Thesis for a Candidate Degree in Mathematics (PhD): Speciality 01.01.04 – Geometry and Topology. – Ivan Franko National University of Lviv, the Ministry of Education and Science of Ukraine, Lviv, 2021.
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 Cm,n of the bicyclic monoid admits only the discrete Hausdorff shift-continuous topology, and if a semigroup S contais Cm,n as a dense proper subsemigroup, then S \Cm,n is an ideal of S. 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 Cm,n in a locally compact semitopological semigroup.
It is proved that the group of automorphisms of the extended bicyclic semigroup CZ is isomorphic to the additive group of integers, all variants of CZ are pairwise isomorphic, and the semigroup CZ and all its variants are not finitely generated. We describe Hausdorff shift-continuous topologies on variants of CZ, and show that there exist non-discrete Hausdorff semigroup topologies on variants of the extended bicyclic semigroup CZ.
We present the construction which implies that there exists a continuum of distinct Hausdorff non-discrete non-compact locally compact shift-continuous topologies on the extended bicyclic semigroup with an adjoined zero C0Z = CZ U {0}. However, we show that every Hausdorff locally compact semigroup topology on C0Z is discrete.
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 shiftcontinuous 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 G0 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.
Keywords: semigroup, interassociate of a semigroup, semitopological semigroup, topological semigroup, bicyclic monoid, locally compact space, discrete space, bicyclic extension, Baire space, variant of a semigroup, extended bicyclic semigroup, group, electorally flexible group, electorally stable group, virtually cyclic group.
