\begin{definition}[Definition:Lower Closure/Element]
Let $\struct {S, \preccurlyeq}$ be an ordered set.
Let $a \in S$.
The '''lower closure of $a$ (in $S$)''' is defined as:
:$a^\preccurlyeq := \set {b \in S: b \preccurlyeq a}$
That is, $a^\preccurlyeq$ is the set of all elements of $S$ that precede $a$.
\end{definition}