

September 9, 2015

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 16, 2015

T. Banakh,
I. Guran,
O. Ravsky


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Active participants of the seminar will present and discuss interesting open problems from various branches of the topological algebra.



September 23, 2015

O. Gutik


On semitopological bicyclic extensions of linearly ordered groups

We discus on topologizations of the bicyclic extensions B(G) and B^{+}(G) of a linearly ordered groups G, as semitopological semigroups
(the definitions of the semigroups see in [O. Gutik, D. Pagon, and K. Pavlyk, Congruences on bicyclic extensions of a linearly ordered group,
Acta Comment. Univ. Tartu. Math. 15 (2011), no. 2, 6180
(MR2961689, Zbl 1257.20059, arXiv:1111.2401)]). We show that for an arbitrary countable linearly ordered group
G every Hausdorff topology τ on B(G) (B^{+}(G)) such that (B(G),τ)
((B^{+}(G),τ)) is a Baire topological semigroup is discrete. Also we prove that for an arbitrary linearly ordered group
G which is not densely ordered every Hausdorff topologyτ on the semigroup B(G) (B^{+}(G)) such that
(B(G),τ) ((B^{+}(G),τ)) is a semitopological semigroup is discrete, and hence B(G)
(B^{+}(G)) is a discrete subspace of any semitopological semigroup which contains B(G)
(B^{+}(G)) as a subsemigroup.



September 30, 2015

O. Gutik


On semitopological bicyclic extensions of linearly ordered groups, II

We discus on topologizations of the bicyclic extensions B(G) and B^{+}(G) of a linearly ordered groups G, as
semitopological semigroups (the definitions of the semigroups see in [O. Gutik, D. Pagon, and K. Pavlyk, Congruences on bicyclic
extensions of a linearly ordered group, Acta Comment. Univ. Tartu. Math. 15 (2011), no. 2, 6180
(MR2961689, Zbl 1257.20059, arXiv:1111.2401)]). We show that for an arbitrary countable
linearly ordered group G every Hausdorff topology τ on B(G) (B^{+}(G)) such that (B(G),τ)
((B^{+}(G),τ)) is a Baire topological semigroup is discrete. Also we prove that for an arbitrary linearly ordered group
G which is not densely ordered every Hausdorff topologyτ on the semigroup B(G) (B^{+}(G)) such that
(B(G),τ) ((B^{+}(G),τ)) is a semitopological semigroup is discrete, and hence B(G)
(B^{+}(G)) is a discrete subspace of any semitopological semigroup which contains B(G)
(B^{+}(G)) as a subsemigroup.



October 7, 2015

S. Bardyla


On a topological αbicyclic semigroup

We give a survey on results of Joseph W. Hogan and Annie A. Selden about topologizations of the αbicyclic semigroup obtained in the papers:
 J. W. Hogan, The αbicyclic semigroup as a topological semigroup,
Semigroup Forum 28 (1984), 265271, (Zbl 0531.22003).
 A. A. Selden, A nonlocally compact nondiscrete topology for the αbicyclic semigroup,
Semigroup Forum 31 (1985), 372374, (Zbl 0567.22002).
 J. W. Hogan, Hausdorff topologies on the αbicyclic semigroup,
Semigroup Forum 36 (1987), 189209, (Zbl 0626.22002).
Some questions on this topics will be posed.



October 21, 2015

T. Banakh,
I. Guran,
O. Gutik,
M. Zarichny


A survey of scientific achievement of Oleksandr Ravsky for the last 40 years
(dedicated to his 40th Birthday)

We give a survey on results of Oleksandr Ravsky obtained in the following topics:
☺ Paratopological Groups;
☺ General Topology;
☺ Combinatorics;
☺ Topological and Semitopological Semigroups.



October 28, 2015

S. Bardyla


On a topological αbicyclic semigroup, II

We give a survey on results of Joseph W. Hogan and Annie A. Selden about topologizations of the αbicyclic semigroup obtained in the papers:
 J. W. Hogan, The αbicyclic semigroup as a topological semigroup,
Semigroup Forum 28 (1984), 265271, (Zbl 0531.22003).
 A. A. Selden, A nonlocally compact nondiscrete topology for the αbicyclic semigroup,
Semigroup Forum 31 (1985), 372374, (Zbl 0567.22002).
 J. W. Hogan, Hausdorff topologies on the αbicyclic semigroup,
Semigroup Forum 36 (1987), 189209, (Zbl 0626.22002).
Some questions on this topics will be posed.



November 4, 2015

I. Guran


On τbounded topological spaces, uniform spaces and topological groups

We consider the notion of the τboundedness in topological spaces and uniform spaces. Let U be the universal uniformity of a topological space X.
A topological spaces X
is said to be τbounded if (X,U) is a τbounded uniform space. It is prove that the free topological group F(X) is
τbounded if and only if the space X
is τbounded.



