Proceedings of the International Geometry Center

ISSN-print: 2072-9812
ISSN-online: 2409-8906
ISO: 26324:2012
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On semigroups which admit only discrete left-continuous Hausdorff topology

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Published Sep 2, 2026
DOI https://doi.org/10.15673/pigc.3328

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Oleg Gutik
Ivan Franko National University of Lviv

Abstract

We give a sufficient condition under which every left-continuous (right-continuous) Hausdorff topology on a semigroup S is discrete. We construct a submonoid C+(a, b) (resp., C−(a, b)) of the bicyclic monoid which contains a family {Sα : α ∈ c} of continuum many subsemigroups with the following properties: (i) every left-continuous (resp., right-continuous) Hausdorff topology on Sα is discrete; (ii) every semigroup Sα admits a non-discrete right-continuous (resp., left-continuous) Hausdorff topology which is not left-continuous (resp., right-continuous); (iii) every semigroup Sα isomorphically embeds into a Hausdorff compact topological semigroup. Also we construct a submonoid C+ℤ (resp., C−ℤ) of the extended bicyclic semigroup which contains a family {Sα : α ∈ c} of continuum many subsemigroups with the above described properties.

Keywords:
Semitopological semigroup, topological semigroup, left topological semigroup, right topological semigroup, discrete topology, bicyclic monoid, compact, extended bicyclic semigroup, embedding

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How to Cite
Gutik, O. (2026). On semigroups which admit only discrete left-continuous Hausdorff topology. Proceedings of the International Geometry Center, 19(2), paper 8, 20 pages. https://doi.org/10.15673/pigc.3328
Issue
Vol 19 No 2 (2026)
Section
Papers
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