Total Partial Order Relation

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A Total Partial Order Relation is a Transitive Antisymmetric Binary Relation (a Partial Order Relation) that is a Total Relation.


  • (Wikipedia, 2009) ⇒
    • 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.