Total order

In mathematics, a total or linear order is a partial order in which any two elements are comparable. That is, a total order is a binary relation on some set , which satisfies the following for all and in :

  1. (reflexive).
  2. If and then (transitive).
  3. If and then (antisymmetric).
  4. or (strongly connected, formerly called total).

Total orders are sometimes also called simple,[1] connex,[2] or full orders.[3]

A set equipped with a total order is a totally ordered set;[4] the terms simply ordered set,[1] linearly ordered set,[2][4] and loset[5][6] are also used. The term chain is sometimes defined as a synonym of totally ordered set,[4] but refers generally to some sort of totally ordered subsets of a given partially ordered set.

An extension of a given partial order to a total order is called a linear extension of that partial order.

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