Let I be an ideal on ?, the notion of I-AD family was introduced in [3].
Analogous to the well studied ideal I(A) generated by almost disjoint
families, we introduce and investigate the ideal I(I-A). It turns out that
some properties of I(I-A) depends on the structure of I. Denoting by a(I)
the minimum of the cardinalities of infinite I-MAD families, several
characterizations for a(I) ? ?1 will be presented. Motivated by the work in
[23], we introduce the cardinality s?,?(I), and obtain a necessary condition
for s?,?(I) = s(I). As an application, we show finally that if a(I) ? s(I),
then BW property coincides with Helly property.