Perfectly supportable semigroups are σ-discrete in each Hausdorff shift-invariant topology

Банах, Тарас; Guran, Igor

doi:10.2478/taa-2013-0001

Cited by 4 publications

(5 citation statements)

References 3 publications

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“…Corollary 6.3 answers a problem posed in [19]. In [2] this corollary is generalized to so-called perfectly supportable semigroups.…”

Section: The Topology Z ′′mentioning

confidence: 73%

Algebraically determined topologies on permutation groups

Banakh,

Guran,

Protasov

2012

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In this paper we answer several questions of Dikran Dikranjan about algebraically determined topologies on the groups S(X) (and Sω(X) ) of (finitely supported) bijections of a set X. In particular, confirming conjecture of Dikranjan, we prove that the topology Tp of pointwise convergence on each subgroup G ⊃ Sω(X) of S(X) is the coarsest Hausdorff group topology on G (more generally, the coarsest T 1 -topology which turns G into a [semi]topological group), and Tp coincides with the Zariski and Markov topologies Z G and M G on G. Answering another question of Dikranjan, we prove that the centralizer topology T G on the symmetric group G = S(X) is discrete if and only if |X| ≤ c. On the other hand, we prove that for a subgroup G ⊃ Sω(X) of S(X) the centralizer topology T G coincides with the topologies Tp = M G = Z G if and only of G = Sω(X). We also prove that the group Sω(X) is σ-discrete in each Hausdorff shift-invariant topology.

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“…Corollary 6.3 answers a problem posed in [19]. In [2] this corollary is generalized to so-called perfectly supportable semigroups.…”

Section: The Topology Z ′′mentioning

confidence: 73%

Algebraically determined topologies on permutation groups

Banakh,

Guran,

Protasov

2012

Preprint

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“…Choose any finite set F ⊂ G such that the restriction h|F : F → F ′ is a bijective map. Then for any points Example 6.6 and Theorem 8.5 yield a measure-theoretic proof of the following known fact (for an alternative proof see [4] and [3]).…”

Section: The Notions Of Solecki Null One and Positive Sets Have Right...mentioning

confidence: 91%

The Solecki submeasures and densities on groups

Banakh

2012

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By definition, the right Solecki density σ R (resp. the Solecki submeasure σ) on a group G is the invariant monotone (subadditive) function assigning to each subset). In this paper we study the properties of the Solecki submeasures and Solecki densities on (topological) groups and establish an interplay between the Solecki submeasure σ and the Haar measure λ on a compact topological group G. In particular, we prove that that every subsetSo, λ and σ coincide on the family of all closed subsets of G and hence the Haar measure λ is completely determined by the Solecki submeasure σ. On the other hand, for any amenable group G the right Solecki density σ R coincides with the upper Banach density d * well-known in Combinatorics of Groups. The right Solecki density yields a convenient tool for studying the difference sets AA −1 and sumsets AB of subsets A, B in groups. Generalizing results of Jin, Beiglböck, Bergelson and Fish, for any subsets A, B ⊂ G of positive right Solecki density σ R (A) and σ R (B) in an amenable group G we prove that (1) G = F AA −1 for some set F ⊂ G of cardinality |F | ≤ 1/σ R (A), (2) the sets AA −1 BB −1 and ABB −1 A −1 contain some Bohr open subset U ∋ 1 G of G, (3) B −1 AA −1 contains some non-empty Bohr open set U in G, (4) AA −1 ⊃ U \ N for some Bohr open set U ∋ 1 G in G and some set N ⊂ G with σ R (N ) = 0, (5) AB ⊃ U ∩ T for some non-empty Bohr open set U in G and some set T ⊂ G with σ R (T ) = 1.

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“…By [2] or [3, 6.1], for every cardinal κ the group Sym fin (κ) is σ-discrete in each shift-invariant Hausdorff topology on Sym fin (κ). The same fact is true for the alternating group Alt(κ).…”

Section: Propositionmentioning

confidence: 99%

“…We say that a topological space X is σ-discrete if X can be written as a countable union of discrete subspaces. By [2] or [3, 6.1], for every cardinal κ the group Sym fin (κ) is σ-discrete in each shift-invariant Hausdorff topology on Sym fin (κ). The same fact is true for the alternating group Alt(κ).…”

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confidence: 99%