On the absolutes of compact spaces with a minimally acting group

Bandlow, Ingo

doi:10.1090/s0002-9939-1994-1246512-x

Cited by 2 publications

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“…Then, by condition (3) we get ∧ = 0 for some ∈ X . This completes the proof of condition (4). By this condition, for every ≤ we can choose ∈ X in such a way that…”

Section: Proposition 22mentioning

confidence: 57%

“…They proved that if B(S) is the clopen algebra of the phase space of the universal minimal dynamical system over a semigroup S (see [2] for definitions) and B(S) is atomless and G is either concellative or has a minimal left ideal or is commutative, then B(S) is a Cohen algebra. In particular, if S is a countable group, then B(S) is a complete Boolean algebra which admits a countable group of automorphisms acting minimally on it (see also Bandlow [4], Turek [14], Geschke [6]). …”

Section: Remark 38mentioning

confidence: 99%

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Boolean algebras admitting a countable minimally acting group

Błaszczyk¹,

Kucharski²,

Turek³

2014

Open Mathematics

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The aim of this paper is to show that every infinite Boolean algebra which admits a countable minimally acting group contains a dense projective subalgebra. MSC:54B35, 54A05, 52A01, 54D30 Regular and relatively complete subalgebrasAll Boolean algebras considered here are assumed to be infinite. Boolean algebraic notions, excluding symbols for Boolean operations, follow the Koppelberg's monograph [9]. In particular, if (B ∧ ∨ − 0 1) is a Boolean algebra, then B + = B \ {0} denotes the set of all non-zero elements of B. A set A ⊆ B is a subalgebra of the Boolean algebra B, A ≤ B for short, if 1 0 ∈ A and A is closed under Boolean operations or, equivalently, − ∈ A for all ∈ A. We shall write A ∼ = B whenever A and B are isomorphic Boolean algebras. A non-empty set X ⊆ B + is called a partition of B whenever ∧ = 0 for distinct ∈ X and B X = 1 *

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“…Then, by condition (3) we get ∧ = 0 for some ∈ X . This completes the proof of condition (4). By this condition, for every ≤ we can choose ∈ X in such a way that…”

Section: Proposition 22mentioning

confidence: 57%

Section: Remark 38mentioning

confidence: 99%

Boolean algebras admitting a countable minimally acting group

Błaszczyk¹,

Kucharski²,

Turek³

2014

Open Mathematics

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“…Generalizing Theorem 1.1, Bandlow showed in [Ba] that if an -bounded group G (every maximal system of pairwise disjoint translates of a neighbourhood in G is countable) has an infinite minimal flow, then the phase space of has the same Gleason cover as for some infinite cardinal (for a simplified proof, see [T2]). Błaszczyk, Kucharski, and Turek demonstrated in [BKT] that every infinite minimal flow of such a group maps irreducibly onto for some infinite .…”

Section: Introductionmentioning

confidence: 99%

Universal minimal flows of extensions of and by compact groups

Bartošová

2022

Ergod. Th. Dynam. Sys.

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Every topological group G has, up to isomorphism, a unique minimal G-flow that maps onto every minimal G-flow, the universal minimal flow $M(G).$ We show that if G has a compact normal subgroup K that acts freely on $M(G)$ and there exists a uniformly continuous cross-section from $G/K$ to $G,$ then the phase space of $M(G)$ is homeomorphic to the product of the phase space of $M(G/K)$ with K. Moreover, if either the left and right uniformities on G coincide or G is isomorphic to a semidirect product $G/K\ltimes K$ , we also recover the action, in the latter case extending a result of Kechris and Sokić. As an application, we show that the phase space of $M(G)$ for any totally disconnected locally compact Polish group G with a normal open compact subgroup is homeomorphic to a finite set, the Cantor set $2^{\mathbb {N}}$ , $M(\mathbb {Z})$ , or $M(\mathbb {Z})\times 2^{\mathbb {N}}.$

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On the absolutes of compact spaces with a minimally acting group

Abstract: Abstract.If an w-bounded group G acts continuously on a compact Hausdorff space X and the orbit of every point is dense in X , then X iscoabsolute to a Cantor cube.

Cited by 2 publications

References 4 publications

Boolean algebras admitting a countable minimally acting group

Boolean algebras admitting a countable minimally acting group

Universal minimal flows of extensions of and by compact groups

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