

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
longstanding 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 C_{m,n} of the bicyclic semigroup C(p,q).
In particular we show that for arbitrary nonnegative integers m and m every Hausdorff topology τ on C_{m,n}
such that (C_{m,n}, τ) is a semitopological semigroup, is discrete and hence C_{m,n}
is a discrete subspace of any topological semigroup containing it. Also, we prove that if C_{m,n} is any interassociate of the bicyclic monoid
such that C_{m,n} is a dense subsemigroup of a Hausdorff semitopological semigroup (S,·) and if
I=S\C_{m,n}≠∅ then I is a twosided ideal of the semigroup S and show that for arbitrary nonnegative integers
m and m any Hausdorff locally compact topology τ on the interassociate C_{m,n} with an adjoined zero 0 of
the bicyclic monoid C^{0}_{m,n} such that
(C^{0}_{m,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 C_{m,n} of the bicyclic semigroup C(p,q).
In particular we show that for arbitrary nonnegative integers m and m every Hausdorff topology τ on C_{m,n}
such that (C_{m,n}, τ) is a semitopological semigroup, is discrete and hence C_{m,n}
is a discrete subspace of any topological semigroup containing it. Also, we prove that if C_{m,n} is any interassociate of the bicyclic monoid
such that C_{m,n} is a dense subsemigroup of a Hausdorff semitopological semigroup (S,·) and if
I=S\C_{m,n}≠∅ then I is a twosided ideal of the semigroup S and show that for arbitrary nonnegative integers
m and m any Hausdorff locally compact topology τ on the interassociate C_{m,n} with an adjoined zero 0 of
the bicyclic monoid C^{0}_{m,n} such that
(C^{0}_{m,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 αbicyclic monoid

For every ordinal α<ω+1 we describe all Hausdorff locally compact topologies τ on the
αbicyclic monoid B_{α} such that (B_{α},τ), is a semitopological semigroup.



November 17, 2016

S. Bardyla


On the locally compact semitopological αbicyclic monoid, II

For every ordinal α<ω+1 we describe all Hausdorff locally compact topologies τ on the
αbicyclic monoid B_{α} such that (B_{α},τ), is a semitopological semigroup.



November 24, 2016

S. Bardyla


On the locally compact semitopological αbicyclic monoid, III

For every ordinal α<ω+1 we describe all Hausdorff locally compact topologies τ on the
αbicyclic monoid B_{α} such that (B_{α},τ), is a semitopological semigroup.



December 1, 2016

S. Bardyla


On the locally compact semitopological αbicyclic monoid, IV

For every ordinal α<ω+1 we describe all Hausdorff locally compact topologies τ on the
αbicyclic monoid B_{α} such that (B_{α},τ), is a semitopological semigroup.



December 8, 2016

I. Posdniakova


On the monoid of monotone injective partial selfmaps of ℕ^{2}_{≤} with cofinite domains and images

Let ℕ^{2}_{≤} be the set ℕ^{2} with the partial order defining as a product of usual order ≤
on the set of positive integers ℕ. We study the semigroup PO_{∞}(ℕ^{2}_{≤}) of monotone injective partial
selfmaps of ℕ^{2}_{≤} having cofinite domain and image. We describe the natural partial order on the semigroup
PO_{∞}(ℕ^{2}_{≤}) and show that it coincides with the natural partial order which is induced from symmetric inverse monoid
I_{ℕ2} over the set ℕ^{2} onto the semigroup PO_{∞}(ℕ^{2}_{≤}).
We proved that the semigroup PO_{∞}(ℕ^{2}_{≤}) is isomorphic to the semidirect product
of the monoid PO^{+}_{∞}(ℕ^{2}_{≤}) of oriental monotone injective partial selfmaps of
ℕ^{2}_{≤} with cofinite domains and images by the group of the cyclic order two ℤ_{2}.
Also we describe the congruence σ on the semigroup PO_{∞}(ℕ^{2}_{≤}), which is generated by the natural order
≼ on the semigroup PO_{∞}(ℕ^{2}_{≤}): ασβ if and only if α
and β are comparable in (PO_{∞}(ℕ^{2}_{≤}),≼). We prove that quotient semigroup
PO^{+}_{∞}(ℕ^{2}_{≤})/σ is isomorphic to the free commutative monoid
FAM_{ω} over an infinite countable set and show that quotient semigroup
PO_{∞}(ℕ^{2}_{≤})/σ is isomorphic to the semidirect product of the free commutative monoid
FAM_{ω} over an infinite countable set by the cyclic group of order two ℤ_{2}.



December 29, 2016

O. Gutik


Topological property of the Taimanov semigroup

For any infinite cardinal κ we construct a semigroup Α_{κ} of cardinality κ such that the following conditions hold:
(i) every T_{1}topology τ on the semigroup Α_{κ} such that (Α_{κ},τ) is a semitopological
semigroup is discrete;
(ii) for every T_{1}topological semigroup S which contains Α_{κ} as a subsemigroup, Α_{κ} is a closed
subsemigroup of S;
(iii) every homomorphic nonisomorphic image of Α_{κ} is a zerosemigroup.



