorff topological semigroup $S$ is called \emph{monothetic} provided it contains a dense cyclic subsemigroup, that is $S=\overline{\langle a^n:n\ge 1\rangle}$ for some element $a\in S$. Compact monothetic topological groups are compact abelian groups whose character groups are semigroups of the unit circle group (endowed with the discrete topology). Pontrjagin alternative states that each locally compact monothetic topological group is either compact, or discrete. Compact monothetic semigroups was described by Hewitt. Whether Pontrjagin alternative holds for locally compact monothetic monoids is a well-known problem, posed by Koch and opened for more than sixty years. In 1988 Zelenyuk constructed a countable locally compact monothetic cancellative semigroup which is neither compact, nor discrete, see [Zel1]. Unfortunately, as it is easy to see, a countable locally compact monothetic monoid is discrete, so we cannot attach a unit to this example. It was hard to overcome this obstacle. It took Zelenyuk thirty years and he constructed a complicated example only in the last year, see [Zel2], and this is a subject of the talk.
[Zel1] Е. Г. Зеленюк, К альтернативе Понтрягина для топологических полугрупп, Матем. заметки, 44:3 (1988), 402–403
[Zel2] E. Zelenyuk A locally compact noncompact monothetic semigroup with identity, Fundamenta Mathematicae, 245 (2019), 101-107 (https://doi.org/10.4064/fm535-3-2018).

Let $\mathcal{I}_{\infty}^{\!\nearrow}(\mathbb{N})$ be the monoid of cofinite monotone partial bijections of the set of positive integers $\mathbb{N}$ and $\mathcal{C}_{\mathbb{N}}$ be a submonoid of $\mathcal{I}_{\infty}^{\!\nearrow}(\mathbb{N})$ which is generated by the transformation $\alpha\colon \mathbb{N}\to \mathbb{N}, i\mapsto i+1$ and its inverse. We show that every automorphism of a full inverse subsemigroup of $\mathcal{I}_{\infty}^{\!\nearrow}(\mathbb{N})$ which contains the semigroup $\mathcal{C}_{\mathbb{N}}$ is trivial.

According to two-month old paper "[Questions on the Structure of Perfect Matchings inspired by Quantum Physics](https://arxiv.org/abs/1902.06023)” by Mario Krenn, Xuemei Gu and Daniel Soltész), "A bridge between quantum physics and graph theory has been uncovered recently [1, 2, 3]. [These are fresh papers, among others, of the first two authors and [Anton Zeilinger](https://en.wikipedia.org/wiki/Anton_Zeilinger), a famous specialist in quantum physics. AR.] It allows to translate questions from quantum physics – in particular about photonic quantum physical experiments – into a purely graph theoretical language. The question can then be analysed using tools from graph theory and the results can be translated back and interpreted in terms of quantum physics. The purpose of this manuscript is to collect and formulate a large class of questions that concern the generation of pure quantum states with photons with modern technology. This will hopefully allow and motivate experts in the field to think about these issues. ... Every progress in any of these purely graph theoretical questions can be immediately translated to new understandings in quantum physics. Apart from the intrinsic beauty of answering purely mathematical questions, we hope that the link to natural science gives additional motivation for having a deeper look on the questions raised above". But it turned out that there are not so much graph theory in the problem, because in fact it is about matchings on a compete graph with possible zero edge weights. So in our talk we shall discuss a linear algebraic approach to the problem.

In the paper [O. Gutik, J. Lawson, and D. Repovš, Semigroup closures of finite rank symmetric inverse semigroups, Semigroup Forum 78 (2009), no. 2, 326-336 (doi: 10.1007/s00233-008-9112-2, MR2486644 (2010f:20066), Zbl 1165.22002, arXiv:0804.1439)] the notion of a semigroup which admits a tight ideal series was intruduced. We introduce the notion of a semigroup which admits a stronly tight ideal series and it more cstronger them above. We discuss on property of such semigroups and show that an $\mathcal{I}_\lambda^n$-extension preserves this property for any cardinal $\lambda$>0 and any positive integer $n\leqslant\lambda$.

