|Definition:|| binary relation ℛ between elements a and b of a given set having the following properties: |
- reflexivity: aℛa,
- antisymmetry: if aℛb and bℛa then a = b,
- transitivity: if aℛb and bℛc then aℛc, for any elements a, b and c of the given set
Note 1 to entry: The given set is said to be ordered by the relation ℛ.
Note 2 to entry: An order relation is a total order if at least one of the relations aℛb and bℛa is true for any elements a and b. The usual order for real numbers is a total order because a ≤ b or b ≤ a.
Note 3 to entry: An order relation is a partial order if, for at least two elements a and b, neither aℛb nor bℛa is true. Examples are the divisibility relation for natural numbers and the inclusion relation for subsets of a set with at least two elements.