Information on the thesis

In the PhD thesis we study topologizations of semigroups, whose algebraic properties are closed to the bicyclic monoid and the structure of the closure of such semigroups and groups in semitopological and topological semigroups. In particular, we consider the extended bicyclic semigroup, the bicyclic extension B(A) of a non-empty shift-set A of a linearly ordered group and variants of the bicyclic monoid and the extended bicyclic semigroup.

We prove that any variant C_m,n of the bicyclic monoid admits only the discrete Hausdorff shift-continuous topology. This is a generalization of well-known Eberhart’s and Selden’s results obtained for the bicyclic monoid. Also we show the following dichotomy: every Hausdorff locally compact shift-continuous topology on the bicyclic monoid with an adjoined zero is either compact or discrete. We describe the adjoining of a compact ideal to an arbitrary variant of the bicyclic monoid C_m,n in a locally compact semitopological semigroup.

It is proved that the group of automorphisms of the extended bicyclic semigroup C_Z is isomorphic to the additive group of integers, all variants of C_Z are pairwise isomorphic, and the semigroup C_Z and all its variants are not finitely generated. We describe Hausdorff shift-continuous topologies on variants of C_Z, and show that there exist non-discrete Hausdorff semigroup topologies on variants of the extended bicyclic semigroup C_Z.

It is shown that for any countable linearly ordered group G and its non-empty shift-set A every Baire shift-continuous T₁-topology on B(A) is discrete, and for any linearly non-dense ordered group G every shift-continuous Hausdorff topology on B(A) is discrete as well.

We prove that every Hausdorff locally compact shift-continuous topology on a discrete electorally flexible infinite group with an adjoined zero $G^0$ is either compact or discrete. Also we show that on any virtually cyclic group with an adjoined zero G⁰ there exist non-discrete non-compact locally compact shift-continuous topologies which induce the discrete topology on G.