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Abstract. We study combinatorial properties of the partial order (Dense(Q), ⊆).To do that we introduce cardinal invariants p Q , t Q , h Q , s Q , r Q , i Q describing properties of Dense(Q). These invariants satisfyWe compare them with their analogues in the well studied Boolean algebra P(ω)/fin. We show that p Q = p, t Q = t and i Q = i, whereas h Q > h and r Q > r are both shown to be relatively consistent with ZFC. We also investigate combinatorics of the ideal nwd of nowhere dense subsets of Q. In particular, we show that non(M) = min{|D| :We use these facts to show that cof(M) ≤ i, which improves a result of S. Shelah. 0. Introduction. The aim of this paper is to point out the similarities and differences between the structure of P(ω)/fin and the structure of the collection of dense subsets of the rationals. Such research was suggested by A. Blass in [Bl] and initiated by J. Cichoń in [Ci]. The basic object studied here is the set Dense(Q) = {D ⊆ Q : D is dense} ordered by inclusion, in comparison with the structure ([ω] ω , ⊆). Neither one of them is a separative partial order. The separative quotient of ([ω] ω , ⊆) augmented with the least element 0 is the well known Boolean algebra P(ω)/fin. The separative quotient of (Dense(Q), ⊆) with added least element is not a Boolean algebra, but just a lattice, with two dense sets being in the same equivalence class if and only if their symmetric difference is a 2000 Mathematics Subject Classification: 03E17, 03E35, 06E15. Key words and phrases: rational numbers, nowhere dense ideal, distributivity of Boolean algebras, cardinal invariants of the continuum.

Let B be a complete ccc Boolean algebra and let τs be the topology on B induced by the algebraic convergence of sequences in B.1. Either there exists a Maharam submeasure on B or every nonempty open set in (B, τs ) is topologically dense.2. It is consistent that every weakly distributive complete ccc Boolean algebra carries a strictly positive Maharam submeasure.3. The topological space (B, τs ) is sequentially compact if and only if the generic extension by B does not add independent reals. Examples are also given of ccc forcings adding a real but not independent reals.

ABSTRACT. We present a proof (without any set-theoretical assumptions) that every infini te complete Boolean algebra includes a free subalgebra of the same cardinality. It follows that the set of all ultrafilters on an infinite complete Boolean algebra B has power 21BI. We shall present proofs (in ZFC only) of the following theorems.

This article investigates the weak distributivity of Boolean -algebras satisfying the countable chain condition. It addresses primarily the question when such algebras carry a -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.