Boolean algebras admitting a countable minimally acting group

Abstract

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 *

Universal minimal flows of extensions of and by compact groups

“…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 .…”