Definition:Ray (Order Theory)/Closed

Definition
Let $\struct {S, \preccurlyeq}$ be a totally ordered set.

Let $a \in S$ be any point in $S$.

The following sets are called closed rays or closed half-lines:


 * $\set {x \in S: a \preccurlyeq x}$ (the upper closure of $a$), denoted $a^\succcurlyeq$
 * $\set {x \in S: x \preccurlyeq a}$ (the lower closure of $a$), denoted $a^\preccurlyeq$.

Also known as
A closed ray is also sometimes referred to as a closed half-line.

The notations:
 * $\hointr a \to$ for $a^\succcurlyeq$
 * $\hointl \gets a$ for $a^\preccurlyeq$

can also be used.

Also see

 * Definition:Open Ray


 * Definition:Upward-Pointing Ray
 * Definition:Downward-Pointing Ray