Definition:Lower Set

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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.