(0,1)-category theory: logic, order theory
proset, partially ordered set (directed set, total order, linear order)
distributive lattice, completely distributive lattice, canonical extension
Let $P$ be a poset such that every directed subset of $P$ has a join; that is, $P$ is a dcpo. A compact element, or finite element, of $P$ is a compact object in $P$ regarded as a thin category; that is, homs out of it commute with these directed joins.
In other words, $c \in P$ is compact precisely if for every directed subset $\{d_i\}$ of $P$ we have
Of course, the $\Leftarrow$ part of this is automatic, so the real condition is the $\Rightarrow$ part. In more elementary terms:
In the case where $P$ has a top element $1$, we say that $P$ is compact if $1$ is a compact element.
Given a set $X$, the finite elements of its power set are precisely the (Kuratowski)-finite subsets of $X$. (This is the origin of the term ‘finite element’.)
Given a topological space (or locale) $X$, the compact elements of its frame of open subspaces are precisely the compact open subspaces of $X$. (This is the origin of the term ‘compact element’.)
If $R$ is a (not necessarily commutative) ring, the lattice $Idl(R)$ of two-sided ideals of $R$ is compact. Indeed, the top element is the ideal generated by $1$, the multiplicative identity, and $1 \in \bigvee_i I_i$ implies $1 \in I_i$ for some index $i$.