Forcing preorders, compatibility and filters

A forcing preorder $(P,\le )$ is a nonempty set with a reflexive transitive relation. The convention is that $q\le p$ means $q$ is stronger than $p$; antisymmetry is not required. Conditions $p,q$ are compatible if there exists $r\in P$ with $r\le p$ and $r\le q$. They are incompatible, written $p\perp q$, if there is no such $r$.

A forcing filter is a nonempty subset $G\subseteq P$ that is upward closed and internally downward directed: if $p\in G$ and $p\le q$, then $q\in G$; if $p,q\in G$, some $r\in G$ satisfies $r\le p,q$. In particular the empty subset is not a forcing filter. A subset $D\subseteq P$ is dense when for every $p\in P$ there exists $q\in D$ with $q\le p$.

There need not be a weakest condition, binary meets, or a zero condition. The directedness condition does not presume a meet operation or demand closure under a nonexistent operation. No genericity requirement is included in the definition of a forcing filter. All conventions are over ZF.