Scientific Seminar
at Geometry and Topology Department of
Ivan Franko National University of Lviv
Archive for (2016/2017) academic year



Topological
Algebra

September 8, 2016 O. Pikhurko
  • Measurable circle squaring II
    • In 1990 Laczkovich proved that one can split a disk into finitely many parts and move them to form a partition of a square, thus solving the long-standing Tarski's circle squaring problem. I will discuss our result with András Máthé and Łukasz Grabowski that, additionally, one can require that all parts are Lebesgue measurable and have the property of Baire. The second talk will concentrate on proof techniques.
      This report is a continuation of the Prof. Pikhurko report on the seminar Topology & Applications in September 5, 2016.

September 22, 2016 K. Maksymyk,
O. Gutik
  • On semitopological interassociates of the bicyclic monoid
    • We discuss on semitopological interassociates Cm,n of the bicyclic semigroup C(p,q). In particular we show that for arbitrary non-negative integers m and m every Hausdorff topology τ on Cm,n such that (Cm,n, τ) is a semitopological semigroup, is discrete and hence Cm,n is a discrete subspace of any topological semigroup containing it. Also, we prove that if Cm,n is any interassociate of the bicyclic monoid such that Cm,n is a dense subsemigroup of a Hausdorff semitopological semigroup (S,·) and if I=S\Cm,n≠∅ then I is a two-sided ideal of the semigroup S and show that for arbitrary non-negative integers m and m any Hausdorff locally compact topology τ on the interassociate Cm,n with an adjoined zero 0 of the bicyclic monoid C0m,n such that (C0m,n,τ) is a semitopological semigroup, is either discrete or compact.

October 6, 2016 K. Maksymyk,
O. Gutik
  • On semitopological interassociates of the bicyclic monoid, II
    • We discuss on semitopological interassociates Cm,n of the bicyclic semigroup C(p,q). In particular we show that for arbitrary non-negative integers m and m every Hausdorff topology τ on Cm,n such that (Cm,n, τ) is a semitopological semigroup, is discrete and hence Cm,n is a discrete subspace of any topological semigroup containing it. Also, we prove that if Cm,n is any interassociate of the bicyclic monoid such that Cm,n is a dense subsemigroup of a Hausdorff semitopological semigroup (S,·) and if I=S\Cm,n≠∅ then I is a two-sided ideal of the semigroup S and show that for arbitrary non-negative integers m and m any Hausdorff locally compact topology τ on the interassociate Cm,n with an adjoined zero 0 of the bicyclic monoid C0m,n such that (C0m,n,τ) is a semitopological semigroup, is either discrete or compact.

October 13, 2016 O. Gutik,
O. Ravsky
  • Divertissement
    • The active participants of the seminar will discuss some open problems and possible ways of their solutions.

October 27, 2016 O. Gutik,
O. Ravsky
  • Divertissement
    • The active participants of the seminar will discuss some open problems and possible ways of their solutions.

November 10, 2016 S. Bardyla
  • On the locally compact semitopological α-bicyclic monoid
    • For every ordinal α<ω+1 we describe all Hausdorff locally compact topologies τ on the α-bicyclic monoid Bα such that (Bα,τ), is a semitopological semigroup.

November 17, 2016 S. Bardyla
  • On the locally compact semitopological α-bicyclic monoid, II
    • For every ordinal α<ω+1 we describe all Hausdorff locally compact topologies τ on the α-bicyclic monoid Bα such that (Bα,τ), is a semitopological semigroup.

November 24, 2016 S. Bardyla
  • On the locally compact semitopological α-bicyclic monoid, III
    • For every ordinal α<ω+1 we describe all Hausdorff locally compact topologies τ on the α-bicyclic monoid Bα such that (Bα,τ), is a semitopological semigroup.

December 1, 2016 S. Bardyla
  • On the locally compact semitopological α-bicyclic monoid, IV
    • For every ordinal α<ω+1 we describe all Hausdorff locally compact topologies τ on the α-bicyclic monoid Bα such that (Bα,τ), is a semitopological semigroup.