January 4, 2017

I. Posdniakova


On the monoid of monotone injective partial selfmaps of ℕ^{2}_{≤} with cofinite domains and images, II

Let ℕ^{2}_{≤} be the set ℕ^{2} with the partial order defining as a product of usual order ≤
on the set of positive integers ℕ. We study the semigroup PO_{∞}(ℕ^{2}_{≤}) of monotone injective partial
selfmaps of ℕ^{2}_{≤} having cofinite domain and image. We describe the natural partial order on the semigroup
PO_{∞}(ℕ^{2}_{≤}) and show that it coincides with the natural partial order which is induced from symmetric inverse monoid
I_{ℕ2} over the set ℕ^{2} onto the semigroup PO_{∞}(ℕ^{2}_{≤}).
We proved that the semigroup PO_{∞}(ℕ^{2}_{≤}) is isomorphic to the semidirect product
of the monoid PO^{+}_{∞}(ℕ^{2}_{≤}) of oriental monotone injective partial selfmaps of
ℕ^{2}_{≤} with cofinite domains and images by the group of the cyclic order two ℤ_{2}.
Also we describe the congruence σ on the semigroup PO_{∞}(ℕ^{2}_{≤}), which is generated by the natural order
≼ on the semigroup PO_{∞}(ℕ^{2}_{≤}): ασβ if and only if α
and β are comparable in (PO_{∞}(ℕ^{2}_{≤}),≼). We prove that quotient semigroup
PO^{+}_{∞}(ℕ^{2}_{≤})/σ is isomorphic to the free commutative monoid
FAM_{ω} over an infinite countable set and show that quotient semigroup
PO_{∞}(ℕ^{2}_{≤})/σ is isomorphic to the semidirect product of the free commutative monoid
FAM_{ω} over an infinite countable set by the cyclic group of order two ℤ_{2}.



January 11, 2017

T. Banakh


Topological semilattices with Hclosed maximal chains are absolutely Hclosed

We prove that each topological semilattice with Hclosed maximal chains is absolutely Hclosed,
thus resolving an open problem posed by Oleg Gutik in 2010.
Also we shall discuss some remaining open problems related to Hclosed topological semilattices.
This is a joint work with Serhiy Bardyla.



February 13, 2017

O. Gutik


On feebly compact shiftcontinuous topologies on the semilattice exp_{n}λ

We discuss feebly compact topologies τ on the semilattice (exp_{n}λ,∩) such that
(exp_{n}λ,τ) is a semitopological semilattice and prove that for any shiftcontinuous T_{1}topology τ on
exp_{n}λ the following conditions are equivalent:
(i) τ is countably pracompact; (ii) τ is feebly compact; (iii) τ is dfeebly compact.
This is a joint work with Oleksandra Sobol.



February 20, 2017

S. Bardyla


On completeness and Hclosedness of topological semilattices

We consider an interplay between completeness and Hclosedness of topological semilattices. We give some sufficient conditions for topological semilattice
to be absolutely Hclosed and posed some open problems.
This is a joint work with Taras Banakh.



March 6, 2017

T. Banakh


kComplete topological semilattices are θmulticlosed and hence absolute Hclosed

A topological semilattice X is kcomplete 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 kcomplete topological semilattice X
is θmulticlosed, which means that for any continuous multimorphism Φ: X⊸Y with θclosed values to a topological
semilattice Y has θclosed image Φ(X). A subset A of a topological space
X is θclosed if each point x∈X∖A has a closed neighborhood
O_{x} disjoint with A.



March 13, 2017

T. Banakh


kComplete topological semilattices are θmulticlosed and hence absolute Hclosed, II

A topological semilattice X is kcomplete 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 kcomplete topological semilattice X
is θmulticlosed, which means that for any continuous multimorphism Φ: X⊸Y with θclosed values to a topological
semilattice Y has θclosed image Φ(X). A subset A of a topological space
X is θclosed if each point x∈X∖A has a closed neighborhood
O_{x} disjoint with A.



March 20, 2017

T. Banakh


kComplete topological semilattices are θmulticlosed and hence absolute Hclosed, III

A topological semilattice X is kcomplete 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 kcomplete topological semilattice X
is θmulticlosed, which means that for any continuous multimorphism Φ: X⊸Y with θclosed values to a topological
semilattice Y has θclosed image Φ(X). A subset A of a topological space
X is θclosed if each point x∈X∖A has a closed neighborhood
O_{x} disjoint with A.



April 3, 2017

O. Gutik, A. Savchuk


On the Bezushchak semigroup ID_{∞}

We show that the Bezushchak semigroup ID_{∞} of partial cofinite 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 cofinite 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 21, 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 21, 2017

I. Stasyuk


On simultaneous extensions of functions and metrics

We discuss on the topics in the title.



July 31, 2017

O. Gutik


On feebly compact semitopological symmetric inverse semigroups of a bounded finite rank

Given a positive integer number $n$, we discuss feebly compact $T_1$topologies $\tau$ on the symmetric inverse semigroup $\mathcal{I}_\lambda^n$ of finite
transformations of the rank $\leqslant n$ such that $(\mathcal{I}_\lambda^n,\tau)$ is a semitopological semigroup. For any positive integer $n\geqslant2$ and any
infinite cardinal $\lambda$ a Hausdorff countably pracompact noncompact shiftcontinuous topology on $\mathcal{I}_\lambda^n$ is constructed. We show that for an
arbitrary positive integer $n$ and an arbitrary infinite cardinal $\lambda$ for a $T_1$topology $\tau$ on $\mathcal{I}_\lambda^n$ the following conditions are
equivalent: (i) $\tau$ is countably pracompact; (ii) $\tau$ is feebly compact; (iii) $\tau$ is dfeebly compact; (iv) $(\mathcal{I}_\lambda^n,\tau)$
is an Hclosed space. Also we prove that for an arbitrary positive integer $n$ and an arbitrary infinite cardinal $\lambda$ every shiftcontinuous semiregular feebly
compact $T_1$topology $\tau$ on $\mathcal{I}_\lambda^n$ is compact.