November 11, 2015

I. Guran


On τbounded topological spaces, uniform spaces and topological groups, II

We consider the notion of the τboundedness in topological spaces and uniform spaces. Let U be the universal uniformity of a topological space
X. A topological spaces X
is said to be τbounded if (X,U) is a τbounded uniform space. It is prove that the free topological group F(X) is
τbounded if and only if the space X
is τbounded.



November 18, 2015

A. Ravsky


Three current open bounty problems from Mathematics Stack Exchange



November 25, 2015

A. Ravsky


Three current open bounty problems from Mathematics Stack Exchange, II



December 9, 2015

S. Bardyla


On the semilattices with compact maximal chains

We introduce some separation axioms in the class of topological
semilattices and
give a sufficient condition of Hclosedness in that class.



December 16, 2015

S. Bardyla


On Lseparation axioms for topological semilattices

We introduce some Lseparation axioms for topological semilattices and construct examples of topological semilattices which show that they are distinct.



January 18, 2016

S. Bardyla


On the closure of the polycyclic monoid in a topological inverse semigroup.

We study the closure of polycyclic monoid in a topological inverse semigroup and find a criterium of Hclosedness of a topological inverse polycyclic monoid in the class
of topological inverse semigroups.



February 10, 2016

S. Bardyla


On the closure of the polycyclic monoid in a topological inverse semigroup, II.

We study the closure of polycyclic monoid in a topological inverse semigroup and find a criterium of Hclosedness of a topological inverse polycyclic monoid in the class of
topological inverse semigroups.



February 17, 2016

I. Pozdniakova


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 properties of elements of the semigroup
PO_{∞}(ℕ^{2}_{≤}) as monotone partial bijection of ℕ^{2}_{≤} and show that the
group of units of PO_{∞}(ℕ^{2}_{≤}) is isomorphic to the cyclic group of the order two. Also we describe the
subsemigroup of idempotents of PO_{∞}(ℕ^{2}_{≤}) and the Green relations on
PO_{∞}(ℕ^{2}_{≤}). In particularly we prove that
D=J in PO_{∞}(ℕ^{2}_{≤}).



February 24, 2016

S. Bardyla


On a topological αbicyclic monoid

We will consider the αbicyclic monoid as a semitopological semigroup and construct some non discrete locally compact topologies on it.



March 2, 2016

S. Bardyla


On a topological αbicyclic monoid, II

We will consider the αbicyclic monoid as a topological semigroup and construct some non discrete locally compact topologies on it.



March 9, 2016

O. Gutik


On semigroups of bijective monotone partial selfmaps of posets with cofinite domains and images

A short survey about results on the topics of the title discussed and reporter put some open problems on this topics.



March 16, 2016

S. Bardyla


On a topological αbicyclic monoid, III

We will consider the αbicyclic monoid as a topological semigroup and construct some non discrete locally compact topologies on it.



March 23, 2016

O. Verbitsky


Universal covers, integral polytopes, and graph isomorphism

I will present two recent results
that are motivated by applications in distributed computing
and isomorphism testing. Both results concern
the classical color refinement algorithm.
Given a connected graph G and its vertex x, let U(G,x) denote the
universal cover of G obtained by unfolding G into a tree starting from x.
Let T=T(n) be the minimum number such that, for graphs G and H with at most n vertices each,
the isomorphism of U(G,x) and U(H,y) surely follows from the isomorphism of these rooted trees truncated at depth T.
Norris [Discrete Appl. Math. 1995] asked if the value of T(n) is bounded by n.
We answer this question in the negative by establishing that T(n)=(2o(1))n.
The graphs G and H we construct for each n to prove the lower bound for
T(n) also show some other tight lower bounds. Both having n vertices, G and H can be distinguished
in 2variable counting logic only with quantifier depth (1o(1))n.
It follows that Color Refinement (CR), the classical procedure used
in isomorphism testing and other areas for computing the coarsest
equitable partition of a graph, needs (1o(1))n rounds to achieve
color stabilization on each of G and H. Somewhat surprisingly,
this number of rounds is not enough for
color stabilization on the disjoint union of G and H,
where (2o(1))n rounds are needed.
Exploring a linear programming approach to Graph Isomorphism,
Tinhofer defined the concept of a compact graph:
A graph is compact if the polytope of its fractional automorphisms is
integral. Tinhofer noted that isomorphism testing for compact graphs
can be done quite efficiently by linear programming. However, the
problem of characterizing and recognizing compact graphs in
polynomial time remains an open question.
We relate this approach to the color refinement algorithm.
We call a graph CRdefinable if the CR procedure
distinguishes it from any nonisomorphic graph.
Babai, Erdős, and Selkow showed that random graphs are
CRdefinable with high probability.
We suggest an efficient characterization of the class of
all CRdefinable graphs.
Using the last result, we prove that all CRdefinable graphs are compact.
In other words, the applicability range for Tinhofer's linear programming
approach to isomorphism testing is at least as large as for the combinatorial
approach based on color refinement.
This is joint work with A.Krebs [1] and V.Arvind, J.Köbler, and G.Rattan
[2,3].
[1] A.Krebs and O.Verbitsky. Universal covers, color refinement, and twovariable counting logic:
Lower bounds for the depth. Proc. LICS, pp. 689700. IEEE Press, 2015.
[2] V.Arvind, J.Köbler, G.Rattan, and O.Verbitsky. On the power of color
refinement.
Proc. FCT'15, LNCS 9210, pp. 339350. Springer, 2015.
[3] V.Arvind, J.Köbler, G.Rattan, and O.Verbitsky. On Tinhofer's
linear programming approach to isomorphism testing.
Proc. MFCS'15, LNCS 9235, pp. 2637. Springer, 2015.



