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).