In the paper [O. Gutik, J. Lawson, and D. Repovš, Semigroup closures of finite rank symmetric inverse semigroups, Semigroup Forum 78 (2009), no. 2, 326-336 (doi: 10.1007/s00233-008-9112-2, MR2486644 (2010f:20066), Zbl 1165.22002, arXiv:0804.1439)] the notion of a semigroup which admits a tight ideal series was intruduced. We introduce the notion of a semigroup which admits a stronly tight ideal series and it more cstronger them above. We discuss on property of such semigroups and show that an $\mathcal{I}_\lambda^n$-extension preserves this property for any cardinal $\lambda>0$ and any positive integer $n\leqslant\lambda$.

A function d:X × X → [0;+∞) calls hemi-metric if d(x,x)=0 and d(x,y) ≤ d(x,z) + d(z,y) for any x,y,z∈X. We call a topological space X hemi-metrizable if there exist a hemi-metric d such that the topology of X is generated by the base consisting of open balls B_d(a,r)={x∈X: d(x,a)< r}. We prove that every hemi-metrizable T₁-space which is β-σ-unfavorable for the Christensen game has a metrizable dense G_δ-subspace. On the other hand we give an example of a normal Baire hereditarily separable space Lindelöf space which is hemi-metrizable but β-σ-favorable for the Saint-Raymond game.

We show that for every compact Hausdorff semitopological semigroup $S$, any cardinal $\lambda>0$ and any positive integer $n\leqslant\lambda$ there exists the unique compact Hausdorff topological $\mathcal{I}_\lambda^n$-extension $\mathcal{I}_\lambda^n(S)$ of $S$ in the class of semitopological semigroups.

1. Олександра Соболь - 2-й р.н.
2. Тарас Мокрицький - 1-й р.н.
3. Ярина Стельмах - 1-й р.н.
4. Христина Сухорукова - 1-й р.н.

We investigate closed subsets (subsemigroups, resp.) of compact-like topological spaces (semigroups, resp.). We prove that each Hausdorff topological space can be embedded as a closed subspace into an H-closed topological space. However, the semigroup of $\omega{\times\omega}$-matrix units cannot be embedded into a topological semigroup which is an H-closed topological space. We give sufficient conditions for closed embeddability of a topological space into countably pracompact topological spaces. Also, we construct a pseudocompact topological semigroup which contains the bicyclic monoid as a closed subsemigroup, providing a positive solution of a problem posed by Banakh, Dimitrova, and Gutik [T. Banakh, S. Dimitrova, and O. Gutik, Embedding the bicyclic semigroup into countably compact topological semigroups, Topology Appl. 157 (2010), no. 18, 2803-2814 (doi: 10.1016/j.topol.2010.08.020)].
This is joint work with A. Ravsky.

We investigate closed subsets (subsemigroups, resp.) of compact-like topological spaces (semigroups, resp.). We prove that each Hausdorff topological space can be embedded as a closed subspace into an H-closed topological space. However, the semigroup of $\omega{\times\omega}$-matrix units cannot be embedded into a topological semigroup which is an H-closed topological space. We give sufficient conditions for closed embeddability of a topological space into countably pracompact topological spaces. Also, we construct a pseudocompact topological semigroup which contains the bicyclic monoid as a closed subsemigroup, providing a positive solution of a problem posed by Banakh, Dimitrova, and Gutik [T. Banakh, S. Dimitrova, and O. Gutik, Embedding the bicyclic semigroup into countably compact topological semigroups, Topology Appl. 157 (2010), no. 18, 2803-2814 (doi: 10.1016/j.topol.2010.08.020)].
This is joint work with A. Ravsky.

We show that every Hausdorff locally compact semigroup topology on the extended bicyclic semigroup with adjoined zero $\mathcal{C}_{\mathbb{Z}}^{\mathbf{0}}$ is discrete, but on $\mathcal{C}_{\mathbb{Z}}^{\mathbf{0}}$ there exist $\mathfrak{c}$ many Hausdorff locally compact shift-continuous topologies.
This is joint work with K. Maksymyk.

A semitopological semilattice $X$ is called em $\bar G_\delta$-separated if for any distinct points $x,y\in X$ there exists a countable family $\{U_n\}_{n\in\omega}$ of closed neighborhoods of $x$ whose intersection $\bigcap_{n\in\omega}U_n$ is a subsemilattice of $X$ that does not contain $y$. We prove that for any complete subsemilattice $X$ of a $\bar G_\delta$-separated semitopological semilattice the partial order of $X$ is a closed subset of $Y\times Y$. This implies that $X$ is closed in $Y$.
More details can be found in this paper (joint with S. Bardyla): https://arxiv.org/abs/1806.02869