A totally ordered set, also called a linearly ordered set, is a set together with a relation on the set called a total order that satisfies the conditions for a partial order plus an additional condition known as the comparability condition. A relation is a total order on a set (" totally orders ") if the following properties hold.

1. Reflexivity: for all .

2. Antisymmetry: and implies .

3. Transitivity: and implies .

4. Comparability (trichotomy law): For any , either or .

The first three are the axioms of a partial order, while addition of the trichotomy law defines a total order.

Every finite totally ordered set is well ordered. Any two totally ordered sets with elements (for a nonnegative integer) are order isomorphic, and therefore have the same order type (which is also an ordinal number).

See also

Order Isomorphic, Order Type, Partial Order, Relation, Total Order, Trichotomy Law, Well Ordered Set

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References

Séroul, R. Programming for Mathematicians. Berlin, Germany: Springer-Verlag, p. 23, 2000.

Referenced on Wolfram|Alpha

Totally Ordered Set

Cite this as:

Weisstein, Eric W. "Totally Ordered Set." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/TotallyOrderedSet.html

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