H-closed quasitopological groups

Bardyla, Serhii; Гутік, Олег; Ravsky, Alex

doi:10.1016/j.topol.2016.12.003

Cited by 9 publications

(16 citation statements)

References 8 publications

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“…The study of e:pTG-closed paratopological groups was initiated by Banakh and Ravsky [13,14], who called them H-closed paratopological groups. In [2,[15][16][17] Hausdorff e: TS-closed (resp. h: TS-closed) topological semigroups are called (absolutely) H-closed.…”

Section: Problemmentioning

confidence: 99%

“…(Banakh, Ravsky). For an Abelian topological group X the following conditions are equivalent: (1) X is e:pTG-closed; (2) X is e: TS-closed; (3) X is e:pTS-closed; (4) X is complete and has compact exponent; (5) X is complete and for every injective continuous homomorphism f : X → Y to a topological group Y the group f (X)/ f (X) is periodic.…”

Section: Problemmentioning

confidence: 99%

“…(2) If X is e: TS-closed, then for any closed normal subgroup N ⊂ X the center Z(X/N) of the quotient topological group X/N has precompact exponent.…”

Section: Theoremmentioning

confidence: 99%

“…For a nilpotent topological group X the following conditions are equivalent: (1) X is complete and has compact exponent; (2) X is e: TS-closed; (3) X is e:pTS-closed.…”

Section: Theoremmentioning

confidence: 99%

“…For a solvable topological group X the following conditions are equivalent: (1) X is compact; (2) X is balanced, locally compact, and h: TG-closed; (3) X is balanced, MAP-solvable and h: TG-closed.…”

Section: Theoremmentioning

confidence: 99%

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Categorically Closed Topological Groups

Банах

2017

Axioms

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Let C be a category whose objects are semigroups with topology and morphisms are closed semigroup relations, in particular, continuous homomorphisms. An object X of the category C is called C-closed if for each morphism Φ ⊂ X × Y in the category C the image Φ(X) = {y ∈ Y : ∃x ∈ X (x, y) ∈ Φ} is closed in Y. In the paper we survey existing and new results on topological groups, which are C-closed for various categories C of topologized semigroups.

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Section: Problemmentioning

confidence: 99%

Section: Problemmentioning

confidence: 99%

“…(2) If X is e: TS-closed, then for any closed normal subgroup N ⊂ X the center Z(X/N) of the quotient topological group X/N has precompact exponent.…”

Section: Theoremmentioning

confidence: 99%

“…For a nilpotent topological group X the following conditions are equivalent: (1) X is complete and has compact exponent; (2) X is e: TS-closed; (3) X is e:pTS-closed.…”

Section: Theoremmentioning

confidence: 99%

Section: Theoremmentioning

confidence: 99%

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Categorically Closed Topological Groups

Банах

2017

Axioms

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Characterizing chain-compact and chain-finite topological semilattices

Банах

Bardyla

2018

Semigroup Forum

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In the paper we present various characterizations of chain-compact and chain-finite topological semilattices. A topological semilattice X is called chain-compact (resp. chain-finite) if each closed chain in X is compact (finite). In particular, we prove that a (Hausdorff) T1-topological semilattice X is chain-finite (chain-compact) if and only if for any closed subsemilattice Z ⊂ X and any continuous homomorphism h :1991 Mathematics Subject Classification. 22A26; 54D30; 54D35; 54H12.

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Absolutely closed semigroups

Banakh,

Bardyla

2023

Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat.

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Let $${\mathcal {C}}$$ C be a class of topological semigroups. A semigroup X is called absolutely $${\mathcal {C}}$$ C -closed if for any homomorphism $$h:X\rightarrow Y$$ h : X → Y to a topological semigroup $$Y\in {\mathcal {C}}$$ Y ∈ C , the image h[X] is closed in Y. Let $$\textsf {T}_{\!\textsf {1}}\textsf {S}$$ T 1 S , $$\textsf {T}_{\!\textsf {2}}\textsf {S}$$ T 2 S , and $$\textsf {T}_{\!\textsf {z}}\textsf {S}$$ T z S be the classes of $$T_1$$ T 1 , Hausdorff, and Tychonoff zero-dimensional topological semigroups, respectively. We prove that a commutative semigroup X is absolutely $$\textsf {T}_{\!\textsf {z}}\textsf {S}$$ T z S -closed if and only if X is absolutely $$\textsf {T}_{\!\textsf {2}}\textsf {S}$$ T 2 S -closed if and only if X is chain-finite, bounded, group-finite and Clifford + finite. On the other hand, a commutative semigroup X is absolutely $$\textsf {T}_{\!\textsf {1}}\textsf {S}$$ T 1 S -closed if and only if X is finite. Also, for a given absolutely $${\mathcal {C}}$$ C -closed semigroup X we detect absolutely $${\mathcal {C}}$$ C -closed subsemigroups in the center of X.

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H-closed quasitopological groups

Cited by 9 publications

References 8 publications

Categorically Closed Topological Groups

Categorically Closed Topological Groups

Characterizing chain-compact and chain-finite topological semilattices

Absolutely closed semigroups

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