

September 5, 2018

O. Gutik


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



September 12, 2018

D. Gavinsky


Entangled simultaneity versus classical interactivity in communication complexity
In 1999 Raz demonstrated a partial function that had an efficient
quantum twoway communication protocol but no efficient classical
twoway protocol and asked, whether there existed a function with an
efficient quantum oneway protocol, but still no efficient classical
twoway protocol. In 2010 Klartag and Regev demonstrated such a
function and asked, whether there existed a function with an efficient
quantum simultaneousmessages protocol, but still no efficient
classical twoway 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 polylogarithmic
size, and whose classical twoway complexity is lower bounded by a
polynomial.



September 19, 2018

S. Bardyla


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



September 26, 2018

S. Bardyla


Embedding of the bicyclic monoid with adjoined zero $\mathcal{C}^0(p,q)$ into compactlike topological semigroups
We study embedding of the bicyclic monoid $\mathcal{C}^0(p,q)=\mathcal{C}(p,q)\sqcup\{0\}$ with adjoined zero into compactlike 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 dcompact topological semigroups.



October 24, 2018

M. Khylynskyi


On an interassociativity of the $\lambda$polycyclic monoid
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.



October 31, 2018

M. Khylynskyi


On an interassociativity of the $\lambda$polycyclic monoid, II
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.



November 7, 2018

M. Khylynskyi


On an interassociativity of the $\lambda$polycyclic monoid, III
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.



November 14, 2018

M. Khylynskyi


On an interassociativity of the $\lambda$polycyclic monoid, IV
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.



November 21, 2018

S. Bardyla


On a semitopological bicyclic monoid with adjoint zero $\mathcal{C}(p,q)^{0}$ which semilattice of idempotents is compact
We describe minimal semigroup topologies on $\mathcal{C}(p,q)^{0}$ and investigate properties
of the semilattice of shiftcontinuous topologies on the bicyclic monoid with adjoint zero.



December 5, 2018

S. Bardyla


On a semitopological bicyclic monoid with adjoint zero $\mathcal{C}(p,q)^{0}$ which semilattice of idempotents is compact, II
We describe minimal semigroup topologies on $\mathcal{C}(p,q)^{0}$ and investigate properties
of the semilattice of shiftcontinuous topologies on the bicyclic monoid with adjoint zero.



January 16, 2019

O. Gutik


The semigroup of star partial homeomorphisms of a finite deminsional Euclidean space
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 nonunit congruence on $\mathbf{PStH}_{\mathbb{R}^n}$ is a group congruence.



January 23, 2019

O. Gutik


The semigroup of star partial homeomorphisms of a finite deminsional Euclidean space, II
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 nonunit congruence on $\mathbf{PStH}_{\mathbb{R}^n}$ is a group congruence.



February 6, 2019

O. Gutik


On inverse submonoids of the monoid of almost monotone injective cofinite partial selfmaps of positive integers
We discuss on the monoid $\mathscr{I}_{\infty}^{\,\Rsh\!\!\!\nearrow}(\mathbb{N})$ of almost monotone injective cofinite 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 nonidentity 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
compactlike topological semigroups.



February 20, 2019

A. Ravsky


Small zerosum subsets of decomposable sets
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 nonempty 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 nonempty 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.



February 27, 2019

T. Mokrytskyi


On the dychotomy of a locally compact semitopological monoid $\mathcal{IPF}(\mathbb{N}^n)^0$ of order isomorphisms of principal filters of a power of the positive integers with adjoined zero
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 shiftcontinuous toplogy on $\mathcal{IPF}(\mathbb{N}^n)^0=\mathcal{IPF}(\mathbb{N}^n)\sqcup\{0\}$ is either compact
or discrete.



