Upper set ($\uparrow A$)

A subset $U$ of a partially ordered set is an upper set when $x\in U$ and $x\le y$ imply $y\in U$. The upward closure $\uparrow A=\{y:\exists a\in A,a\le y\}$ is the smallest upper set containing $A$. Both union and intersection preserve this property.

An upper set generated by a finite set $A$ is represented by the minimal elements of $A$. Removing a nonminimal generator preserves the upward closure. If each pair has a least upper bound, then $\uparrow A\cap \uparrow B=\uparrow \{a\vee b:a\in A,b\in B\}$, so finite generators support intersection without enumerating the potentially infinite represented sets.