\begin{definition}[Definition:Ray (Order Theory)/Upward-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 $S$.
An '''upward-pointing ray''' is a ray which is bounded below:
: an open ray $a^\succ:= \left\{{x \in S: a \prec x}\right\}$
: a closed ray $a^\succcurlyeq: \left\{{x \in S: a \preccurlyeq x}\right\}$
\end{definition}