# Definition:Total Ordering

## Definition

Let $\RR \subseteq S \times S$ be a relation on a set $S$.

### Definition 1

$\RR$ is a **total ordering** on $S$ if and only if:

That is, $\RR$ is an ordering with no non-comparable pairs:

- $\forall x, y \in S: x \mathop \RR y \lor y \mathop \RR x$

### Definition 2

$\RR$ is a **total ordering** on $S$ if and only if:

\(\text {(1)}: \quad\) | \(\ds \RR \circ \RR\) | \(\subseteq\) | \(\ds \RR\) | |||||||||||

\(\text {(2)}: \quad\) | \(\ds \RR \cap \RR^{-1}\) | \(\subseteq\) | \(\ds \Delta_S\) | |||||||||||

\(\text {(3)}: \quad\) | \(\ds \RR \cup \RR^{-1}\) | \(=\) | \(\ds S \times S\) |

### Class Theory

In the context of class theory, the definition follows the same lines:

Let $V$ be a basic universe.

Let $\RR \subseteq V \times V$ be a relation.

Let $\RR$ be such that:

- $(1): \quad \RR$ is an ordering on $\Field \RR$
- $(2): \quad \forall x, y \in \Field \RR: x \mathop \RR y \lor y \mathop \RR x$ (that is, $x$ and $y$ are comparable)

where $\Field \RR$ denotes the field of $\RR$.

Then $\RR$ is a **total ordering**.

## Also known as

Some sources refer to a **total ordering** as a **linear ordering**, or a **simple ordering**.

If it is necessary to emphasise that a **total ordering** $\preceq$ is *not* strict, then the term **weak total ordering** may be used.

## Examples

### Usual Ordering on Real Numbers

Let $\R$ denote the set of real numbers.

The usual ordering $\le$ on $\R$ and its dual $\ge$ are total orderings on $\R$.

### Monarchy

Let $K$ denote the set of British monarchs.

Let $\MM$ denote the relation on $K$ defined as:

- $a \mathrel \MM b$ if and only if $a$ was monarch after or at the same time as $b$.

Its dual $\MM^{-1}$ is defined as:

- $a \mathrel {\MM^{-1} } b$ if and only if $a$ was monarch before or at the same time as $b$.

Then $\MM$ and $\MM^{-1}$ are total orderings on $K$.

## Also see

- Results about
**total orderings**can be found here.