# Definition:Lower Set

Let $\left({S, \preceq}\right)$ be an ordered set.
Let $L \subseteq S$.
Then $L$ is a lower set in $S$ if and only if:
For all $l \in L$ and $s \in S$: if $s \preceq l$ then $s \in L$.
That is, $L$ is a lower set if and only if it contains its own lower closure.