March 6, 2019

T. Mokrytskyi


On the dychotomy of a locally compact semitopological monoid $\mathcal{IPF}(\mathbb{N}^n)^0$ of order isomorphisms of principal filters of a power of the positive integers with adjoined zero, II
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 shiftcontinuous toplogy on $\mathcal{IPF}(\mathbb{N}^n)^0=\mathcal{IPF}(\mathbb{N}^n)\sqcup\{0\}$ is either compact
or discrete.



April 3, 2019

A. Ravsky


Zelenyuk's example of a locally compact noncompact monothetic semigroup with identity
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 wellknown 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), 101107 (https://doi.org/10.4064/fm53532018).



April 10, 2019

O. Gutik


On automorphisms of a full inverse subsemigroup of $\mathcal{I}_{\infty}^{\!\nearrow}(\mathbb{N})$ which contains the semigroup $\mathcal{C}_{\mathbb{N}}$
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.



April 17, 2019

A. Ravsky


An algebraic problem from quantum physics
According to twomonth 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.



All participants


Common Photo



May 8, 2019

O. Sobol


Semigroups with strongly tight ideal series and their $\mathcal{I}_\lambda^n$extensions
In the paper [O. Gutik, J. Lawson, and D. Repovš, Semigroup closures of finite rank symmetric inverse semigroups, Semigroup Forum 78 (2009), no. 2, 326336 (doi: 10.1007/s0023300891122, 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 $n\leqslant\lambda$.



May 15, 2019

O. Sobol


Semigroups with strongly tight ideal series and their $\mathcal{I}_\lambda^n$extensions, II
In the paper [O. Gutik, J. Lawson, and D. Repovš, Semigroup closures of finite rank symmetric inverse semigroups, Semigroup Forum 78 (2009), no. 2, 326336 (doi: 10.1007/s0023300891122, 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 $n\leqslant\lambda$.



O. Maslyuchenko (Univ. of Silesia in Katowice, Poland)


Hemimetrizable spaces and an analogue of Kenderov's and Debs' theorems
A function d:X × X → [0;+∞) calls hemimetric 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 hemimetrizable if there exist a hemimetric 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 hemimetrizable T_{1}space which is βσunfavorable 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 hemimetrizable but βσfavorable for the SaintRaymond game.



June 5, 2019

O. Gutik


Compact semitopological semigroups and their compact topological $\mathcal{I}_\lambda^n$extensions
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.



June 5, 2019

Аспіранти кафедри геометрії і топології


Звіти аспірвнтів кафедри геометрії і топології
1. Олександра Соболь  2й р.н.
2. Тарас Мокрицький  1й р.н.
3. Ярина Стельмах  1й р.н.
4. Христина Сухорукова  1й р.н.



July 19, 2019

S. Bardyla (Kurt Gödel Research Center, University of Vienna)


Closed subspaces of compactlike spaces
We investigate closed subsets (subsemigroups, resp.) of compactlike topological spaces
(semigroups, resp.). We prove that each Hausdorff topological space can be embedded as a closed
subspace into an Hclosed topological space. However, the semigroup of
$\omega{\times\omega}$matrix units cannot be embedded into a topological semigroup which is an Hclosed
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, 28032814
(doi: 10.1016/j.topol.2010.08.020)].
This is joint work with A. Ravsky.



July 26, 2019

S. Bardyla (Kurt Gödel Research Center, University of Vienna)


Closed subspaces of compactlike spaces, II
We investigate closed subsets (subsemigroups, resp.) of compactlike topological spaces
(semigroups, resp.). We prove that each Hausdorff topological space can be embedded as a closed
subspace into an Hclosed topological space. However, the semigroup of
$\omega{\times\omega}$matrix units cannot be embedded into a topological semigroup which is an Hclosed
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, 28032814
(doi: 10.1016/j.topol.2010.08.020)].
This is joint work with A. Ravsky.



August 21, 2019

O. Gutik


On a semitopological extended bicyclic semigroup with adjoined zero
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 shiftcontinuous topologies.
This is joint work with K. Maksymyk.



August 29, 2019

T. Banakh


On $\bar G_\delta$separated semitopological semilattices
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

