Scientific Seminar
at Geometry and Topology Department of
Ivan Franko National University of Lviv
Archive for (2018/2019) academic year



Topological
Algebra

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 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.

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 non-second countable topology will be constructed.

September 26, 2018 S. Bardyla
  • Embedding of the bicyclic monoid with adjoined zero $\mathcal{C}^0(p,q)$ into compact-like topological semigroups

    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.

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 shift-continuous 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 shift-continuous 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 non-unit 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 non-unit 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 co-finite partial selfmaps of positive integers

    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.

February 20, 2019 A. Ravsky
  • Small zero-sum 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 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.

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 shift-continuous 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 shift-continuous toplogy on $\mathcal{IPF}(\mathbb{N}^n)^0=\mathcal{IPF}(\mathbb{N}^n)\sqcup\{0\}$ is either compact or discrete.