In mathematics, a total order or linear order on a set X is any binary relation on X that is antisymmetric, transitive, and total. This means that, if we denote the relation by ≤, the following statements hold for all a, b and c in X: History Main article: History of mathematics In addition to recognizing how to count concrete objects, prehistoric peoples also recognized how to count abstract quantities, like time -- days, seasons, years. ... The notion of a set is one of the most important and fundamental concepts in modern mathematics. ... In mathematics, the concept of binary relation is exemplified by such ideas as is greater than and is equal to in arithmetic, or is congruent to in geometry, or is an element of or is a subset of in set theory. ... In mathematics, a binary relation R on a set X is antisymmetric if it holds for all a and b in X that if a is related to b and b is related to a then a = b. ... In mathematics, a binary relation R over a set X is transitive if it holds for all a, b, and c in X, that if a is related to b and b is related to c, then a is related to c. ... In mathematics, a binary relation R over a set X is total if it holds for all a and b in X that a is related to b or b is related to a. ...

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.

Notice that the totalness condition 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. It follows that a total order can also be defined as a partial order that is total. In mathematics, a binary relation R over a set X is reflexive if for all a in X, a is related to itself. ... In mathematics, a partially ordered set (or poset for short) is a set equipped with a special binary relation which formalizes the intuitive concept of an ordering. ...

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 . See lattice for other mathematical as well as non-mathematical meanings of the term. ...

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. Generally, a trichotomy is a splitting into three disjoint parts. ...

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. We can use open intervals to define a topology on any ordered set, the order topology. In elementary algebra, an interval is a set that contains every real number between two indicated numbers, and possibly the two numbers themselves. ... Topology (Greek topos, place and logos, study) is a branch of mathematics concerned with the study of topological spaces. ... In mathematics, the order topology is a topology that can be defined on any totally ordered set. ...

Examples

The following is valid up to order isomorphism: In mathematics, the 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. ... In 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. ...

The set of natural numbers is the unique smallest totally ordered set with no upper bound. Similarly, the unique smallest totally ordered set with neither an upper nor a lower bound is the integers. 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 numbers. The unique smallest unbounded connected totally ordered set is the real numbers. Natural 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 (this is... In 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. ... In 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. ... The integers consist of the positive natural numbers (1, 2, 3, …) the negative natural numbers (−1, −2, −3, ...) and the number zero. ... In 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. ... In topology and related branches of mathematics, a topological space X is said to be disconnected if it is the union of two disjoint nonempty open sets. ... The text or formatting below is generated by a template which has been proposed for deletion. ...

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. In mathematics, any integer (whole number) is either even or odd. ... Decimal, or less commonly, denary, usually refers to the base 10 numeral system. ...

Any set of cardinal numbers or ordinal numbers is totally ordered (in fact, even well-ordered). In linguistics, cardinal numbers is the name given to number words that are used for quantity (one, two, three), as opposed to ordinal numbers, words that are used for order (first, second, third). ... Ordinal numbers, or ordinals for short, are numbers used to denote the position in an ordered sequence: first, second, third, fourth, etc. ... In mathematics, a well-order (or well-ordering) on a set S is a total order on S with the property that every non-empty subset of S has a least element in this ordering. ...

See also happened-before. The happened-before relationship is important in figuring out partial ordering of events and in producing and synchronizing logical clocks for asynchronous distributed systems. ...