Let $\kappa$ be a cardinal (I'm most interested in $\kappa=\aleph\_{\omega+1}$ but I suspect a general answer is known). What is the cofinality of $(P(\kappa)/NS,\subseteq)$? By this I mean the least cardinal $\lambda$ such that there exists a subcollection $X\subseteq P(\kappa)/NS$ of size $\lambda$ such that for any $A\in P(\kappa)$ there is $B\in X$ with $A\subseteq B$.


Here we consider $P(\kappa)/NS$ to be without the largest element.


This may be trivial but it's been bugging me all day!


Drake 



cofinality of $(P(\kappa)/NS,\subseteq)$
I'm going to assume that when you write $\subseteq$, you really mean $\subseteq\_{NS}$. We say $A \subseteq\_{NS} B$ when $A$ is contained in $B$ except for a nonstationary set, i.e. $A \setminus B \in NS$. If we don't do this, then as Joel said, $\lambda = \kappa$, since the collection of "coatoms" is cofinal.


If we take the set of equivalence classes of stationary sets modulo $NS$, (i.e. $A \sim B$ when $A \triangle B \in NS$) we get an atomless boolean algebra under the set operations modulo $NS$, and this is what is commonly meant by $\mathcal{P}(\kappa)/NS$. Now in general an "upwardly dense" subset $A$ of a boolean algebra $B$ ("cofinal" as you call it) generates a (downwardly) dense set by just taking the complement of every element of $A$.


Now, it is known to be equiconsistent with infinitely many Woodin cardinals that $NS\_{\omega\_1}$ is $\omega\_1$-dense. But by results of Gitik and Shelah, $\mathcal{P}(\kappa)/NS$ never has the $\kappa^+$-chain condition for regular $\kappa > \omega\_1$. Hence the density of this algebra is always greater than $\kappa$. Under GCH, it must be equal to $\kappa^+$.


Woodin showed relative to an almost-huge cardinal that it is consistent for a successor cardinal $\kappa > \omega\_1$ to have a stationary subset $S$ such that $\mathcal{P}(S)/NS$ has the $\kappa^+$-c.c. (See Foreman's article in Handbook of Set Theory.) I'm pretty sure that the stronger property of being $\kappa$-dense is not known to be consistent from any large cardinal assumption. Under a fragment of GCH, if $\kappa$ is the successor of singular cardinal (for example $\aleph\_{\omega+1}$) $NS\_\kappa$ is provably not $\kappa$-dense below any stationary set. (This is my result, to appear in my thesis soon.)