April 6, 2016

O. Gutik


Divertissement

Active participants of the seminar will present and discuss interesting open problems from various branches of topological algebra and its applications.



April 13, 2016

I. Гуран


Про проблему тотожності слів у групах

Цикл доповідей буде присвяченим проблемі тотожності слів у групах.



April 20, 2016

S. Bardyla


Zariski topology on the topological semilattices

We will introduce a notion of Zariski topology on topological semilattices and
give necessary and sufficient conditions for Zariski topology to be compact.



April 27, 2016

O. Gutik


Compactlike semilattices topologies on the semilattice exp_{n}(λ)

We discuss about compactlike semilattices topologies on the semilattice exp_{n}(λ), i.e., the
semilattice of finite subsets of a bounded rank n of an arbirary infinite cardinal λ with the operation ∩. Particularly, we
describe all feebly compact Hausdorff semilattice topologies on exp_{n}(λ).



May 4, 2016

O. Gutik


The extension I_{λ}^{n}(S) of a semigroup S by symmetric inverse semigroup of a bounded finite
rank n preserves the property to have tight ideal series

Let I_{λ}^{n}(S) be the extension of a semigroup S by symmetric inverse semigroup of a bounded finite
rank n, where λ is any nonzero cardinal and n is an arbitrary positive integer ≤ λ. We describe all ideals of the semigroup
I_{λ}^{n}(S) up to modulo of a monoid S and show if the semigroup S has tight ideal series then
I_{λ}^{n}(S) has tight ideal series too.



May 11, 2016

О. Равський, O. Гутік


Відповідь на питання Salvo Tringali з MathOwerflow

Побудовано приклад гаусдорфової абельової топологічної напівтрупи S, яка не допускає гомоморфного ущільнення у гаусдорфову (пара)топологічну групу, таким чином
відповідь на питання Salvo Tringali з
MathOwerflow.
У якості напівгрупи S можна взяти вільну абельову напівгрупу над гаусдорфовим простором, який не є функціонально гаусдорфовим.



May 18, 2016

I. Гуран


Про проблему тотожності слів у групах, II

Цикл доповідей буде присвяченим проблемі тотожності слів у групах.



May 25, 2016

S. Bardyla


On the dychotomy of a locally comapct semitopological λpolycyclic monoid

For every nonzero cardinal λ we will consider the λpolycyclic monoid P_{λ} as
a semitopological semigroup and describe all locally compact topologies on P_{λ}.



June 1, 2016

S. Bardyla


On the dychotomy of a locally comapct semitopological λpolycyclic monoid, II

For every nonzero cardinal λ we will consider the λpolycyclic monoid P_{λ} as
a semitopological semigroup and describe all locally compact topologies on P_{λ}.



June 22, 2016

S. Bardyla


Some topological properties of a semitopological αbicyclic semigroup

We will consider locally compact topologizations of the αbicyclic monoid as a semitopological semigroup
and investigate some completions of a topological inverse αbicyclic semigroup.



July 27, 2016

O. Gutik


On locally compact semitopological 0bisimple inverse ωsemigroups

We discuss on the structure of Hausdorff locally compact semitopological 0bisimple inverse ωsemigroups with a
compact maximal subgroup. In particular, we show that a Hausdorff locally compact semitopological 0bisimple inverse
ωsemigroup with a compact maximal subgroup is either compact or topologically isomorphic to the topological sum of
its Hclasses.



July 29, 2016

O. Gutik


On locally compact semitopological 0bisimple inverse ωsemigroups, II

We discuss on the structure of Hausdorff locally compact semitopological 0bisimple inverse ωsemigroups with a
compact maximal subgroup. In particular, we show that a Hausdorff locally compact semitopological 0bisimple inverse
ωsemigroup with a compact maximal subgroup is either compact or topologically isomorphic to the topological sum of
its Hclasses.



August 17, 2016

O. Gutik


On locally compact semitopological 0bisimple inverse ωsemigroups, III

We discuss on the structure of Hausdorff locally compact semitopological 0bisimple inverse ωsemigroups with a
compact maximal subgroup. In particular, we show that a Hausdorff locally compact semitopological 0bisimple inverse
ωsemigroup with a compact maximal subgroup is either compact or topologically isomorphic to the topological sum of
its Hclasses.

