Definition:Totally Ordered Set

Definition

Let $\left({S, \preceq}\right)$ be a poset.

Then $\left({S, \preceq}\right)$ is a totally ordered set (or toset) if $\preceq$ is a total ordering.

Different authors refer to partially and totally ordered sets in different ways. Some use ordering to mean an ordering which may or may not be partial, while others use ordering exclusively to mean a total ordering.

As tosets are still posets, all results applying to posets also apply to tosets.

So it is usual to use the term poset to mean a set which may be partially or totally ordered, and partially ordered set when we want to make it clear that the set under discussion is definitely not totally ordered.

Also see

• Partially ordered set (poset)
• Well-ordered set (woset)

• Chain

Sources

• Paul R. Halmos: Naive Set Theory (1960)... (previous)... (next): $\S 14$: Order
• A.N. Kolmogorov and S.V. Fomin‎: Introductory Real Analysis (1968): $\S 3.3$
• T.S. Blyth: Set Theory and Abstract Algebra (1975): $\S 7$
• Keith Devlin: The Joy of Sets: Fundamentals of Contemporary Set Theory (1993): $\S 1.5$