Just a few minutes ago I conceived a definition of generalized Fréchet filters with definition for every poset on which filters are considered (however, I have not yet calculated the class of posets for which generalized Fréchet filter is defined; it should be easy but I am busy with other business).

Generalized Fréchet filter on a poset $latex \mathfrak{A}$ is a filter $latex \Omega$ such that $latex \partial \Omega = \left\{ x \in \mathfrak{A} \hspace{0.5em} | \hspace{0.5em} \mathrm{atoms}\, x \text{ is infinite} \right\} . $

See my book for a definition of $latex \partial$.