Suppose κ is a regular uncountable cardinal, λ is a cardinal > 1, and
is a κ-complete uniform ideal on κ. This paper deals with a saturation property Sat(κ, λ,
) of
, which is a weakening of usual λ-saturatedness. Roughly speaking, Sat(κ, λ,
) means that
can be densely extended to λ-saturated ideals on small fields of subsets of κ. We will show that some consequences of the existence of a λ-saturated ideal on κ follow from weaker ∃
: Sat(κ, λ,
), and that ∃
: Sat(κ, λ,
) is connected with weak compactness and complete ineffability of κ in much the same way as the existence of a saturated ideal on κ is connected with measurability of κ.
In §2, we define Sat(κ, λ,
), mention a few results that can be proved by straightforward adaptation of known methods, and discuss generic ultrapowers of ZFC−, which will be used repeatedly in the subsequent sections as a main technical tool. A related concept Sat(κ, λ) is also defined and shown to be equivalent to ∃
: Sat(κ, λ,
) under a certain condition.
In §3, we show that ∃
: Sat(κ, κ,
) implies that κ is highly Mahlo, improving results in [KT] and [So].