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).