Filter in an ordered set ($G\subseteq P$)

A filter is a nonempty upward-closed subset $G$ of a partial order such that every two members have a common lower bound in $G$. With standard notation for forcing, lower means stronger, so these are exactly the directed filters used to define a generic filter. The projection of a product-generic filter uses these properties before genericity of its factor filters is proved.

The principal filter in an ordered set generated by $p$ contains exactly the elements above $p$. It is upward closed, and $p$ is a common lower bound in the filter for any two of its members. With standard notation for forcing, these are the conditions weaker than $p$. The principal filter need not be a generic filter: a dense set can require a strict strengthening. The compatible-condition filter in a forcing atom determines a ground-model generic filter includes strengthenings as well as weakenings and can therefore be strictly larger.