# 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 $\left({S, \preccurlyeq}\right)$ 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 $a^\preccurlyeq : \left\{{x \in S: x \preccurlyeq a}\right\}$

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

This category contains only the following page.