Solution
= Solution
For a <regular cardinal> $\kappa$,
$$
\operatorname{Fn}_\kappa(I,J)=\{p:p:I\rightharpoonup J,\ |\operatorname{dom}p|<\kappa\}.
$$
Again $q\le p$ means $q\supseteq p$, and the maximal element is the empty function. The regularity of $\kappa$ ensures that the union of a descending sequence of fewer than $\kappa$ conditions still has domain of cardinality below $\kappa$ whenever the conditions form a compatible increasing chain of partial functions.