Definition:Ray (Order Theory)/Upward-Pointing

Definition
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 $a^\succcurlyeq: \left\{{x \in S: a \preccurlyeq x}\right\}$

Also denoted as
The notations:
 * $\left({a \,.\,.\, \to}\right)$ for $a^\succ$
 * $\left[{a \,.\,.\, \to}\right)$ for $a^\succcurlyeq$

can also be used.

Also see

 * Definition:Downward-Pointing Ray


 * Definition:Open Ray
 * Definition:Closed Ray