# Definition:Ray (Order Theory)

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*This page is about Ray in the context of Order Theory. For other uses, see Ray.*

## 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 $S$.

### Open Ray

The following sets are called **open rays** or **open half-lines**:

- $\set {x \in S: a \prec x}$ (the strict upper closure of $a$), denoted $a^\succ$
- $\set {x \in S: x \prec a}$ (the strict lower closure of $a$), denoted $a^\prec$.

### Closed Ray

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 $a^\preccurlyeq$.

### Upward-Pointing Ray

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 $a^\succcurlyeq: \set {x \in S: a \preccurlyeq x}$

### Downward-Pointing Ray

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 known as

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

The Wirth interval notation is another notation used widely on $\mathsf{Pr} \infty \mathsf{fWiki}$:

- $\openint a \to$ 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.

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