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

• Topological semigroups of cofinite monotone bijective partial transformations of positive integers
  • We show that the semigroup of partial cofinal monotone bijective transformations of the set of positive integers has algebraic properties similar to the bicyclic semigroup: it is bisimple and all of its non-trivial group homomorphisms are either isomorphisms or group homomorphisms. We also prove that every locally compact topology on the semigroup of partial cofinal monotone bijective transformations of the set of positive integers S such that S is a topological inverse semigroup is discrete and we describe the closure of S in a topological semigroup.

• On H-closed Clifford topological inverse semigroups
  • We show that an arbitrary Clifford topological inverse semigroup with an algebraic closed maximal subsemilattice and H-closed maximal subgroups is H-closed in the class of topological inverse semigroups.

• On H-closed inverse semigroup topologies on the semigroup of finite partial bijections of a bounded finite rank
  • We show that the topological inverse semigroup of finite partial bijections of a bounded finite rank I_λⁿ is H-closed if and only if its band E(I_λⁿ) is compact. Also we construct H-closed semigroup topology on I_λⁿ such that the band E(I_λⁿ) is a discrete subspace of I_λⁿ.

• On chains in H-closed topological pospaces
  • We study chains in an H-closed topological partially ordered space. We give sufficient conditions for a maximal chain L in an H-closed topological partially ordered space (H-closed topological semilattice) under which L contains a maximal (minimal) element. We also give sufficient conditions for a linearly ordered topological partially ordered space to be H-closed. We prove that a linearly ordered H-closed topological semilattice is an H-closed topological pospace and show that in general, this is not true. We construct an example of an H-closed topological pospace with a non-H-closed maximal chain and give sufficient conditions under which a maximal chain of an H-closed topological pospace is an H-closed topological pospace.

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

• On a metrizability of cancellative topological semigroups
  • We discuss on a metrizability of cancellative topological semigroups.

• Koch Problem on monothetic topological semigroups
  • We give the sufficient conditions on a locally compact topological semigroup under which the Pontryagin Alternative holds.

• On a finite symmetric inverse semigroup of bounded rank
  • We describe all congruences on a finite symmetric inverse semigroup of bounded rank and show that it is algebraically h-closed in the class in semitopological inverse semigroups with continuous inversion.

• On maximal almost disjoint families
  • The talk will be devoted to different types of maximal almost disjoint families. In particular, several constructions of strongly maximal almost disjoint families of functions will be presented.

• On compact topologies on a finite symmetric inverse semigrou of bounded rank
  • We describe all compact and countable compact Hausdorff topologies on a finite symmetric inverse semigroup of bounded rank I_λⁿ such that I_λⁿ is a semitopological semigroup.

• Continuously cancellative Clifford inverse topological semigroups.
  • We prove that a Clifform inverse topological semigroup S whose idempotent band E is a U-semilattice embeds into a compact Clifford topological inverse semigroup with zero-dimensional band if and only if
    1. E embeds into a zero-dimensional compact semilattice;
    2. the maximal subgroups H_e of S are totally bounded;
    3. S is continuously cancellative.
    A Clifford topological inverse semigroup S is called continuously cancellative is for each point x in S and a neighborhood O(x) there are neighborhoods U(x) and O(e) of the idempotent e=xx^-1 such that a point y of S belongs to the neighborhood O(x) provided yy^-1 lies in O(e) and ye lies in U(x).

• On a semigroup of almost monotone partial bijective transformations of positive intergers.
  • We discuss on algebraic properties of the semigroup of almost monotone partial bijective transformations of positive intergers ant its topologizations as semitopological and topological semigroups.