

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.