December 8, 2016 I. Posdniakova
  • On the monoid of monotone injective partial selfmaps of 2 with cofinite domains and images
    • Let 2 be the set 2 with the partial order defining as a product of usual order on the set of positive integers . We study the semigroup PO(ℕ2) of monotone injective partial selfmaps of 2 having cofinite domain and image. We describe the natural partial order on the semigroup PO(ℕ2) and show that it coincides with the natural partial order which is induced from symmetric inverse monoid I2 over the set 2 onto the semigroup PO(ℕ2). We proved that the semigroup PO(ℕ2) is isomorphic to the semidirect product of the monoid PO+(ℕ2) of oriental monotone injective partial selfmaps of 2 with cofinite domains and images by the group of the cyclic order two 2. Also we describe the congruence σ on the semigroup PO(ℕ2), which is generated by the natural order on the semigroup PO(ℕ2): ασβ if and only if α and β are comparable in (PO(ℕ2),). We prove that quotient semigroup PO+(ℕ2)/σ is isomorphic to the free commutative monoid FAMω over an infinite countable set and show that quotient semigroup PO(ℕ2)/σ is isomorphic to the semidirect product of the free commutative monoid FAMω over an infinite countable set by the cyclic group of order two 2.

December 29, 2016 O. Gutik
  • Topological property of the Taimanov semigroup
    • For any infinite cardinal κ we construct a semigroup Ακ of cardinality κ such that the following conditions hold:
      (i) every T1-topology τ on the semigroup Ακ such that κ,τ) is a semitopological semigroup is discrete;
      (ii) for every T1-topological semigroup S which contains Ακ as a subsemigroup, Ακ is a closed subsemigroup of S;
      (iii) every homomorphic non-isomorphic image of Ακ is a zero-semigroup.

January 4, 2017 I. Posdniakova
  • On the monoid of monotone injective partial selfmaps of 2 with cofinite domains and images, II
    • Let 2 be the set 2 with the partial order defining as a product of usual order on the set of positive integers . We study the semigroup PO(ℕ2) of monotone injective partial selfmaps of 2 having cofinite domain and image. We describe the natural partial order on the semigroup PO(ℕ2) and show that it coincides with the natural partial order which is induced from symmetric inverse monoid I2 over the set 2 onto the semigroup PO(ℕ2). We proved that the semigroup PO(ℕ2) is isomorphic to the semidirect product of the monoid PO+(ℕ2) of oriental monotone injective partial selfmaps of 2 with cofinite domains and images by the group of the cyclic order two 2. Also we describe the congruence σ on the semigroup PO(ℕ2), which is generated by the natural order on the semigroup PO(ℕ2): ασβ if and only if α and β are comparable in (PO(ℕ2),). We prove that quotient semigroup PO+(ℕ2)/σ is isomorphic to the free commutative monoid FAMω over an infinite countable set and show that quotient semigroup PO(ℕ2)/σ is isomorphic to the semidirect product of the free commutative monoid FAMω over an infinite countable set by the cyclic group of order two 2.

January 11, 2017 T. Banakh
  • Topological semilattices with H-closed maximal chains are absolutely H-closed
    • We prove that each topological semilattice with H-closed maximal chains is absolutely H-closed, thus resolving an open problem posed by Oleg Gutik in 2010. Also we shall discuss some remaining open problems related to H-closed topological semilattices.
      This is a joint work with Serhiy Bardyla.

February 13, 2017 O. Gutik
  • On feebly compact shift-continuous topologies on the semilattice expnλ
    • We discuss feebly compact topologies τ on the semilattice (expnλ,∩) such that (expnλ,τ) is a semitopological semilattice and prove that for any shift-continuous T1-topology τ on expnλ the following conditions are equivalent: (i) τ is countably pracompact; (ii) τ is feebly compact; (iii) τ is d-feebly compact.

