Scientific Seminar
at Geometry and Topology Department of
Ivan Franko National University of Lviv
Archive for (2017/2018) academic year



Topological
Algebra

September 14, 2017 All participants
September 27, 2017 M. Khylynskyi
  • On the λ-polycyclic estension of a monoid
    • We introduce the λ-polycyclic estension of a monoid and discuss its properties.

October 4, 2017 M. Khylynskyi
  • On the λ-polycyclic estension of a monoid, II
    • We introduce the λ-polycyclic estension of a monoid and discuss its algebraic properties.

October 18, 2017 T. Mokrytskyi
  • The monoid $\mathcal{IPF}(\mathbb{N}^n)$ of order isomorphisms of principal filters of a power of the positive integers

    • 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 study algebraic properties of the semigroup $\mathcal{IPF}(\mathbb{N}^n)$. In particular, we show that $\mathcal{IPF}(\mathbb{N}^n)$ bisimple inverse semigroup, describe Green's relations on $\mathcal{IPF}(\mathbb{N}^n)$ and its maximal subgroups. We prove that every non-identity congruence $\mathfrak{C}$ on the semigroup $\mathcal{IPF}(\mathbb{N}^n)$ is group.

October 25, 2017 T. Mokrytskyi
  • The monoid $\mathcal{IPF}(\mathbb{N}^n)$ of order isomorphisms of principal filters of a power of the positive integers, 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 study algebraic properties of the semigroup $\mathcal{IPF}(\mathbb{N}^n)$. In particular, we show that $\mathcal{IPF}(\mathbb{N}^n)$ bisimple inverse semigroup, describe Green's relations on $\mathcal{IPF}(\mathbb{N}^n)$ and its maximal subgroups. We prove that every non-identity congruence $\mathfrak{C}$ on the semigroup $\mathcal{IPF}(\mathbb{N}^n)$ is group.

November 1, 2017 A. Ravsky
  • A few open problems from Mathematics StackExchange

November 8, 2017 O. Sobol
November 15, 2017 O. Sobol
November 22, 2017 O. Sobol
November 29, 2017 O. Sobol
December 6, 2017 S. Bardyla
  • On topological semilattices with compact maximal chains

    • We investigate topological semilattices with compact maximal chains. In particular, we prove that a topological semilattice X is multiclosed in the class of topological semigroups iff X is a semilattice with compact maximal chains. Some open problems will be posed.

December 13, 2017 S. Bardyla
  • On topological semilattices with compact maximal chains, II

    • We investigate topological semilattices with compact maximal chains. In particular, we prove that a topological semilattice X is multiclosed in the class of topological semigroups iff X is a semilattice with compact maximal chains. Some open problems will be posed.

December 19, 2017 All participants
January 3, 2018 S. Bardyla
  • On topological semilattices with compact maximal chains, III

    • We investigate topological semilattices with compact maximal chains. In particular, we prove that a topological semilattice X is multiclosed in the class of topological semigroups iff X is a semilattice with compact maximal chains. Some open problems will be posed.

January 24, 2018 A. Ravsky
February 14, 2018 O. Sobol,
O. Gutik
  • On feebly compact semitopological semilattice $\exp_n\lambda$

    • We discuss on feebly compact shift-continous topologies $\tau$ on the semilattice $\left(\exp_n\lambda,\cap\right)$. It is proved that for any shift-continuous $T_1$-topology $\tau$ on $\exp_n\lambda$ the following conditions are equivalent: $(i)$ $\tau$ is countably pracompact; $(ii)$ $\tau$ is feebly compact; $(iii)$ $\tau$ is $d$-feebly compact; $(iv)$~$\left(\exp_n\lambda,\tau\right)$ is an $H$-closed space; $(v)$~$\left(\exp_n\lambda,\tau\right)$ is sequentially countably pracompact; $(vi)$~$\left(\exp_n\lambda,\tau\right)$ is selectively sequentially feebly compact; $(vii)$~$\left(\exp_n\lambda,\tau\right)$ is selectively feebly compact; $(viii)$~$\left(\exp_n\lambda,\tau\right)$ is sequentially feebly compact; $(ix)$~$\left(\exp_n\lambda,\tau\right)$ is $\mathfrak{D}(\omega)$-compact; $(x)$~$\left(\exp_n\lambda,\tau\right)$ is $\mathbb{R}$-compact; $(xi)$~$\left(\exp_n\lambda,\tau\right)$ is infra H-closed.

February 21, 2018 O. Sobol,
O. Gutik
  • On feebly compact semitopological semilattice $\exp_n\lambda$, II

    • We discuss on feebly compact shift-continous topologies $\tau$ on the semilattice $\left(\exp_n\lambda,\cap\right)$. It is proved that for any shift-continuous $T_1$-topology $\tau$ on $\exp_n\lambda$ the following conditions are equivalent: $(i)$ $\tau$ is countably pracompact; $(ii)$ $\tau$ is feebly compact; $(iii)$ $\tau$ is $d$-feebly compact; $(iv)$~$\left(\exp_n\lambda,\tau\right)$ is an $H$-closed space; $(v)$~$\left(\exp_n\lambda,\tau\right)$ is sequentially countably pracompact; $(vi)$~$\left(\exp_n\lambda,\tau\right)$ is selectively sequentially feebly compact; $(vii)$~$\left(\exp_n\lambda,\tau\right)$ is selectively feebly compact; $(viii)$~$\left(\exp_n\lambda,\tau\right)$ is sequentially feebly compact; $(ix)$~$\left(\exp_n\lambda,\tau\right)$ is $\mathfrak{D}(\omega)$-compact; $(x)$~$\left(\exp_n\lambda,\tau\right)$ is $\mathbb{R}$-compact; $(xi)$~$\left(\exp_n\lambda,\tau\right)$ is infra H-closed.

