Totally Ordered Set

A total order (or "totally ordered set," or "linearly ordered set") is a set plus 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).

REFERENCES:

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