A **total order** is a relation from a set to itself that satisfies the following properties for all :

- Antisymmetry — If and , then ;
- Transitivity — If and , then ;
- Totality — Either or .

The totality property implies the reflexive property:

Since is antisymmetric, transitive, and reflexive, it is also a partial order.

If (less than or equal to) is a total order on a set , then we can define the following relations:

- Greater than or equal to: define by for all ;
- Less than: define by , but for all ;
- Greater than: define by , but for all .

The following results can be derived from the previous definitions:

- The relation is also a total order;
- For any , exactly one of the following is true:

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