Let I be a definable (e.g., an analytic) ideal on
the natural numbers omega, and consider the quotient P(omega)/I,
equipped with the ordering induced by the inclusion relation on omega.
If I = Fin is the ideal of finite sets, this structure
has been intensively investigated, and a number of cardinal invariants
which describe its combinatorial properties have been defined.
In this talk I will present a few results on analogous cardinal
invariants for the case where I is an ideal distinct from Fin.
If I is not F_sigma, the quotient is not sigma-closed in general,
and some cardinal invariants may become countable. Therefore I will
concentrate on the case where I is an F_sigma-ideal. It turns out that
P(omega)/I looks rather similar to P(omega)/Fin
in this case, that many inequalities between cardinal invariants which
hold in the classical case can be generalized, and similarly for consistency
results. In fact, cardinal invariants of P(omega)/I are rather
hard to distinguish from their classical counterparts. However, we
know a few consistency results. For example, the splitting number of
P(omega)/I where I is any summable ideal may be
strictly smaller than the splitting number of P(omega)/Fin.