Total Partial Order Relation
- AKA: Total Order Relation, Chain, Chain Relation, Linear Order, Total Order, Linear Order Relation, Total Ordering Relation.
- It can be associated with a Total Partially Ordered Set.
- See: Non-Total Partial Order Relation, Total Strict Order Relation, Total Weak Order Relation.
- (Wikipedia, 2009) ⇒ http://en.wikipedia.org/wiki/Total_order
- In mathematics and set theory, a total order, linear order, simple order, or (non-strict) ordering is a binary relation (here denoted by infix ≤) on some set X. The relation is transitive, antisymmetric, and total. A set paired with a total order is called a totally ordered set, a linearly ordered set, a simply ordered set, or a chain.
- If X is totally ordered under ≤, then the following statements hold for all a, b and c in X:
- If a ≤ b and b ≤ a then a = b (antisymmetry);
- If a ≤ b and b ≤ c then a ≤ c (transitivity);
- a ≤ b or b ≤ a (totality).
- Contrast with a partial order, which lacks the third condition. A relation having the property of "totality" means that any pair of elements in the set of the relation are mutually comparable under the relation.
- Totality implies reflexivity, that is, a ≤ a. Thus a total order is also a partial order, that is, a binary relation which is reflexive, antisymmetric and transitive. Hence a total order is also a partial order satisfying the "totality" condition.