On locally compact shift continuous topologies on the semigroup \boldsymbolB_[0,∞) with an adjoined compact ideal

2024/01/12 by Олег Гутік, Gutik, Oleg, Markian Khylynskyi +1
Mathematics · Decision Sciences · #Advanced Topology and Set Theory #Fuzzy and Soft Set Theory #Advanced Banach Space Theory

paper · pdf · doi:10.48550/arxiv.2401.06636

Abstract

Let \boldsymbolB_[0,∞) be the semigroup which is defined in the Ahre paper \citeAhre=1981. The semigroup \boldsymbolB_[0,∞) with the induced usual topology τ_u from ℝ², with the topology τ_L which is generated by the natural partial order on \boldsymbolB_[0,∞), and the discrete topology are denoted by \boldsymbolB¹_[0,∞), \boldsymbolB²_[0,∞), and \boldsymbolB^\mathfrakd_[0,∞), respectively. We show that if S₁^I (S₂^I) is a Hausdorff locally compact semitopological semigroup \boldsymbolB¹_[0,∞) (\boldsymbolB²_[0,∞)) with an adjoined compact ideal I then either I is an open subset of S₁^I (S₂^I) or the semigroup S₁^I (S₂^I) is compact. Also, we proved that if S_\mathfrakd^I is a Hausdorff locally compact semitopological semigroup \boldsymbolB^\mathfrakd_[0,∞) with an adjoined compact ideal I then I is an open subset of S_\mathfrakd^I.

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