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 : (reflexive). If and then (transitive). If and then (antisymmetric). or (strongly connected, formerly called total). Reflexivity (1.) already follows from connectedness (4.), but is required explicitly by many authors nevertheless, to indicate the kinship to partial orders. Total orders are sometimes also called simple, connex, or full orders. A set equipped with a total order is a totally ordered set; the terms simply ordered set, linearly ordered set, and loset are also used. The term chain is sometimes defined as a synonym of totally ordered set, 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. Strict and non-strict total orders A on a set is a strict partial order on in which any two distinct elements are comparable. That is, a total order is a binary relation on some set , which satisfies the following for all and in : Not (irreflexive). If then not (asymmetric). If and then (transitive). If , then or (connected). Asymmetry follows from transitivity and irreflexivity; moreover, irreflexivity follows from asymmetry. For each (non-strict) total order there is an associated relation , called the strict total order associated with that can be defined in two equivalent ways: if and (reflexive reduction). if not (i.e., is the complement of the converse of ). Conversely, the reflexive closure of a strict total order is a (non-strict) total order. Examples Any subset of a totally ordered set is totally ordered for the restriction of the order on . The unique order on the empty set, , is a total order. Any set of cardinal numbers or ordinal numbers (more strongly, these are well-orders). If is any set and an injective function from to a totally ordered set then induces a total ordering on by setting if and only if . The lexicographical order on the Cartesian product of a family of totally ordered sets, indexed by a well ordered set, is itself a total order. The set of real numbers ordered by the usual "less than or equal to" (≤) or