Normal filters on a regular cardinal

Let $\kappa$ be regular uncountable. A proper tail-containing filter $F\subseteq \mathcal{P}(\kappa )$ contains $\kappa$, excludes $\mathrm{\varnothing }$, is upward closed, is closed under finite intersections, and contains every $[\beta ,\kappa )$, $\beta <\kappa$. It is normal if the diagonal intersection of every $\kappa$-sequence of its members belongs to $F$.

A set $S$ is $F$-positive if $\kappa \setminus S\mathrm{\notin }F$, equivalently if it meets every member of $F$: disjointness from $A\in F$ puts $A\subseteq \kappa \setminus S$ in the filter by upward closure, and the converse uses that complement itself. The dual ideal consists of sets whose complements belong to $F$; complementation proves its downward and finite-union closure. Positive need not mean membership in the filter. $\kappa$-complete means closed under intersections of fewer than $\kappa$ members.