Dualization of the van Douwen diagram

Abstract

We make a more systematic study of the van Douwen diagram for cardinal coefficients related to combinatorial properties of partitions of natural numbers.

Subgroups of $$SF(\omega )$$ S F ( ω ) and the relation of almost containedness

“…Here t d is defined by the same scheme as t for the lattice P (ω) of all subsets of ω (see [7]). Moreover it is proved in [14] that the tower cardinal for partitions is ω 1 in ZFC and it is proved in [6] that the size of a maximal almost orthogonal family of partitions must be 2 ω .…”

Ramseyan ultrafilters

“…The dual forms of the classical cardinal characteristics were introduced and investigated in [CW00] and further investigated in [Ha98 2 ]. Concerning the dual-shattering cardinal H, one easily gets ℵ 1 ≤ H ≤ h, and in [Ha98 2 ] it is shown that H > ℵ 1 is consistent with ZFC and H = add(R 0 ) = cov(R 0 ), where R 0 denotes the ideal of dual Ramsey-null sets.…”

“…In this dualization process, a lot of work is already done. We refer to: [HL∞ 1 ] for a dualization of βN; [CS84], [Ha98 1 ] and [HL∞ 2 ] for the dualization of the Ramsey property and of Mathias forcing; [CS84] for a dualization of Ramsey's Theorem; [CW00] and [Ha98 2 ] for the dualization of some cardinal characteristics of the continuum.…”

Martin's Axiom and the Dual Distributivity Number

“…Using these notions, one can redefine in the present context those cardinal invariants in van Douwen's diagram [11] which deal with P(ω), almost containedness and almost intersection. These dual invariants were introduced mainly by Cichoń, Krawczyk, Majcher-Iwanow and Wȩglorz in [4], and studied as well by Carlson, Matet, Halbeisen and Spinas. For example Carlson proved that the dual tower number T is ℵ 1 [8, Proposition 4.3], and Krawczyk showed that the (dual) almost orthogonality number A, the size of the least maximal almost orthogonal ("mao" for short) family of infinite partitions, is c, the cardinality of the continuum [4, Section 4].…”

“…These dual invariants were introduced mainly by Cichoń, Krawczyk, Majcher-Iwanow and Wȩglorz in [4], and studied as well by Carlson, Matet, Halbeisen and Spinas. For example Carlson proved that the dual tower number T is ℵ 1 [8, Proposition 4.3], and Krawczyk showed that the (dual) almost orthogonality number A, the size of the least maximal almost orthogonal ("mao" for short) family of infinite partitions, is c, the cardinality of the continuum [4, Section 4].Recall that the distributivity number h of P(ω)/fin is the smallest κ such that forcing with P(ω)/fin is not κ-distributive or, equivalently, the size of the least family …”