the quotient boolean algebra of $P(\kappa)$ over the nonstationary ideal

Question

Let $\kappa$ be a regular cardinal. Then the quotient boolean algebra over the nonstationary ideal, $P(\kappa)/I_{NS}$ is $\kappa^+$-complete. Specifically, any $S \subseteq P(\kappa)/I_{NS}$ of cardinality $\kappa$ has both supremum and infimum, which are given by the diagonal intersection and diagonal union of the representatives of the family respectively. How does one prove this result?

Answer 1

Here's a nice generalization. Suppose $I$ is a normal and fine ideal on $Z \subseteq \mathcal{P}(X)$. Let $\{ A_x : x \in X \} \subseteq \mathcal{P}(Z)$. Then the least upper bound of $\{ A_x : x \in X \}$ mod $I$ is the diagonal union, $\nabla A_x =_{def} \{ z : (\exists x) x \in z \in A_x\}$.

(In your case, $I = NS_\kappa$, and $Z = X = \kappa$).

Proof: It is clear the the diagonal union is an upper bound, so it suffices to show that for all $I$-positive $B$ (i.e. stationary in your case) such that $B \subseteq \nabla A_x$, there is $x$ such that $A_x \cap B$ is $I$-positive. Suppose $B$ is such a set, and for all $z \in B$, choose an $x$ such that such that $x \in z \in A_x$. By normality, there is an $I$-positive $C \subseteq B$ and some $x_0$ such that for all $z \in C$, $z \in A_{x_0}$. This does it.