Related papers: On incomparability and related cardinal functions …

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,…

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 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…

Using Koszmider's strongly unbounded functions, we show the following consistency result: Suppose that $\kappa,\lambda$ are infinite cardinals such that $\kappa^{+++} \leq \lambda$, $\kappa^{<\kappa}=\kappa$ and $2^{\kappa}= \kappa^+$, and…

This paper deals with variety of problems in pcf theory and infinitary combinatorics. We look at normal filters and prc, measures of the size of [lambda]^{<kappa}, pcf-inaccessibility, entangled orders (and narrow Boolean Algebras),…

We deal with some of problems posed by Monk and related to cardinal invariant of ultraproducts of Boolean algebras. We also introduce and investigate some new cardinal invariants.

We study the question which Boolean algebras have the property that for every generating set there is an ultrafilter selecting maximal number of its elements. We call it the ultrafilter selection property. For cardinality aleph-one the…

We prove that assuming suitable cardinal arithmetic, if B is a Boolean algebra every homomorphic image of which is isomorphic to a factor, then B has locally small density. We also prove that for an (infinite) Boolean algebra B, the number…

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 characterize exactly the compactness properties of the product of \kappa\ copies of the space \omega\ with the discrete topology. The characterization involves uniform ultrafilters, infinitary languages, and the existence of nonstandard…

We prove that if there are $\mathfrak c$ incomparable selective ultrafilters then, for every infinite cardinal $\kappa$ such that $\kappa^\omega=\kappa$, there exists a group topology on the free Abelian group of cardinality $\kappa$…

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 establish some similarities/analogies between uncountable cardinals or powersets and the class $V$ of all sets. They concern mainly the Boolean algebras ${\cal P}(\kappa)$, for a regular cardinal $\kappa$, and ${\cal C}(V)$ (the class of…

We prove that consistently there is a singular cardinal $\kappa$ of uncountable cofinality such that $2^\kappa$ is weakly inaccessible, and every regular cardinal strictly between $\kappa$ and $2^\kappa$ is the character of some uniform…

We prove a Theorem about the relationship between the Depth of the ultraproduct of Boolean algebras, divided by an ultrafilter, and the products of the depths of each component. This answers (partly) an open problem of Monk.

Complete Boolean algebras proved to be an important tool in topology and set theory. Two of the most prominent examples are B(kappa), the algebra of Borel sets modulo measure zero ideal in the generalized Cantor space {0,1}^kappa equipped…

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…

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

We characterize the situation of small cardinality for a product of cardinals divided by an ultrafilter. We develop the notion of weak normality. We include an application to Boolean Algebras.

Motivated by recent results and questions of D. Raghavan and S. Shelah, we present ZFC theorems on the bounding and various almost disjointness numbers, as well as on reaping and dominating families on uncountable, regular cardinals. We…