• 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.

• We discuss on semitopological interassociates C_m,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 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 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 C_m,n with an adjoined zero 0 of the bicyclic monoid C⁰_m,n such that (C⁰_m,n,τ) is a semitopological semigroup, is either discrete or compact.

• We discuss on semitopological interassociates C_m,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 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 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 C_m,n with an adjoined zero 0 of the bicyclic monoid C⁰_m,n such that (C⁰_m,n,τ) is a semitopological semigroup, is either discrete or compact.

• The active participants of the seminar will discuss some open problems and possible ways of their solutions.

• The active participants of the seminar will discuss some open problems and possible ways of their solutions.

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

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

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

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

• 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 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 I_ℕ² over the set ℕ² 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 ℕ²_≤ with cofinite domains and images by the group of the cyclic order two ℤ₂. 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 FAM_ω over an infinite countable set and show that quotient semigroup PO_∞(ℕ²_≤)/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 ℤ₂.

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

• 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 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 I_ℕ² over the set ℕ² 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 ℕ²_≤ with cofinite domains and images by the group of the cyclic order two ℤ₂. 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 FAM_ω over an infinite countable set and show that quotient semigroup PO_∞(ℕ²_≤)/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 ℤ₂.

• 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.

• We discuss feebly compact topologies τ on the semilattice (exp_nλ,∩) such that (exp_nλ,τ) is a semitopological semilattice and prove that for any shift-continuous T₁-topology τ on exp_nλ 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.

• 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.

• 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_∞.

• 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_∞.

• We discuss topologizations and embeddings of graph inverse semigroups.

• We discuss topologizations and embeddings of graph inverse semigroups.

• 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."

• We discuss on the topics in the title.

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 non-compact shift-continuous 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 d-feebly compact; (iv) $(\mathcal{I}_\lambda^n,\tau)$ is an H-closed space. Also we prove that for an arbitrary positive integer $n$ and an arbitrary infinite cardinal $\lambda$ every shift-continuous semiregular feebly compact $T_1$-topology $\tau$ on $\mathcal{I}_\lambda^n$ is compact.