On the semigroup BωF which is generated by the family F of atomic subsets of ω

Oleg Gutik, Oleksandra Lysetska


We study the semigroup BωF, which is introduced in [O. Gutik and M. Mykhalenych, On some generalization of the bicyclic monoid, Visnyk Lviv. Univ. Ser. Mech.-Mat. 90 (2020), 5-19], in the case when the family F of subsets of cardinality 1 in ω. We show that BωF is isomorphic to the  subsemigroup Bω(Fmin) of the Brandt ω-extension of the semilattice Fmin and describe all shift-continuous feebly compact T1-topologies on the semigroup Bω(Fmin). In particulary we prove that every shift-continuous feebly compact T1-topology τ on Bω(Fmin) is compact and moreover in this case  the space (Bω(Fmin),τ) is homeomorphic to the one-point Alexandroff compactification of the discrete countable space Dω. We study the closure of BωF in a semitopological semigroup. In particularly we show that BωF is algebraically complete in the class of Hausdorff semitopological inverse semigroups with continuous inversion, and a Hausdorff topological inverse semigroup BωF is closed in any Hausdorff topological semigroup if and only if the band E(BωF) is compact.

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DOI: http://dx.doi.org/10.30970/vmm.2021.92.034-050


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