Definition:Lower Section

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

Let $L \subseteq S$.

Then $L$ is a lower set in $S$ iff
 * For all $l \in L$ and $s \in S$: if $s \preceq L$ then $s \in L$.

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