# Definition:Ray (Order Theory)

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*This page is about upper closure or lower closure in a totally ordered set. For other uses, see Definition:Ray.*

## Contents

## Definition

Let $\left({S, \preccurlyeq}\right)$ 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**:

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

### Closed Ray

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

- $\left\{{x \in S: a \preccurlyeq x}\right\}$ (the upper closure of $a$), denoted $a^\succcurlyeq$
- $\left\{{x \in S: x \preccurlyeq a}\right\}$ (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 $a^\succ:= \left\{{x \in S: a \prec x}\right\}$
- a closed ray $a^\succcurlyeq: \left\{{x \in S: a \preccurlyeq x}\right\}$

### Downward-Pointing Ray

A **downward-pointing ray** is a ray which is bounded above:

- an open ray $a^\prec := \left\{{x \in S: x \prec a}\right\}$
- a closed ray $a^\preccurlyeq : \left\{{x \in S: x \preccurlyeq a}\right\}$

## Also known as

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

The notations:

- $\left({a \,.\,.\, \to}\right)$ for $a^\succ$
- $\left({\gets \,.\,.\, a}\right)$ for $a^\prec$
- $\left[{a \,.\,.\, \to}\right)$ for $a^\succcurlyeq$
- $\left[{\gets \,.\,.\, a}\right)$ 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.