Definition:Total Ordering/Definition 2

Definition

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

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

\(\text {(1)}: \quad\)

\(\ds \RR \circ \RR\)

\(\subseteq\)

\(\ds \RR\)

\(\text {(2)}: \quad\)

\(\ds \RR \cap \RR^{-1}\)

\(\subseteq\)

\(\ds \Delta_S\)

\(\text {(3)}: \quad\)

\(\ds \RR \cup \RR^{-1}\)

\(=\)

\(\ds S \times S\)

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.

Also see

• Equivalence of Definitions of Total Ordering

• Definition:Ordering/Definition 2

• Results about total orderings can be found here.

Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {III}$: The Natural Numbers: $\S 14$: Orderings: Exercise $14.27 \ \text {(b)}$