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  • The active participants of the seminar were discuss new results and posed some open problems.

• Divertisement
  • The active participants of the seminar were discuss new results and posed some open problems.

• On closures in semitopological inverse semigroups with continuous inversion
  • We discuss on the closures of subgroups, semilattices and different kinds of semigroup extensions in semitopological inverse semigroups with continuous inversion. In particularly we show that a topological group G is H-closed in the class of semitopological inverse semigroups with continuous inversion if and only if G is compact, a Hausdorff linearly ordered topological semilattice E is H-closed in the class of semitopological semilattices if and only if E is H-closed in the class of topological semilattices, and a topological Brandt λ⁰-extension of S is (absolutely) H-closed in the class of semitopological inverse semigroups with continuous inversion if and only if so is S. Also, we construct an example of an H-closed semitopological semilattice in the class of semitopological semilattices which is not absolutely H-closed in the class of semitopological semilattices.

• On H-closed quasitopological groups
  • We should prove that each Cauchy compete Hausdorff quasitopological group is H-closed in the class of Hausdorff quasitopological groups. In particular, a Hausdorff topological group G is H-closed in the class of Hausdorff quasitopological groups iff G is Raikov complete.

• Cowide topologies and their applications
  • TBA

• On the semigroup of co-finite momotone partial bijections of the real line
  • We discusss on the structure of the semigroup of co-finite momotone partial bijections of the real line.

• On H-closed pospaces without infinite antichains
  • We describe the topology of an H-closed pospace without infinite antichains and show that an H-closed pospace without infinite antichains is monotone normal.

• An introduction to H-closed quasitopological groups
  • See the title. :-) Also we show that a topological group G is H-closed in the class of quasitopological groups if and only if G is Raikov complete and present some examples.

• On the k-polycyclic monoid
  • For every cardinal k we intriduce a notion of the k-polycyclic monoid and study it's topological properties as a topological inverse semigroup. For evere positive integer n we constract a minimal topology on the n-polycyclic monoid and prove that with this topology it is a H-complete topological inverse semigroup.

• On some asymptotic bounds for Ramsey numbers
  • We will discuss some new bounds for diagonal and off-diagonal Ramsey numbers.

• A natural uniformity on a paratopological group
  • We prove that each (Hausdorff) paratopological group G carries a natural (separated) uniformity turning G into a uniform quasi-topological group. This uniformity is generated by the base consisting of the sets {(x,y)∈ G×G: y∈ VxV^-1∩V^-1xV} where V runs over neighborhoods of the unit in G. The presence of this uniqformity guarantees that each Hausdorff paratopologial group G condences onto a Tychonoff quasi-topological group and so G is functionally Hausdorff. If G is countably Hausdorff, then G condences onto a metrizable quasi-topological group and hence is submetrizable. This resolves two probelsm of Arkhangelski and Tkachenko. More details can be found here.

• On an embedding of the polycyclic monoid
  • We prove that for every non zero cardinal k the k-Polycyclic monoid does not embed into any countably compact topologycal semigroup.

• Divertisement
  • We started a new spring session of our seminar from the Divertisment. The active participants of the seminar are discussed new results and they posed some open problems.

• Some remarks on topologizations of the polycyclic monoid
  • We built locally compact non discrete topology Ƭ on the k-polycyclic monoid P such that (P,Ƭ) is topologycal inverse semigroup. Also we proved, that Y\P is a ideal for every semitopological semigroup Y such that cl(P)=Y, and construct a compact topology Ʈ on P such that (P,Ʈ) is a semitopologycal semigroup.

• On boundedness of topological rings
  • We consider various types of boundedness in topological rings: precompact, τ-narrow, ets. Cardinal functions are investigated.

• On some problems on topological graph semigroups and infinite word semigroups
  • Some problems on topological graph semigroups and infinite word semigroups were posed.

• On boundedness of topological rings
  • We consider various types of boundedness in topological rings: precompact, τ-narrow, ets. Cardinal functions are investigated.

