Suppose $(\mathcal O, \leq)$ is an arbitrary poset. Let us say that $\mathcal O$ is compact if every $\mathcal C\subseteq\mathcal O$ which is centered (any finite subset of $\mathcal C$ has a lower bound in $\mathcal O$) can be extended to a filter in $\mathcal O$.
This is equivalent to the condition that $\mathcal C$ is contained in a directed set (one containing lower bounds for all finite subsets).
For example, if $\mathcal O$ is a meet-semilattice, then it is compact --- it suffices to close $\mathcal C$ under meets.
Not every poset is compact: for instance, if we let $\mathcal O$ consist of the nonempty finite sets of integers and put $A\leq B$ when $A=B$ or $B\subseteq A$ and $B$ is a singleton, then the set $\mathcal C$ of singletons in $\mathcal O$ is centered, but it does not have any centered proper supersets (and it is not a filter).
I was wondering if there is some simpler criterion for this "compactness", or whether it has an established name?