March 14, 2018 A. Savchuk,
O. Gutik
  • The semigroup of partial co-finite isometries of positive integers

    • The semigroup $\mathbf{I}\mathbb{N}_{\infty}$ of all partial co-finite isometries of positive integers is discussed. We describe the Green relations on the semigroup $\mathbf{I}\mathbb{N}_{\infty}$, its band and proved that $\mathbf{I}\mathbb{N}_{\infty}$ is a simple $E$-unitary $F$-inverse semigroup. It is described the least group congruence $\mathfrak{C}_{\mathbf{mg}}$ on $\mathbf{I}\mathbb{N}_{\infty}$ and proved that the quotient-semigroup $\mathbf{I}\mathbb{N}_{\infty}/\mathfrak{C}_{\mathbf{mg}}$ is isomorphic to the additive group of integers $\mathbb{Z}(+)$. An example of a non-group congruence on the semigroup $\mathbf{I}\mathbb{N}_{\infty}$ is presented.

March 21, 2018 O. Gutik
  • The semigroup of partial co-finite isometries of positive integers, II

    • The semigroup $\mathbf{I}\mathbb{N}_{\infty}$ of all partial co-finite isometries of positive integers is discussed. We proved that a congruence on the semigroup $\mathbf{I}\mathbb{N}_{\infty}$ is group if and only if its restriction onto an isomorphic copy of the bicyclic semigroup in $\mathbf{I}\mathbb{N}_{\infty}$ is a group congruence.

March 28, 2018 O. Gutik
  • On group congruences of inverse semigroups whose contain the semigroup of partial co-finite isometries of positive integers

    • Let $\mathcal{I}_{\infty}^{\!\nearrow}(\mathbb{N})$ be the the semigroup of cofinite monotone partial bijections of the set of pusitive integers $\mathbb{N}$ and $\mathbf{I}\mathbb{N}_{\infty}$ be the semigroup of all partial co-finite isometries of $\mathbb{N}$. We give a criterium when an inverse subsemigroup $S$ of $\mathcal{I}_{\infty}^{\!\nearrow}(\mathbb{N})$ such that $S$ contains $\mathbf{I}\mathbb{N}_{\infty}$ has the following property: every non-identity congruence of $S$ is a group congrunce. We describe minimal inviverse subsemigroups with such property.

April 4, 2018 L. Plachta
  • On mesures of nonplanarity of cubic graphs

    • We study two measures of nonplanarity of cubic graphs $G$, the genus $\gamma(G)$ and the edge deletion number $\mathbf{ed}(G)$. For cubic graphs of small order these parameters are compared with another measure of nonplanarity, the (rectiliniar) crossing number $\overline{\mathbf{cr}}(G)$. We introduce operations of connected sum, specified for cubic graphs $G$, and show that under certain conditions the parameters $\gamma(G)$ and $ed(G)$ are additive (subadditive) with respect to them.
      The minimal genus graphs (i.e. the cubic graphs of minimum order with given value of genus $\gamma$) and the minimal edge deletion graphs (i.e. cubic graphs of minimum order with given value of edge deletion number $\mathbf{ed}$) are introduced and studied. We provide upper bounds for the order of minimal genus and minimal edge deletion graphs.

April 11, 2018 O. Gutik
  • On group congruences of inverse semigroups whose contain the semigroup of partial co-finite isometries of positive integers, II

    • Let $\mathcal{I}_{\infty}^{\!\nearrow}(\mathbb{N})$ be the the semigroup of cofinite monotone partial bijections of the set of pusitive integers $\mathbb{N}$ and $\mathbf{I}\mathbb{N}_{\infty}$ be the semigroup of all partial co-finite isometries of $\mathbb{N}$. We give a criterium when an inverse subsemigroup $S$ of $\mathcal{I}_{\infty}^{\!\nearrow}(\mathbb{N})$ such that $S$ contains $\mathbf{I}\mathbb{N}_{\infty}$ has the following property: every non-identity congruence of $S$ is a group congrunce. We describe minimal inviverse subsemigroups with such property.

April 18, 2018 S. Bardyla
  • Embedding of graph inverse semigroups into compact-like topological semigroups

    • We investigate graph inverse semigroups which are subsemigroups of compact-like topological semigroups. More precisely, we characterise graph inverse semigroups which embeds densely into d-compact topological semigroups.

April 25, 2018 S. Bardyla
  • Embedding of graph inverse semigroups into compact-like topological semigroups, II

    • We investigate graph inverse semigroups which are subsemigroups of compact-like topological semigroups. More precisely, we characterise graph inverse semigroups which embeds densely into d-compact topological semigroups.