Related papers: Cellularity of free products of Boolean algebras (…

We address a number of problems on Boolean Algebras. For example, we construct, in ZFC, for any BA B, and cardinal kappa BAs B_1,B_2 extending B such that the depth of the free product of B_1,B_2 over B is strictly larger than the depths of…

We consider homogeneity properties of Boolean algebras that have nonprincipal ultrafilters which are countably generated.It is shown that a Boolean algebra B is homogeneous if it is the union of countably generated nonprincipal ultrafilters…

The original theme of the paper is the existence proof of ``there is < eta_alpha : alpha < lambda > which is a (lambda,J)-sequence for < I_i:i<delta >, a sequence of ideals. This can be thought of as in a generalization to Luzin sets and…

We prove that colouring of pairs from aleph_2 with strong properties exists. The easiest to state (and quite a well known problem) it solves: there are two topological spaces with cellularity aleph_1 whose product has cellularity aleph_2 ;…

Monk asks (problems 13, 15 in his list; pi is the algebraic density):''For a Boolean algebra B, aleph_0 <= theta <= pi (B), does B have a subalgebra B' with pi (B')= theta ?'' If theta is regular the answer is easily positive, we show that…

We address several questions of Donald Monk related to irredundance and spread of Boolean algebras, gaining both some ZFC knowledge and consistency results. We show in ZFC that irr(B_0 times B_1)= max(irr(B_0),irr(B_1)). We prove…

We prove the consistency of irr(prod limits_{i<kappa}B_i/D)< prod limits_{i<kappa}irr(B_i)/D, where D is an ultrafilter on kappa and each B_i is a Boolean Algebra. This solves the last problem of this form from the Monk's list of problems,…

Let X be the moduli space of SL(n,C), SU(n), GL(n,C), or U(n)-valued representations of a rank r free group. We classify the algebraic singular stratification of X. This comes down to showing that the singular locus corresponds exactly to…

We answer three problems by J. D. Monk on cardinal invariants of Boolean algebras. Two of these are whether taking the algebraic density pi(A), resp. the topological density d(A), of a Boolean algebra A commutes with formation of…

We give a simplified proof of a theorem of M. Rabus and S. Shelah claiming that for each cardinal mu there is a c.c.c Boolean algebra with topological density mu.

We prove the consistency of: if B_1, B_2 are Boolean algebra satisfying the c.c.c. and the 2^{aleph_0}-c.c. respectively then B_1 x B_2 satisfies the 2^{aleph_0}-c.c.

We compare the forcing related properties of a complete Boolean algebra B with the properties of the convergences $\lambda_s$ (the algebraic convergence) and $\lambda_{ls}$ on B generalizing the convergence on the Cantor and Aleksandrov…

Let C denote any of the following cardinal characteristics of Boolean algebras: incomparability, spread, character, pi-character, hereditary Lindelof number, hereditary density. It is shown to be consistent that there exists a sequence…

For every uncountable cardinal mu there is a ccc Boolean algebra whose topological density is mu .

We construct Boolean Algebras answering questions of Monk on cardinal invariants. The results are proved in ZFC (rather than giving consistency results). We deal with the existence of superatomic Boolean Algebras with ``few automorphisms'',…

We try to build, provably in ZFC, for a first order T a model in which any isomorphism between two Boolean algebras is definable. The problem, compared to [Sh:384], is with pseudo-finite Boolean algebras. A side benefit is that we do not…

We examine the question of when the *-homomorphism of full amalgamated free product C*-algebras \lambda: A *_D B --> A' *_{D'} B', arising from compatible inclusions of C*-algebras A in A', B in B' and D in D', is an embedding. Results…

Known and new results on free Boolean topological groups are collected. An account of properties which these groups share with free or free Abelian topological groups and properties specific of free Boolean groups is given. Special emphasis…

We prove two results intended to streamline proofs about cellularity that pass through mutual algebraicity. First, we show that a countable structure $M$ is cellular if and only if $M$ is $\omega$-categorical and mutually algebraic. Second,…

We show that for every $d\ge 1$, if $L_1,\ldots, L_d$ are linearly ordered compact spaces and there is a continuous surjection \[ L_1\times L_2\times \dots\times L_d\to K_1\times K_2\times\ldots\times K_{d}\times K_{d+1},\] where all the…