Remarks on gaps in Dense(q)/nwd

The structure Dense(Q)/nwd and gaps in analytic quotients of P(ω) have been studied in the literature. We prove a that in ZFC you can prove that structures Dense(Q)/nwd and P(Q)/nwd have gaps of type (add(M ), ω) and that there are no (λ, ω)-gaps for λ < add(M ), where add(M ) is the additivity of the meager ideal. We also give a direct proof of the existence of (ω1, ω1)-gaps. In this paper we study some aspects of the structure of the dense subsets of rational numbers Dense(Q) ordered by almost inclusion with respect to nowhere dense sets of rationals. The interest in this structure was raised by Blass [Bla] and it was later investigated by Cichoń [Cic01] and Balcar, Hernández-Hernández and Hrušák [BHHH04]. It can be viewed as a counterpart to the well-studied structure of subsets of natural numbers ordered by almost inclusion with respect to finite sets, and the same kinds of concepts can be used to investigate its structure. This paper concentrates on the concept of a gap, which is well known for P(ω)/fin (see eg. [Sch93]). For P(Q)/nwd gaps have been studied in a more general setting of analytic ideals of N by Todorčević [Tod98] and Farah [Far00]. First we answer positively a question in [Tod98] if other cardinal