• On a completion of quasitopological grups
  • We consider a completion G* of a quasitopologycal group G and prove that G* is an H-closed quasitopologycal group. Moreover G is a dense subgroup of G*.

• Characteristic subsets of locally compact abelian topological groups
  • A subset $A$ of a locally compact abelian topological group $G$ we shall call *characteristic*, if there is no sequence $\{\phi_k\}$ of non-trivial characters of the group $G$ such that $\{\phi_k(a)\}$ converges to the unit for each element $x\in A$. Investigating the problem which subsets $A$ are characteristic, we obtain some beginning results and pose some problems and directions of investigation.

• On a completion of quasitopological grups
  • We consider a completion G* of a quasitopologycal group G and prove that G* is an H-closed quasitopologycal group. Moreover G is a dense subgroup of G*.

• On boundedness of topological rings
  • We consider various types of boundedness in topological rings: precompact, τ-narrow, ets. Cardinal functions are investigated.

• Про наболіле та вимучене. Bardyla Theorem.
  • Епіграф: "Видел Ленин пашни, избы деревень, видел он вчерашний и грядущий день!"
    (Це - не реклама комунізму, це - ностальгія за недоведеною Серьоженькою теоремою!)
    Inspired and filling the small gap by the first speaker, he and the second speaker proved that if (G,τ) is a Cauchy full quasitopological group, (G,σ) is a quasitopological group, σͻτ, and the space (G,σ) has a π-base, consisting of open in the space (G,τ) subets then (G,σ) is a Cauchy full quasitopological group too.

• k-Spaces, their generalizations and applications
  • We consider some generalizations of the k-space property and discuss their applications to free constructions of topological algebra.

• On metrization of functors
  • We discuss the problem of metrization of functional functors and functors with finite supports.

• Cardinal invariants distinguishing permutation groups
  • We prove that for infinite cardinals κ<λ the alternating group Alt(λ) (of even permutations) of λ is not embeddable into the symmetric group Sym(κ) (of all permutations) of κ. To prove this fact we introduce and study several monotone cardinal group invariants which take value κ on groups Alt(κ) and Sym(κ).

• On a completeness of factorizable topological inverse semigroups
  • We study a completeness of factorizable topological inverse semigroups and introduce a method how to construct such semigroups.

• Некоторые связи между топологией и динамикой
  • Мы рассмотрим некоторые связи между топологией дифференцируемого многообразия и свойствами индивидуальных диффеоморфизмов и групп диффеоморфизмов этого многообразия. Классические результаты такого рода относятся к свойствам периодических орбит. Примеры: формула следа Лефшеца, теория Нильсена. С развитием гиперболической динамики в последние полвека появились результаты, которые гарантируют существование сложных инвариантных множеств с весьма нетривиальной динамикой. В последние 10 лет обнаружились весьма удивительные свойства коммутативных групп диффеоморфизмов ранга больше единицы.
    Необходимые понятия из теории динамических систем будут объяснены.

• On the dichotomy of a locally compact semitopological bicyclic monoid with adjoined zero
  • We show that that if τ be a Hausdorff locally compact topology on the bicyclic semigroup with adjoined zero C ⁰ = C(p,q)∪{0} such that (C ⁰,τ) is a semitopological semigroup, then τ is either compact or discrete.

• Нові пригоди коня та віслючка
  • Віслючок розкаже про свою важку долю без коня. Він надибав яблучко та усе від нього відкушує, відкушує, але ніяк не проковтне. Віслючок тягнеться до яблучка та підстрибує, але треба коня, бо у мене не кінські сили, а лише віслючі.
    

• On the structure of the semigroup of co-finite momotone partial homeomorphisms of the real line
  • We describe the structure of the semigroup of co-finite momotone partial homeomorphisms of the real line.

• K-теорія: Q-конструкція Квилена
  • Переповідається Q-конструкція Квилена для точних категорій A. По такій будується категорія QA і її класифікуючий простір BQA. Вищі K-групи визначаються як гомотопічні групи BQA.

• On extent of weakly submeta-Lindelöf spaces
  • We discuss on the topic of the title.