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

Гутік, Олег; Khylynskyi, Pavlo

doi:10.1515/taa-2022-0130

Cited by 3 publications

(4 citation statements)

References 36 publications

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“…The obtained contradiction implies the statement of the lemma. By Lemma 3.16 of [24] there exists an open neighbourhood U (I) of the ideal I with the compact closure U (I).…”

Section: Proof It Is Obvious That τmentioning

confidence: 99%

“…This result was extended by Bardyla onto the a polycyclic monoid [8] and graph inverse semigroups [9], and by Mokrytskyi onto the monoid of order isomorphisms between principal filters of N n with adjoined zero [34]. In [24] the results of the paper [23] were extended to the monoid IN ∞ of all partial cofinite isometries of positive integers with adjoined zero. In [27] the similar dichotomy was proved for so called bicyclic extensions B F ω when a family F consists of inductive non-empty subsets of ω. Algebraic properties on a group G such that if the discrete group G has these properties then every locally compact shift continuous topology on G with adjoined zero is either compact or discrete studied in [33].…”

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confidence: 99%

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On locally compact shift continuous topologies on the semigroup $\boldsymbol{B}_{[0,\infty)}$ with an adjoined compact ideal

Gutik,

Khylynskyi

2024

Mat. Stud.

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Let $[0,\infty)$ be the set of all non-negative real numbers. The set $\boldsymbol{B}_{[0,\infty)}=[0,\infty)\times [0,\infty)$ with the following binary operation $(a,b)(c,d)=(a+c-\min\{b,c\},b+d-\min\{b,c\})$ is a bisimple inverse semigroup.In the paper we study Hausdorff locally compact shift-continuous topologies on the semigroup $\boldsymbol{B}_{[0,\infty)}$ with an adjoined compact ideal of the following tree types.The semigroup $\boldsymbol{B}_{[0,\infty)}$ with the induced usual topology $\tau_u$ from $\mathbb{R}^2$, with the topology $\tau_L$ which is generated by the natural partial order on the inverse semigroup $\boldsymbol{B}_{[0,\infty)}$, and the discrete topology are denoted by $\boldsymbol{B}^1_{[0,\infty)}$, $\boldsymbol{B}^2_{[0,\infty)}$, and $\boldsymbol{B}^{\mathfrak{d}}_{[0,\infty)}$, respectively. We show that if $S_1^I$ ($S_2^I$) is a Hausdorff locally compact semitopological semigroup $\boldsymbol{B}^1_{[0,\infty)}$ ($\boldsymbol{B}^2_{[0,\infty)}$) with an adjoined compact ideal $I$ then either $I$ is an open subset of $S_1^I$ ($S_2^I$) or the topological space $S_1^I$ ($S_2^I$) is compact. As a corollary we obtain that the topological space of a Hausdorff locally compact shift-continuous topology on $S^1_{\boldsymbol{0}}=\boldsymbol{B}^1_{[0,\infty)}\cup\{\boldsymbol{0}\}$ (resp. $S^2_{\boldsymbol{0}}=\boldsymbol{B}^2_{[0,\infty)}\cup\{\boldsymbol{0}\}$) with an adjoined zero $\boldsymbol{0}$ is either homeomorphic to the one-point Alexandroff compactification of the topological space $\boldsymbol{B}^1_{[0,\infty)}$ (resp. $\boldsymbol{B}^2_{[0,\infty)}$) or zero is an isolated point of $S^1_{\boldsymbol{0}}$ (resp. $S^2_{\boldsymbol{0}}$).Also, we proved that if $S_{\mathfrak{d}}^I$ is a Hausdorff locally compact semitopological semigroup $\boldsymbol{B}^{\mathfrak{d}}_{[0,\infty)}$ with an adjoined compact ideal $I$ then $I$ is an open subset of $S_{\mathfrak{d}}^I$.

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“…The obtained contradiction implies the statement of the lemma. By Lemma 3.16 of [24] there exists an open neighbourhood U (I) of the ideal I with the compact closure U (I).…”

Section: Proof It Is Obvious That τmentioning

confidence: 99%

mentioning

confidence: 99%

On locally compact shift continuous topologies on the semigroup $\boldsymbol{B}_{[0,\infty)}$ with an adjoined compact ideal

Gutik,

Khylynskyi

2024

Mat. Stud.

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“…Lemma 6 was proved in [17] and it shows that on the semigroup T with the F-property there exists a Hausdorff compact shift-continuous topology τ Ac . Lemma 6 ([17]).…”

Section: Definition 1 ([9]mentioning

confidence: 99%

“…However, this dichotomy does not hold for the McAlister semigroup M 2 and moreover, M 2 admits continuum many different Hausdorff locally compact inverse semigroup topologies [5]. Also, different locally compact semitopological semigroups with zero were studied in [16,17,26].…”

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 Locally Compact Shift-Continuous Topologies on Semigroups C+(a;b) and C

Gutik

2024

BMJ

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Let $\mathscr{C}_{+}(a,b)$ and $\mathscr{C}_{-}(a,b)$ be upper and down subsemigroups of the bicyclic semigroup defined in \cite{Makanjuola-Umar=1997}. Let $\mathscr{C}_{+}(p,q)^0$ and $\mathscr{C}_{-}(p,q)^0$ be the semigroups $\mathscr{C}_{+}(a,b)$ and $\mathscr{C}_{-}(a,b)$ with the adjoined zero. We show that the semigroups $\mathscr{C}_{+}(p,q)^0$ and $\mathscr{C}_{-}(p,q)^0$ admit continuum many different Hausdorff locally compact shift-continuous topologies up to topological isomorphism.

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

Cited by 3 publications

References 36 publications

On locally compact shift continuous topologies on the semigroup $\boldsymbol{B}_{[0,\infty)}$ with an adjoined compact ideal

On locally compact shift continuous topologies on the semigroup $\boldsymbol{B}_{[0,\infty)}$ with an adjoined compact ideal

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

On Locally Compact Shift-Continuous Topologies on Semigroups C+(a;b) and C

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