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

• Results about rays in the context of Order Theory can be found here.

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