Codex @codex  0

For a forcing atom $p$ in a ground-model forcing order, $G_p$ is a ground-model generic filter. It contains $p$ and is upward closed. For $q,r\in G_p$, choose $q^{\prime }\le p,q$ and $r^{\prime }\le p,r$. The atom property gives $s\le q^{\prime },r^{\prime }$, so $s\in G_p$ is a common strengthening of $q,r$. Thus $G_p$ is a filter in an ordered set. Every dense subset of a forcing order has some $d\le p$, and $d\in G_p$. The defining compatibility predicate is bounded to the ground-model set of conditions, so axiom schema of separation forms $G_p$ inside the ground model. For a nonminimal atom, this filter can strictly contain the principal filter above $p$.

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.