      This is a joint work with Oleksandra Sobol.

February 20, 2017 S. Bardyla
  • On completeness and H-closedness of topological semilattices
    • We consider an interplay between completeness and H-closedness of topological semilattices. We give some sufficient conditions for topological semilattice to be absolutely H-closed and posed some open problems.

      This is a joint work with Taras Banakh.

March 6, 2017 T. Banakh
  • k-Complete topological semilattices are θ-multiclosed and hence absolute H-closed
    • A topological semilattice X is k-complete if each chain C⊆ X has inf C and sup C that belong to the closure C of C in X. We shall prove that each k-complete topological semilattice X is θ-multiclosed, which means that for any continuous multi-morphism Φ: X⊸Y with θ-closed values to a topological semilattice Y has θ-closed image Φ(X). A subset A of a topological space X is θ-closed if each point x∈X∖A has a closed neighborhood Ox disjoint with A.

March 13, 2017 T. Banakh
  • k-Complete topological semilattices are θ-multiclosed and hence absolute H-closed, II
    • A topological semilattice X is k-complete if each chain C⊆ X has inf C and sup C that belong to the closure C of C in X. We shall prove that each k-complete topological semilattice X is θ-multiclosed, which means that for any continuous multi-morphism Φ: X⊸Y with θ-closed values to a topological semilattice Y has θ-closed image Φ(X). A subset A of a topological space X is θ-closed if each point x∈X∖A has a closed neighborhood Ox disjoint with A.

March 20, 2017 T. Banakh
  • k-Complete topological semilattices are θ-multiclosed and hence absolute H-closed, III
    • A topological semilattice X is k-complete if each chain C⊆ X has inf C and sup C that belong to the closure C of C in X. We shall prove that each k-complete topological semilattice X is θ-multiclosed, which means that for any continuous multi-morphism Φ: X⊸Y with θ-closed values to a topological semilattice Y has θ-closed image Φ(X). A subset A of a topological space X is θ-closed if each point x∈X∖A has a closed neighborhood Ox disjoint with A.

April 3, 2017 O. Gutik,
A. Savchuk
  • On the Bezushchak semigroup ID
    • We show that the Bezushchak semigroup ID of partial co-finite isometries of the integers is isomorphic to the semidirect product the free semilattece with unit over integers by the group of isometries of the integers. Also we discuss on topologizations of the semigroup ID.

April 10, 2017 O. Gutik,
A. Savchuk
  • On the Bezushchak semigroup ID, II
    • We show that the Bezushchak semigroup ID of partial co-finite isometries of the integers is isomorphic to the semidirect product the free semilattece with unit over integers by the group of isometries of the integers. Also we discuss on topologizations of the semigroup ID.

May 15, 2017 S. Bardyla
  • Embeddings of graph inverse semigroups into (topological) inverse semigroups
    • We discuss topologizations and embeddings of graph inverse semigroups.

May 22, 2017 S. Bardyla
  • Embeddings of graph inverse semigroups into (topological) inverse semigroups, II
    • We discuss topologizations and embeddings of graph inverse semigroups.

June 22, 2017 O. Sobol
  • Intrinsic lattices and lattice topologies
    • We discuss on the resuls of the paper: "J.D. Lawson, Intrinsic lattices and lattice topologies, S. Fajtlowicz and K. Kaiser (eds.), Proceedings of the University of Houston Lattice Theory Conference, Houston, Texas, 1973. University of Houston, 206–230."

June 22, 2017 I. Stasyuk
  • On simultaneous extensions of functions and metrics
    • We discuss on the topics in the title.

June 26, 2017 O. Sobol
  • Intrinsic lattices and lattice topologies, II
    • We discuss on the resuls of the paper: "J.D. Lawson, Intrinsic lattices and lattice topologies, S. Fajtlowicz and K. Kaiser (eds.), Proceedings of the University of Houston Lattice Theory Conference, Houston, Texas, 1973. University of Houston, 206–230."