When a locally compact monothetic semigroup is compact

Zelenyuk, Yevhen; Zelenyuk, Yuliya

doi:10.1515/jgth-2019-0184

Cited by 3 publications

(2 citation statements)

References 8 publications

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“…From the other side, Zelenyuk in [53] constructed a countable monothetic locally compact topological semigroup without unit which is neither compact nor discrete and in [54] he constructed a monothetic locally compact topological monoid with the same property. The topological properties of monothetic locally compact (semi)topological semigroups studied in [3,20,55,56].…”

Section: Introductionmentioning

confidence: 99%

On a locally compact monoid of cofinite partial isometries of $\mathbb{N}$ with adjoined zero

Гутік¹,

Khylynskyi²

2021

Preprint

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Let C N be a monoid which is generated by the partial shift α : n → n+1 of the set of positive integers N and its inverse partial shift β : n+1 → n. In this paper we prove that if S is a submonoid of the monoid IN ∞ of all partial cofinite isometries of positive integers which contains C N as a submonoid then every Hausdorff locally compact shift-continuous topology on S with adjoined zero is either compact or discrete. Also we show that the similar statement holds for a locally compact semitopological semigroup S with an adjoined compact ideal.

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Section: Introductionmentioning

confidence: 99%

On a locally compact monoid of cofinite partial isometries of $\mathbb{N}$ with adjoined zero

Гутік¹,

Khylynskyi²

2021

Preprint

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“…From the other hand, Zelenyuk in [33] constructed a countable monothetic locally compact topological semigroup without an identity which is neither compact nor discrete and in [34] he constructed a monothetic locally compact topological monoid with the same property. The topological properties of monothetic locally compact (semi)topological semigroups are studied in [2,14,35,36]. In the paper [15] it is proved that every Hausdorff locally compact shift-continuous topology on the bicyclic monoid with an adjoined zero is either compact or discrete.…”

mentioning

confidence: 99%

On a semitopological semigroup $\boldsymbol{B}_{\omega}^{\mathscr{F}}$ when a family $\mathscr{F}$ consists of inductive non-empty subsets of $\omega$

Гутік

Mykhalenych

2023

Mat. Stud.

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Let $\boldsymbol{B}_{\omega}^{\mathscr{F}}$ be the bicyclic semigroup extension for the family $\mathscr{F}$ of ${\omega}$-closed subsets of $\omega$ which is introduced in \cite{Gutik-Mykhalenych=2020}.We study topologizations of the semigroup $\boldsymbol{B}_{\omega}^{\mathscr{F}}$ for the family $\mathscr{F}$ of inductive ${\omega}$-closed subsets of $\omega$. We generalize Eberhart-Selden and Bertman-West results about topologizations of the bicyclic semigroup \cite{Bertman-West-1976, Eberhart-Selden=1969} and show that every Hausdorff shift-continuous topology on the semigroup $\boldsymbol{B}_{\omega}^{\mathscr{F}}$ is discrete and if a Hausdorff semitopological semigroup $S$ contains $\boldsymbol{B}_{\omega}^{\mathscr{F}}$ as a proper dense subsemigroup then $S\setminus\boldsymbol{B}_{\omega}^{\mathscr{F}}$ is an ideal of $S$. Also, we prove the following dichotomy: every Hausdorff locally compact shift-continuous topology on $\boldsymbol{B}_{\omega}^{\mathscr{F}}$ 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 $\boldsymbol{B}_{\omega}^{\mathscr{F}}$ with an adjoined zero is discrete and every Hausdorff locally compact shift-continuous topology on the semigroup $\boldsymbol{B}_{\omega}^{\mathscr{F}}\sqcup I$ with an adjoined compact ideal $I$ is either compact or the ideal $I$ is open, which extent many results about locally compact topologizations of some classes of semigroups onto extensions of the semigroup $\boldsymbol{B}_{\omega}^{\mathscr{F}}$.

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On a locally compact monoid of cofinite partial isometries of ℕ with adjoined zero

Гутік

Khylynskyi

2022

Topological Algebra and Its Applications

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Let 𝒞ℕ be a monoid which is generated by the partial shift α : n↦n +1 of the set of positive integers ℕ and its inverse partial shift β : n + 1 ↦n. In this paper we prove that if S is a submonoid of the monoid Iℕ∞ of all partial cofinite isometries of positive integers which contains Cscr;ℕ as a submonoid then every Hausdorff locally compact shift-continuous topology on S with adjoined zero is either compact or discrete. Also we show that the similar statement holds for a locally compact semitopological semigroup S with an adjoined compact ideal.

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When a locally compact monothetic semigroup is compact

Cited by 3 publications

References 8 publications

On a locally compact monoid of cofinite partial isometries of $\mathbb{N}$ with adjoined zero

On a locally compact monoid of cofinite partial isometries of $\mathbb{N}$ with adjoined zero

On a semitopological semigroup $\boldsymbol{B}_{\omega}^{\mathscr{F}}$ when a family $\mathscr{F}$ consists of inductive non-empty subsets of $\omega$

On a locally compact monoid of cofinite partial isometries of ℕ with adjoined zero

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