Definition:Totally Ordered Set/Also known as

Totally Ordered Set: Also known as

A totally ordered set is also called a simply ordered set or linearly ordered set.

It is also known as a toset.

This term may be encountered on $\mathsf{Pr} \infty \mathsf{fWiki}$.

Some sources refer to a totally ordered set as an ordered set, using the term partially ordered set for what goes as an ordered set on $\mathsf{Pr} \infty \mathsf{fWiki}$.

Some sources use the term chain, but this word is generally restricted to mean specifically a totally ordered subset of a given ordered set.

The term permutation is an older term for totally ordered set, but has since been changed to mean the bijection that can be applied on such a totally ordered set in order to redefine its ordering.

Sources

• 1955: John L. Kelley: General Topology ... (previous) ... (next): Chapter $0$: Orderings
• 1960: Paul R. Halmos: Naive Set Theory ... (previous) ... (next): $\S 14$: Order
• 1964: A.M. Yaglom and I.M. Yaglom: Challenging Mathematical Problems With Elementary Solutions: Volume $\text { I }$ ... (previous) ... (next): Problems
• 1968: A.N. Kolmogorov and S.V. Fomin: Introductory Real Analysis ... (previous) ... (next): $\S 3.3$: Ordered sets. Order types
• 1975: T.S. Blyth: Set Theory and Abstract Algebra ... (previous) ... (next): $\S 7$
• 1996: Winfried Just and Martin Weese: Discovering Modern Set Theory. I: The Basics ... (previous) ... (next): Part $1$: Not Entirely Naive Set Theory: Chapter $2$: Partial Order Relations
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): partial order
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): partial order