Total order


In mathematics, a total order, simple order, linear order, connex order, or full order is a binary relation on some set, which is antisymmetric, transitive, and a connex relation. A set paired with a total order is called a chain, a totally ordered set, a simply ordered set, a linearly ordered set, or a loset.
Formally, a binary relation is a total order on a set if the following statements hold for all and in :
; Antisymmetry: If and then ;
; Transitivity: If and then ;
; Connexity: or.
Antisymmetry eliminates uncertain cases when both precedes and precedes. A relation having the connex property means that any pair of elements in the set of the relation are comparable under the relation. This also means that the set can be diagrammed as a line of elements, giving it the name linear. The connex property also implies reflexivity, i.e., a ≤ a. Therefore, a total order is also a partial order, as, for a partial order, the connex property is replaced by the weaker reflexivity property. An extension of a given partial order to a total order is called a linear extension of that partial order.

Strict total order

For each total order ≤ there is an associated asymmetric transitive semiconnex relation <, called a strict total order or strict semiconnex order, which can be defined in two equivalent ways:
Properties:
We can work the other way and start by choosing < as a transitive trichotomous binary relation; then a total order ≤ can be defined in two equivalent ways:
Two more associated orders are the complements ≥ and >, completing the quadruple.
We can define or explain the way a set is totally ordered by any of these four relations; the notation implies whether we are talking about the non-strict or the strict total order.

Examples

Lattice theory

One may define a totally ordered set as a particular kind of lattice, namely one in which we have
We then write a ≤ b if and only if. Hence a totally ordered set is a distributive lattice.

Finite total orders

A simple counting argument will verify that any non-empty finite totally ordered set has a least element. Thus every finite total order is in fact a well order. Either by direct proof or by observing that every well order is order isomorphic to an ordinal one may show that every finite total order is order isomorphic to an initial segment of the natural numbers ordered by <. In other words, a total order on a set with k elements induces a bijection with the first k natural numbers. Hence it is common to index finite total orders or well orders with order type ω by natural numbers in a fashion which respects the ordering.

Category theory

Totally ordered sets form a full subcategory of the category of partially ordered sets, with the morphisms being maps which respect the orders, i.e. maps f such that if a ≤ b then f ≤ f.
A bijective map between two totally ordered sets that respects the two orders is an isomorphism in this category.

Order topology

For any totally ordered set X we can define the open intervals =, =, = and = X. We can use these open intervals to define a topology on any ordered set, the order topology.
When more than one order is being used on a set one talks about the order topology induced by a particular order. For instance if N is the natural numbers, < is less than and > greater than we might refer to the order topology on N induced by < and the order topology on N induced by >.
The order topology induced by a total order may be shown to be hereditarily normal.

Completeness

A totally ordered set is said to be complete if every nonempty subset that has an upper bound, has a least upper bound. For example, the set of real numbers R is complete but the set of rational numbers Q is not.
There are a number of results relating properties of the order topology to the completeness of X:
A totally ordered set which is a complete lattice is compact. Examples are the closed intervals of real numbers, e.g. the unit interval , and the affinely extended real number system. There are order-preserving homeomorphisms between these examples.

Sums of orders

For any two disjoint total orders and, there is a natural order on the set, which is called the sum of the two orders or sometimes just :
Intuitively, this means that the elements of the second set are added on top of the elements of the first set.
More generally, if is a totally ordered index set, and for each the structure is a linear order, where the sets are pairwise disjoint, then the natural total order on is defined by

Orders on the Cartesian product of totally ordered sets

In order of increasing strength, i.e., decreasing sets of pairs, three of the possible orders on the Cartesian product of two totally ordered sets are:
All three can similarly be defined for the Cartesian product of more than two sets.
Applied to the vector space Rn, each of these make it an ordered vector space.
See also examples of partially ordered sets.
A real function of n real variables defined on a subset of Rn defines a strict weak order and a corresponding total preorder on that subset.

Related structures

A binary relation that is antisymmetric, transitive, and reflexive is a partial order.
A group with a compatible total order is a totally ordered group.
There are only a few nontrivial structures that are reducts of a total order. Forgetting the orientation results in a betweenness relation. Forgetting the location of the ends results in a cyclic order. Forgetting both data results in a separation relation.