# Category:Rays (Order Theory)

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This category contains results about **rays** in the context of **Order Theory**.

Definitions specific to this category can be found in Definitions/Rays (Order Theory).

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

## Pages in category "Rays (Order Theory)"

This category contains only the following page.