The almost disjointness cardinal invariant in the quotient algebra of the rationals modulo the nowhere dense subsets.
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.
scientific article; zbMATH DE number 2084013