\begin{definition}[Definition:Lower Closure/Set]
Let $\struct {S, \preccurlyeq}$ be an ordered set or preordered set.
Let $T \subseteq S$.
The '''lower closure of $T$ (in $S$)''' is defined as:
:$T^\preccurlyeq := \bigcup \set {t^\preccurlyeq: t \in T}$
where $t^\preccurlyeq$ is the lower closure of $t$.
That is:
:$T^\preccurlyeq := \set {l \in S: \exists t \in T: l \preccurlyeq t}$
\end{definition}