Bohuslav Balcar

Weak distributivity, a problem of Von Neumann and the mystery of measurability

�DedicatedtoDorothyMaharamStone This article investigates the weak distributivity of Booleano-algebras satisfying the countable chain condition. It addresses primarily the question when such algebras carry ao-additive measure. We use as a starting point the problem of John von Neumann stated in 1937 in the Scottish Book. He asked if the countable chain condition and weak distributivity are sufficient for the existence of such a measure. Subsequent research has shown that the problem has two aspects: one set theoretic and one combinatorial. Recent results provide a complete solution of both the set theoretic and the combinatorial problems. We shall survey the history of von Neumann’s Problem and outline the solution of the set theoretic problem. The technique that we describe owes much to the early work of Dorothy Maharam to whom we dedicate this article. §1. CompleteBooleanalgebrasandweakdistributivity. ABooleanalgebra

On minimal π-character of points in extremally disconnected compact spaces

Abstract We consider the relationship between π-character, refinement number (=weak density) and π-weight in complete Boolean algebras. As an application we shall show that every extremally disconnected compact space contains a point which is not an accumulation point of any countable discrete subset, provided that minimal π-character and π-weight coincide.

Structural properties of universal minimal dynamical systems for discrete semigroups

We show that for a discrete semigroup S there exists a uniquely de- termined complete Boolean algebra B(S) - the algebra of clopen subsets of M(S). M(S) is the phase space of the universal minimal dynamical system for S and it is an extremally disconnected compact Hausdor space. We deal with this connection of semigroups and complete Boolean algebras focusing on structural properties of these algebras. We show that B(S) is either atomic or atomless; that B(S) is weakly homogenous provided S has a minimal left ideal; and that for countable semigroups B(S) is semi-Cohen. We also present a class of what we call group-like semigroups that includes commutative semigroups, inverse semigroups, and right groups. The group reection G(S) of a group-like semigroup S can be constructed via universal minimal dynamical system for S and, moreover, B(S) and B(G(S)) are the same.

Semi-Cohen Boolean algebras☆

Abstract We investigate classes of Boolean algebras related to the notion of forcing that adds Cohen reals. A Cohen algebra is a Boolean algebra that is dense in the completion of a free Boolean algebra. We introduce and study generalizations of Cohen algebras: semi-Cohen algebras, pseudo-Cohen algebras and potentially Cohen algebras. These classes of Boolean algebras are closed under completion.

Sequential continuity and submeasurable cardinals

Abstract Submeasurable cardinals are defined in a similar way as measurable cardinals are. Their characterizations are given by means of sequentially continuous pseudonorms (or homomorphisms) on topological groups and of sequentially continuous (or uniformly continuous) functions on Cantor spaces (for that purpose it is proved that if a complete Boolean algebra admits a nonconstant sequentially continuous function, it admits a Maharam submeasure).

Reaping number and p-character of Boolean algebras

For every Boolean algebra B the minimal π-character of an ultrafilter on B is at most 2r2, where r2 is the reaping number of B. An example of B is given for which r2(B)< min{πχ(U): U ϵ Ult(B)}.

Quotients of Boolean algebras and regular subalgebras

Let

Dynamical systems on compact extremally disconnected spaces

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