# saturated ideals

Is it possible to have a saturated ideal on a successor cardinal which does not extend the nonstationary ideal? (i.e. some nonstationary set is positive for this ideal)

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## 1 Answer

Yes, it is. The reason is that an ideal $I$ on $P(\kappa)$ is saturated just in case the quotient Boolean algebra $P(\kappa)/I$ satisfies the $\kappa^+$-chain condition, and this is a property that is preserved by permutations of the underlying set $\kappa$. But the property of extending the non-stationary ideal is not preserved by such permutations, since we can perform a permutation of $\kappa$ that takes a nonstationary set to a club set. Thus, there are isomorphic versions of any saturated ideal that remain saturated, but which do not extend the non-stationary ideal.

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But if you add a normality condition, the story becomes more complicated. –  Joel David Hamkins Jan 19 '12 at 1:42
I also just realized, if you take the induced ideal: $\{ X : 1 \Vdash \delta \in j(X) \}$, then this is saturated for any choice of delta, so we may pick a successor ordinal less than $\kappa^+$ and the get set of limit ordinals to be in the induced ideal. –  Monroe Eskew Jan 19 '12 at 1:44
Yes, I think that is right. –  Joel David Hamkins Jan 19 '12 at 1:48