Categorically compact topological groups

Dikranjan, Dikran; Uspenskij, Vladimir V.

doi:10.1016/s0022-4049(96)00139-9

Cited by 46 publications

(45 citation statements)

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“…In [36] Raikov proved that a topological group G is absolutely closed if and only if it is Raikov complete, i.e., G is complete with respect to the two-sided uniformity. A topological group G is called h-complete if for every continuous homomorphism h : G → H the subgroup f (G) of H is closed [13]. In our terminology such topological groups are called absolutely H-closed in the class of topological groups.…”

Section: ])mentioning

confidence: 99%

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On a complete topological inverse polycyclic monoid

Bardyla

Гутік

2016

Carpathian Math. Publ.

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We give sufficient conditions when a topological inverse λ-polycyclic monoid P λ is absolutely Hclosed in the class of topological inverse semigroups. For every infinite cardinal λ we construct the coarsest semigroup inverse topology τ mi on P λ and give an example of a topological inverse monoid S which contains the polycyclic monoid P 2 as a dense discrete subsemigroup.Key words and phrases: inverse semigroup, bicyclic monoid, polycyclic monoid, free monoid, semigroup of matrix units, topological semigroup, topological inverse semigroup, minimal topology.Ivan Franko National University, 1 Universytetska str., 79000, Lviv, Ukraine E-mail: sbardyla@yahoo.com (Bardyla S.O.), o_gutik@franko.lviv.ua, ovgutik@yahoo.com (Gutik O.V.) In this paper all topological spaces will be assumed to be Hausdorff. We shall follow the terminology of [10,12,16,31]. If A is a subset of a topological space X, then we denote the closure of the set A in X by cl X (A). By N we denote the set of all positive integers and by ω the first infinite cardinal.A semigroup S is called an inverse semigroup if every a in S possesses a unique inverse, i.e. if there exists a unique element a −1 in S such thatA map that associates to any element of an inverse semigroup its inverse is called the inversion.A band is a semigroup of idempotents. If S is a semigroup, then we shall denote the subset of idempotents in S by E(S). If S is an inverse semigroup, then E(S) is closed under multiplication. The semigroup operation on S determines the following partial order on E(S): e f if and only if e f = f e = e. This order is called the natural partial order on E(S). A semilattice is a commutative semigroup of idempotents. A semilattice E is called linearly ordered or a chain if its natural order is a linear order. A maximal chain of a semilattice E is a chain which is properly contained in no other chain of E. The Axiom of Choice implies the existence of maximal chains in any partially ordered set. According to [35, Definition II.5.12] a chain L is called ω-chain if L is order isomorphic to {0, −1, −2, −3, . . .} with the usual order . Let E be a semilattice and e ∈ E. We denote ↓e = { f ∈ E | f e} andIf S is a semigroup, then we shall denote by R, L , D and H the Green relations on S (see [17] The R-class (resp., L -, H -, or D-class) of the semigroup S which contains an element a of S will be denoted by R a (resp., L a , H a , or D a ).

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Section: ])mentioning

confidence: 99%

“…In our terminology such topological groups are called absolutely H-closed in the class of topological groups. The h-completeness is preserved under taking products and closed central subgroups [13]. H-closed paratopological and topological groups in the class of paratopological groups were studied in [37].…”

Section: ])mentioning

confidence: 99%

On a complete topological inverse polycyclic monoid

Bardyla

Гутік

2016

Carpathian Math. Publ.

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“…Our next example shows that this theorem does not generalize to solvable topological groups and thus resolves in negative Question 3.13 in [10] and Question 36 in [12]. Proof.…”

Section: Some Counterexamplesmentioning

confidence: 95%

“…e: TS-Closed and h: TS-closed topological semigroups were introduced in 1969 by Stepp [7,8] who called them maximal and absolutely maximal semigroups, respectively. The study of h: TG-closed, p: TG-closed and c: TG-closed topological groups (called h-complete, hereditarily h-complete and c-compact topological groups, respectively) was initiated by Dikranjan and Tonolo [9] and continued by Dikranjan, Uspenskij [10], see the monograph of Lukàcs [11] and survey ( [12] §4) of Dikranjan and Shakhmatov. The study of e:pTG-closed paratopological groups was initiated by Banakh and Ravsky [13,14], who called them H-closed paratopological groups.…”

Section: Problemmentioning

confidence: 99%

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Categorically Closed Topological Groups

Банах

2017

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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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Topological Features of Topological Groups

Tkachenko

2001

Handbook of the History of General Topology

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Categorically compact topological groups

Cited by 46 publications

References 17 publications

On a complete topological inverse polycyclic monoid

On a complete topological inverse polycyclic monoid

Categorically Closed Topological Groups

Topological Features of Topological Groups

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