Definition:Lower Set

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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.