Definition of normal M-ultrafilter

Question

When you define what is a normal ideal, will it make a difference if

1. you require all functions which are regressive on a positive set with respect to the ideal to be constant or if
2. you require that that all functions which are regressive every where except for a set in the ideal?

In what I have read the first option is taken. But then what about when defining what is a normal M-ultrafilter? What definition to take?

Answer 1

I'm assuming that $M$ is a transitive model of (some sufficiently large fragment) of $\mathrm{ZF}$ and that $U$ is a filter on some ordinal $\kappa$. A $M$-ultrafilter $U$ on $\kappa \in |M|$ is a subset $U \subseteq \mathcal P^{M}(\kappa)$ s.t.

1. $\emptyset \not \in U$,
2. $y,z \in U \implies y \cap z \in U$,
3. $y \in U \wedge y \subseteq z \in \mathcal P^M(\kappa) \implies z \in U$,
4. for all $y \in \mathcal P^M(\kappa) \colon y \in U \wedge (\kappa \setminus y) \in U$ and
5. for all functions $f \colon \kappa \to \kappa$ such that $f \in |M|$ and $\{ \alpha \in \kappa \mid f(\alpha) < \alpha \} \in U$ there is some $\beta < \kappa$ such that $\{ \alpha < \kappa \mid f(\alpha) = \beta \} \in U$.

Item 5. is what you were asking for. Any function $f \colon \kappa \to \kappa$ which exists in $M$ and is regressive on a $U$-positive set (which is the same a being in $U$, as $U$ is ultra,) is constant on a $U$-positive set.