Definition:Ray (Order Theory)/Downward-Pointing

Definition
Let $\struct {S, \preccurlyeq}$ 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 $a^\prec := \set {x \in S: x \prec a}$
 * a closed ray $a^\preccurlyeq : \set {x \in S: x \preccurlyeq a}$

Also denoted as
The notations:
 * $\openint \gets a$ for $a^\prec$
 * $\hointl \gets a$ for $a^\preccurlyeq$

can also be used.

Also see

 * Definition:Upward-Pointing Ray


 * Definition:Open Ray
 * Definition:Closed Ray