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- Title
ON A SEMITOPOLOGICAL SEMIGROUP B<sup>F</sup><sub>ω</sub> WHEN A FAMILY F CONSISTS OF INDUCTIVE NON-EMPTY SUBSETS OF ω.
- Authors
GUTIK, O. V.; MYKHALENYCH, M. S.
- Abstract
Let BFω be the bicyclic semigroup extension for the family F of ω-closed subsets of ω which is introduced in [19]. We study topologizations of the semigroup BFω for the family F of inductive ω-closed subsets of ω. We generalize Eberhart-Selden and Bertman-West results about topologizations of the bicyclic semigroup [6,12] and show that every Hausdorff shift-continuous topology on the semigroup BFω is discrete and if a Hausdorff semitopological semigroup S contains BFω as a proper dense subsemigroup then S\BFω is an ideal of S. Also, we prove the following dichotomy: every Hausdorff locally compact shift-continuous topology on BFω with an adjoined zero is either compact or discrete. As a consequence of the last result we obtain that every Hausdorff locally compact semigroup topology on BFω with an adjoined zero is discrete and every Hausdorff locally compact shift-continuous topology on the semigroup BFω⊔ I with an adjoined compact ideal I is either compact or the ideal I is open, which extends many results about locally compact topologizations of some classes of semigroups onto extensions of the semigroup BFω.
- Subjects
GROUP theory; SET theory; ALGEBRAIC topology; DISCRETE groups; RING extensions (Algebra)
- Publication
Matematychni Studii, 2023, Vol 59, Issue 1, p20
- ISSN
1027-4634
- Publication type
Article
- DOI
10.30970/ms.59.1.20-28