\begin{definition}[Definition:Ray (Order Theory)/Downward-Pointing]
Let $\left({S, \preccurlyeq}\right)$ be a totally ordered set.
Let $\prec$ be the reflexive reduction of $\preccurlyeq$.
Let $a \in S$ be any point in $S$.
A '''downward-pointing ray''' is a ray which is bounded above:
: an open ray $a^\prec := \left\{{x \in S: x \prec a}\right\}$
: a closed ray $a^\preccurlyeq : \left\{{x \in S: x \preccurlyeq a}\right\}$
\end{definition}