Equivalence of Definitions of Total Ordering

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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:

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

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.

By definition 2 of ordering:

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

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.

$\blacksquare$