Combinatorics of dense subsets of the rationals

Abstract

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.

Parametrized $\diamondsuit $ principles

“…Recall that i is the smallest cardinality of a maximal independent family. In [1] the rational reaping number…”

“…1 The main interest in it comes from the following fact relating to the question of whether d = ω 1 implies a = ω 1 . Theorem 1.1 ([16]).…”

“…The meaning of | · | should always be clear from the context. If B is a Borel subset of a Polish space, we will 1 Another ♦-like principle in this spirit is the statement ♦(ω <ω 1 ) presented in Section 6.2 of [36] (Definition 6.37). To draw an analogy, this principle might also be called ♦ non (M) in the language of [16] (see Theorem 6.49 of [36]).…”

On some ideal related to the ideal (v0)

“…Another natural structure of the cardinality continuum, consisting of infinite countable sets and ordered by inclusion is the family Dense.Q/ D fD Â QW D is denseg. In [1] there were investigated similarities and differences between combinatorial properties of structures OE! !…”

Remarks on gaps in \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}${\mathrm{Dense}(\mathbb {Q})/\mathbf {nwd}}$\end{document}

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Key words Rational numbers, nowhere dense ideal, cardinal invariants of the continuum.

MSC (2010) 03E05, 03E17The structure Dense(Q)/nwd and gaps in analytic quotients of P(ω) have been studied in the literature [1][2][3]. We prove that the structures Dense(Q)/nwd and P(Q)/nwd have gaps of type (add(M ), ω), and there are no (λ, ω)-gaps for λ < add(M ), where add(M ) is the additivity number of the meager ideal.

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