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

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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 $S$.

An **upward-pointing ray** is a ray which is bounded below:

- an open ray $a^\succ:= \set {x \in S: a \prec x}$
- a closed ray $a^\succcurlyeq: \set {x \in S: a \preccurlyeq x}$

## Also denoted as

The notations:

- $\openint a \to$ for $a^\succ$
- $\hointr a \to$ for $a^\succcurlyeq$

can also be used.

## Also see

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

## Sources

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