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