= Principal filter in an ordered set
{title2=$\uparrow p=\{q:p\leq q\}$}
= Principal filter
{synonym}
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.
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