Related papers: Constructing Boolean algebras for cardinal invaria…

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 prove the consistency with ZFC of ``the length of an ultraproduct of Boolean algebras is smaller than the ultraproduct of the lengths''. Similarly for some other cardinal invariants of Boolean algebras.

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 investigate sigma-entangled linear orders and narrowness of Boolean algebras. We show existence of sigma-entangled linear orders in many cardinals, and we build Boolean algebras with neither large chains nor large pies. We study the…

We show that it is consistent with ZFC (relative to large cardinals) that every infinite Boolean algebra B has an irredundant subset A such that 2^{|A|} = 2^{|B|}. This implies in particular that B has 2^{|B|} subalgebras. We also discuss…

We construct, in ZFC, a sequence of Boolean algebras for which the product of Lengths is strictly smaller than the Length of the product algebra.

We prove that for any superatomic Boolean Algebra of cardinality >beth_omega there is an automorphism moving uncountably many atoms. Similarly for larger cardinals. Any of those results are essentially best possible.

Let inv denote the cardinal invariants Depth^+ and Length^+ on Boolean algebras. For many singular cardinals we create a strict inequality between the product of the inv values and the inv of the product algebra. The proof holds in ZFC.

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…

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…

The structure of quotient Boolean algebras in terms of cardinal invariants is investigated. Some results of Gitik and Shelah regarding atomless ideals are reproved and proofs are significantly simplified.

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…

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…

How many endomorphisms does a Boolean algebra have? Can we find Boolean algebras with as few endomorphisms as possible? Of course from any ultrafilter of the Boolean algebra we can define an endomorphism, and we can combine finitely many…

This paper develops a rich theory of cardinality in the paraconsistent and paracomplete set theory $\mathrm{BZFC}$, where sets can be inconsistent ($A$ such that ``$x\in A$'' is both true and false for some $x$) or incomplete ($A$ such that…

We give a definition of some classes of boolean algebras generalizing free boolean algebras; they satisfy a universal property that certain functions extend to homomorphisms. We give a combinatorial property of generating sets of these…

We give a definition of some classes of boolean algebras generalizing free boolean algebras; they satisfy a universal property that certain functions extend to homomorphisms. We give a combinatorial property of generating sets of these…

We investigate the structure of ultrafilters on Boolean algebras in the framework of Tukey reducibility. In particular, this paper provides several techniques to construct ultrafilters which are not Tukey maximal. Furthermore, we connect…

In [Sh:89] we, answering a question of Monk, have explicated the notion of ``a Boolean algebra with no endomorphisms except the ones induced by ultrafilters on it'' (see section 2 here) and proved the existence of one with character density…

For each regular cardinal k > w we show the consistent existence of a thin very tall superatomic Boolean algebra of width k.