# Equivalence of Definitions of Total Ordering

## Contents

## Theorem

The following definitions of the concept of **Total Ordering** are equivalent:

### Definition 1

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

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

- $\forall x, y \in S: x \mathop {\mathcal R} y \lor y \mathop {\mathcal R} x$

### Definition 2

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

- $(1): \quad \mathcal R \circ \mathcal R = \mathcal R$
- $(2): \quad \mathcal R \cap \mathcal R^{-1} = \Delta_S$
- $(3): \quad \mathcal R \cup \mathcal R^{-1} = S \times S$

## Proof

### Definition 1 implies Definition 2

Let $\mathcal R$ be an ordering which is also connected.

- $(1): \quad \mathcal R \circ \mathcal R = \mathcal R$
- $(2): \quad \mathcal R \cap \mathcal R^{-1} = \Delta_S$

By Relation is Connected and Reflexive iff Total, $\mathcal R$ is a total relation.

Thus by Relation is Total iff Union with Inverse is Trivial Relation:

- $(3): \quad \mathcal R \cup \mathcal R^{-1} = S \times S$

Hence $\mathcal R$ is a total ordering by definition 2.

$\Box$

### Definition 2 implies Definition 1

Let $\mathcal R$ be a relation which fulfils the conditions:

- $(1): \quad \mathcal R \circ \mathcal R = \mathcal R$
- $(2): \quad \mathcal R \cap \mathcal R^{-1} = \Delta_S$
- $(3): \quad \mathcal R \cup \mathcal R^{-1} = S \times S$

By Relation is Total iff Union with Inverse is Trivial Relation it follows that $\mathcal R$ is a total relation.

From $(1)$ and $(2)$, $\mathcal R$ is an ordering by definition 2.

Thus by definition 1 of ordering, $\mathcal R$ is a reflexive relation.

From Relation is Connected and Reflexive iff Total, $\mathcal R$ is a connected relation.

Hence $\mathcal R$ is a total ordering by definition 1.

$\blacksquare$