# Strong ideals that are not pre-saturated

An ideal on $\omega_1$ is strong if it is precipitous and the associated generic elementary embedding always maps $\omega_1$ to $\omega_2$. This definition is from Baumgartner and Taylor, Saturation Properties of Ideals in Generic Extensions II (available online at http://www.jstor.org/stable/1998900.) Every pre-saturated ideal on $\omega_1$ is strong (this was well-known even before the terminology was introduced, I think) and in this paper the authors ask whether the converse is true.

Does anyone know the status of this question: "is every strong ideal on $\omega_1$ pre-saturated"?

-

Unfortunately, I don't think Trevor's question is addressed by either of the papers that Matteo mentions.

-
Welcome to MathOverflow, Sean! –  Trevor Wilson Sep 8 '12 at 21:30
It is consistent that the nonstationary ideal on $\omega_1$ is strong but not pre-saturated. Baumgartner and Taylor proved in the aforementioned paper that strong ideals are preserved by c.c.c. forcing and asked whether the same is true for pre-saturated ideals. The answer to this question is negative, implying a negative answer to the question I posted above. Apparently this was first proved by Veličković in the paper Forcing axioms and stationary sets (which I cannot seem to access online) from ZFC + SPFA. Another example of a c.c.c. forcing that destroys pre-saturation may be found in a more recent paper by Larson and Yorioka, Another c.c.c. forcing that destroys presaturation, assuming the consistency of ZF + AD.