# Definition:Lower Set

Let $\struct {S, \preceq}$ be an ordered set.
Let $L \subseteq S$.
Then $L$ is a lower set in $S$ if and only if:
$\forall l \in L, s \in S: s \preceq l \implies s \in L$
That is, $L$ is a lower set if and only if it contains its own lower closure.