Definition:Ray (Order Theory)/Downward-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 $S$.

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

Also denoted as
The notations:
 * $\left({\gets \,.\,.\, a}\right)$ for $a^\prec$
 * $\left({\gets \,.\,.\, a}\right]$ for $a^\preccurlyeq$

can also be used.

Also see

 * Definition:Upward-Pointing Ray


 * Definition:Open Ray
 * Definition:Closed Ray