Definition:Ray (Order Theory)

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 $S$.

Also known as
A ray (either open or closed is also sometimes referred to as a half-line (either open or closed).

The notations:
 * $\openint a \to$ for $a^\succ$
 * $\openint \gets a$ for $a^\prec$
 * $\hointr a \to$ for $a^\succcurlyeq$
 * $\hointr \gets a$ for $a^\preccurlyeq$

can also be used.

Also see

 * Definition:Order Topology: a topology whose sub-basis consists of open rays.