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Home > Total order

a ≤ a (reflexivity)

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 (totalness)

A set with a total order on it is called a totally ordered set, a linearly ordered set, or a chain. The totalness property can be stated thus: that any pair of elements in the chain are mutually comparable.

Because a binary relation that is reflexive, antisymmetric and transitive is called a partial order, a total order can also be defined as a partial order that is total. Alternatively, one may define a totally ordered set as a particular kind of lattice, namely one in which we have for all a, b. We then write a ≤ b if and only if .

If a and b are members of a totally ordered set, we may write a < b if a ≤ b and a ≠ b. The binary relation < is then transitive (a < b and b < c implies a < c) and trichotomous (one and only one of a < b, b < a and a = b is true). In fact, we can define a total order to be a transitive trichotomous binary relation <, and then define a ≤ b to mean a < b or a = b, and this definition can be shown to be equivalent to the one given at the beginning of this article.

For any totally ordered set X we can define the open intervals (a, b) = {x : a < x and x < b}, (−∞, b) = {x : x < b}, (a, ∞) = {x : a < x} and (−∞, ∞) = X. The totally ordered set X turns into a topological space if we define a subset to be open if and only if it is a union of (possibly infinitely many) such open intervals. This is called the order topology on X; it is always a normal Hausdorff space. Unless otherwise stated, it is understood that this topology is being used on a totally ordered set.

Examples

The following is valid up toIn mathematics, the jargon term up to xxxx" is used to describe a situation in which members of an equivalence class can be regarded as a single entity for some purpose. xxxx" describes a property or process which transforms an element into one from the s order isomorphismIn the mathematical field of order theory an order isomorphism is a special kind of monotone function that constitutes a suitable notion of isomorphism for partially ordered sets. Whenever two partially ordered sets are order isomorphic they can be consid:

The set of natural numberNatural number can mean either a positive integer ( 1, 2, 3, 4,. or a non-negative integer ( 0, 1, 2, 3, 4,. Natural numbers have two main purposes: they can be used for counting ("there are 3 apples on the table"), or they can be used for ordering ("thiss is the unique smallest totally ordered set with no upper boundIn mathematics, especially in order theory, an upper bound of a subset S of some partially ordered set is an element which is greater than or equal to every element of S''. The term lower bound is defined dually. Formally, given a partially ordered set P. Similarly, the unique smallest totally ordered set with neither an upper nor a lower bound is the integerThe integers consist of the positive natural numbers (1, 2, 3, …) the negative natural numbers (−1, −2, −3,. and the number zero. The set of all integers is usually denoted in mathematics by Z (or Z in blackboard bold, ), which sts. The unique smallest unbounded totally ordered set which also happens to be dense in the sense that (a, b) is non-empty for every a < b, is the rational numberIn mathematics, a rational number (or informally fraction is a ratio of two integers, usually written as the vulgar fraction a ''b where b is not zero. The set of all rational numbers is denoted by Q or in blackboard bold. Using the set-builder notation is. The unique smallest unbounded connected totally ordered set is the real numbers.

Note that subsets are possible, which in a way are smaller, but that they are order isomorphic and therefore not counting as smaller. For example, instead of natural numbers and integers we can take the even ones, and instead of all rational numbers we can take those with a finite decimal expansion.

Any set of cardinal numbers or ordinal numbers is totally ordered (in fact, even well-ordered).

See also: happened-before

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