Definition:Total Ordering/Definition 1

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:

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

$(2): \quad \RR$ is connected

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

$\forall x, y \in S: x \mathop \RR y \lor y \mathop \RR x$

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

• Results about total orderings can be found here.

Sources

• 1955: John L. Kelley: General Topology ... (previous) ... (next): Chapter $0$: Orderings
• 1960: Paul R. Halmos: Naive Set Theory ... (previous) ... (next): $\S 14$: Order
• 1963: George F. Simmons: Introduction to Topology and Modern Analysis ... (previous) ... (next): $\S 1$: Sets and Set Inclusion
• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {III}$: The Natural Numbers: $\S 14$: Orderings
• 1968: Ian D. Macdonald: The Theory of Groups ... (previous) ... (next): Appendix: Elementary set and number theory
• 1982: P.M. Cohn: Algebra Volume 1 (2nd ed.) ... (previous) ... (next): Chapter $1$: Sets and mappings: $\S 1.5$: Ordered Sets
• 1993: Keith Devlin: The Joy of Sets: Fundamentals of Contemporary Set Theory (2nd ed.) ... (previous) ... (next): $\S 1$: Naive Set Theory: $\S 1.5$: Relations
• 2008: Paul Halmos and Steven Givant: Introduction to Boolean Algebras ... (previous) ... (next): Appendix $\text{A}$: Set Theory: Order
• 2010: Steve Awodey: Category Theory (2nd ed.) ... (previous) ... (next): $\S 1.4.3$