- 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: