# Equivalence of Definitions of Total Ordering

## Theorem

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

### 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:

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

## Proof

### Definition 1 implies Definition 2

Let $\RR$ be an ordering which is also connected.

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

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

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

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

Hence $\RR$ is a total ordering by definition 2.

$\Box$

### Definition 2 implies Definition 1

Let $\RR$ be a relation which fulfils the conditions:

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

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

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

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

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

Hence $\RR$ is a total ordering by definition 1.

$\blacksquare$