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

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

• 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, 61-80 (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.

• 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, 61-80 (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.

• On a topological α-bicyclic semigroup
  • We give a survey on results of Joseph W. Hogan and Annie A. Selden about topologizations of the α-bicyclic semigroup obtained in the papers:
    1. J. W. Hogan, The α-bicyclic semigroup as a topological semigroup, Semigroup Forum 28 (1984), 265-271, (Zbl 0531.22003).
    2. A. A. Selden, A nonlocally compact nondiscrete topology for the α-bicyclic semigroup, Semigroup Forum 31 (1985), 372-374, (Zbl 0567.22002).
    3. J. W. Hogan, Hausdorff topologies on the α-bicyclic semigroup, Semigroup Forum 36 (1987), 189-209, (Zbl 0626.22002).
    Some questions on this topics will be posed.

• 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.

• On a topological α-bicyclic semigroup, II
  • We give a survey on results of Joseph W. Hogan and Annie A. Selden about topologizations of the α-bicyclic semigroup obtained in the papers:
    1. J. W. Hogan, The α-bicyclic semigroup as a topological semigroup, Semigroup Forum 28 (1984), 265-271, (Zbl 0531.22003).
    2. A. A. Selden, A nonlocally compact nondiscrete topology for the α-bicyclic semigroup, Semigroup Forum 31 (1985), 372-374, (Zbl 0567.22002).
    3. J. W. Hogan, Hausdorff topologies on the α-bicyclic semigroup, Semigroup Forum 36 (1987), 189-209, (Zbl 0626.22002).
    Some questions on this topics will be posed.

• On τ-bounded topological spaces, uniform spaces and topological groups
  • We consider the notion of the τ-boundedness 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 τ-bounded if (X,U) is a τ-bounded uniform space. It is prove that the free topological group F(X) is τ-bounded if and only if the space X is τ-bounded.

• On τ-bounded topological spaces, uniform spaces and topological groups, II
  • We consider the notion of the τ-boundedness 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 τ-bounded if (X,U) is a τ-bounded uniform space. It is prove that the free topological group F(X) is τ-bounded if and only if the space X is τ-bounded.

• Three current open bounty problems from Mathematics Stack Exchange
  • The speaker talk about his advances in the solution of these problems. The first of them is from topological algebra, the second from measure theory and the third from the theory of finite metric spaces and graph theory.

• Three current open bounty problems from Mathematics Stack Exchange, II
  • The speaker talk about his advances in the solution of these problems. The first of them is from topological algebra, the second from measure theory and the third from the theory of finite metric spaces and graph theory.

• On the semilattices with compact maximal chains
  • We introduce some separation axioms in the class of topological semilattices and give a sufficient condition of H-closedness in that class.

• On L-separation axioms for topological semilattices
  • We introduce some L-separation axioms for topological semilattices and construct examples of topological semilattices which show that they are distinct.

• 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 H-closedness of a topological inverse polycyclic monoid in the class of topological inverse semigroups.

• 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 H-closedness of a topological inverse polycyclic monoid in the class of topological inverse semigroups.

• On the monoid of monotone injective partial selfmaps of ℕ²_≤ with cofinite domains and images
  • Let ℕ²_≤ be the set ℕ² with the partial order defining as a product of usual order ≤ on the set of positive integers ℕ. We study the semigroup PO_∞(ℕ²_≤) of monotone injective partial selfmaps of ℕ²_≤ having cofinite domain and image. We describe properties of elements of the semigroup PO_∞(ℕ²_≤) as monotone partial bijection of ℕ²_≤ and show that the group of units of PO_∞(ℕ²_≤) is isomorphic to the cyclic group of the order two. Also we describe the subsemigroup of idempotents of PO_∞(ℕ²_≤) and the Green relations on PO_∞(ℕ²_≤). In particularly we prove that D=J in PO_∞(ℕ²_≤).

• On a topological α-bicyclic monoid
  • We will consider the α-bicyclic monoid as a semitopological semigroup and construct some non discrete locally compact topologies on it.

• On a topological α-bicyclic monoid, II
  • We will consider the α-bicyclic monoid as a topological semigroup and construct some non discrete locally compact topologies on it.

• On semigroups of bijective monotone partial self-maps 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.

• On a topological α-bicyclic monoid, III
  • We will consider the α-bicyclic monoid as a topological semigroup and construct some non discrete locally compact topologies on it.

• 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)=(2-o(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 2-variable counting logic only with quantifier depth (1-o(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 (1-o(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 (2-o(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 CR-definable if the CR procedure distinguishes it from any non-isomorphic graph. Babai, Erdős, and Selkow showed that random graphs are CR-definable with high probability. We suggest an efficient characterization of the class of all CR-definable graphs. Using the last result, we prove that all CR-definable 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. 689-700. 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. 339-350. 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. 26-37. Springer, 2015.

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

• Про проблему тотожності слів у групах
  • Цикл доповідей буде присвяченим проблемі тотожності слів у групах.

• 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.

• Compact-like semilattices topologies on the semilattice exp_n(λ)
  • We discuss about compact-like 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).

• The extension I_λⁿ(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_λⁿ(S) be the extension of a semigroup S by symmetric inverse semigroup of a bounded finite rank n, where λ is any non-zero cardinal and n is an arbitrary positive integer ≤ λ. We describe all ideals of the semigroup I_λⁿ(S) up to modulo of a monoid S and show if the semigroup S has tight ideal series then I_λⁿ(S) has tight ideal series too.

• 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.

• Про проблему тотожності слів у групах, II
  • Цикл доповідей буде присвяченим проблемі тотожності слів у групах.

• On the dychotomy of a locally comapct semitopological λ-polycyclic monoid
  • For every non-zero cardinal λ we will consider the λ-polycyclic monoid P_λ as a semitopological semigroup and describe all locally compact topologies on P_λ.

• On the dychotomy of a locally comapct semitopological λ-polycyclic monoid, II
  • For every non-zero cardinal λ we will consider the λ-polycyclic monoid P_λ as a semitopological semigroup and describe all locally compact topologies on P_λ.

• Some topological properties of a semitopological α-bicyclic semigroup
  • We will consider locally compact topologizations of the α-bicyclic monoid as a semitopological semigroup and investigate some completions of a topological inverse α-bicyclic semigroup.

• On locally compact semitopological 0-bisimple inverse ω-semigroups
  • We discuss on the structure of Hausdorff locally compact semitopological 0-bisimple inverse ω-semigroups with a compact maximal subgroup. In particular, we show that a Hausdorff locally compact semitopological 0-bisimple inverse ω-semigroup with a compact maximal subgroup is either compact or topologically isomorphic to the topological sum of its H-classes.

• On locally compact semitopological 0-bisimple inverse ω-semigroups, II
  • We discuss on the structure of Hausdorff locally compact semitopological 0-bisimple inverse ω-semigroups with a compact maximal subgroup. In particular, we show that a Hausdorff locally compact semitopological 0-bisimple inverse ω-semigroup with a compact maximal subgroup is either compact or topologically isomorphic to the topological sum of its H-classes.

• On locally compact semitopological 0-bisimple inverse ω-semigroups, III
  • We discuss on the structure of Hausdorff locally compact semitopological 0-bisimple inverse ω-semigroups with a compact maximal subgroup. In particular, we show that a Hausdorff locally compact semitopological 0-bisimple inverse ω-semigroup with a compact maximal subgroup is either compact or topologically isomorphic to the topological sum of its H-classes.