On almost precipitous ideals.

We answer questions concerning an existence of almost precipitous ideals raised in [5]. It is shown that every successor of a regular cardinal can carry an almost precipitous ideal in a generic extension of L. In L[µ] every regular cardinal which is less than the measurable carries an almost precipitous non-precipitous ideal. Also, results of [4] are generalized- thus assumptions on precipitousness are replaced by those on ∞-semi precipitousness. 1 On semi precipitous and almost precipitous ideals Definition 1.1 Let κ be a regular uncountable cardinal, τ a ordinal and I a κ-complete ideal over κ. We call I τ-almost precipitous iff every generic ultrapower of I is wellfounded up to the image of τ. Clearly, any such I is τ-almost precipitous for each τ < κ. Also, note if τ ≥ (2κ)+ and I is τ-almost precipitous, then I is precipitous. Definition 1.2 Let κ be a regular uncountable cardinal. We call κ almost precipitous iff for each τ < (2κ)+ there is τ-almost precipitous ideal over κ