The almost disjointness cardinal invariant in the quotient algebra of the rationals modulo the nowhere dense subsets. (English) Zbl 1052.03027

Steprāns, Juris

Real Anal. Exch. 27(2001-2002), No. 2, 795-800 (2002).

Let \(B\) be a Boolean algebra that does not have the countable chain condition. Then \({\mathfrak a}(B)\) denotes the least uncountable cardinal of an uncountable maximal antichain. This generalizes the almost disjointness number of the continuum. Let \(\mathcal N\) be the ideal of nowhere dense subsets of \(\mathbb{Q}\). Then \({\mathfrak a}{{\mathcal N}}\) denotes \({\mathfrak a}({\mathcal P}(\mathbb{Q})/{\mathcal N})\). The author shows that in the iterated Laver model \({\mathfrak a} =\aleph_2\) but \({\mathfrak a}({\mathcal N}) =\aleph_1\). Further on he shows that \({\mathfrak a}({\mathcal N})\geq {\mathfrak p}\), thus \({\mathfrak a}({\mathcal N})\) can be arbitrarily large.

Reviewer: Martin Weese (Potsdam)

MSC:

03E17	Cardinal characteristics of the continuum
03E35	Consistency and independence results

Keywords:

almost disjointness number; nowhere dense set; Laver forcing