Equivalence of Definitions of Total Ordering

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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:

$(1): \quad \mathcal R$ is an ordering on $S$
$(2): \quad \mathcal R$ is connected

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.

By definition 2 of ordering:

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

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.

$\blacksquare$