# Definition:Ray (Order Theory)/Downward-Pointing

< Definition:Ray (Order Theory)(Redirected from Definition:Downward-Pointing Ray)

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## 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

- Results about
**rays in the context of order theory**can be found**here**.

## Sources

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