Definition:Upper Section

Definition
Let $(S, \preceq)$ be an ordered set.

Let $U \subseteq S$.

Then $U$ is an upper set in $S$ iff
 * For all $u \in U$ and $s \in S$: if $u \preceq s$ then $s \in U$.

That is, $U$ is an upper set iff it contains its own upper closure.