Order in mathematics may refer to:

Set theory

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 \leq on some set X, which satisfies the following for all a, b and c in X: # a \leq a ( reflexive ...

and

partial order In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binary ...

, a binary relation generalizing the usual ordering of numbers and of words in a dictionary *

Ordered set In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binary r ...

* Order in

Ramsey theory Ramsey theory, named after the British mathematician and philosopher Frank P. Ramsey, is a branch of mathematics that focuses on the appearance of order in a substructure given a structure of a known size. Problems in Ramsey theory typically ask ...

, uniform structures in consequence to critical set cardinality

Algebra

Order (group theory) In mathematics, the order of a finite group is the number of its elements. If a group is not finite, one says that its order is ''infinite''. The ''order'' of an element of a group (also called period length or period) is the order of the sub ...

, the cardinality of a group or period of an element *

Order of a polynomial (disambiguation) In mathematics, the order of a polynomial may refer to: *the degree of a polynomial, that is, the largest exponent (for a univariate polynomial) or the largest sum of exponents (for a multivariate polynomial) in any of its monomials; *the multiplica ...

* Order of a square matrix, its dimension *

Order (ring theory) In mathematics, an order in the sense of ring theory is a subring \mathcal of a ring A, such that #''A'' is a finite-dimensional algebra over the field \mathbb of rational numbers #\mathcal spans ''A'' over \mathbb, and #\mathcal is a \mathbb ...

, an algebraic structure *

Ordered group In abstract algebra, a partially ordered group is a group (''G'', +) equipped with a partial order "≤" that is ''translation-invariant''; in other words, "≤" has the property that, for all ''a'', ''b'', and ''g'' in ''G'', if ''a'' ≤ ''b'' ...

Ordered field In mathematics, an ordered field is a field together with a total ordering of its elements that is compatible with the field operations. The basic example of an ordered field is the field of real numbers, and every Dedekind-complete ordered fiel ...

Analysis

Order (differential equation) In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, and ...

or order of highest derivative, of a differential equation * Leading-order terms *NURBS order, a number one greater than the degree of the polynomial representation of a

non-uniform rational B-spline Non-uniform rational basis spline (NURBS) is a mathematical model using basis splines (B-splines) that is commonly used in computer graphics for representing curves and surfaces. It offers great flexibility and precision for handling both analy ...

Order of convergence In numerical analysis, the order of convergence and the rate of convergence of a convergent sequence are quantities that represent how quickly the sequence approaches its limit. A sequence (x_n) that converges to x^* is said to have ''order of co ...

, a measurement of convergence *

Order of derivation In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. ...

*Order of an

entire function In complex analysis, an entire function, also called an integral function, is a complex-valued function that is holomorphic on the whole complex plane. Typical examples of entire functions are polynomials and the exponential function, and any fin ...

* Order of a power series, the lowest degree of its terms *Ordered list, a

sequence In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is called ...

tuple In mathematics, a tuple is a finite ordered list (sequence) of elements. An -tuple is a sequence (or ordered list) of elements, where is a non-negative integer. There is only one 0-tuple, referred to as ''the empty tuple''. An -tuple is defi ...

* Orders of approximation in Big O notation * Z-order (curve), a space-filling curve

Arithmetic

Multiplicative order In number theory, given a positive integer ''n'' and an integer ''a'' coprime to ''n'', the multiplicative order of ''a'' modulo ''n'' is the smallest positive integer ''k'' such that a^k\ \equiv\ 1 \pmod n. In other words, the multiplicative ord ...

in modular arithmetic *

Order of operations In mathematics and computer programming, the order of operations (or operator precedence) is a collection of rules that reflect conventions about which procedures to perform first in order to evaluate a given mathematical expression. For examp ...

Orders of magnitude An order of magnitude is an approximation of the logarithm of a value relative to some contextually understood reference value, usually 10, interpreted as the base of the logarithm and the representative of values of magnitude one. Logarithmic dis ...

, a class of scale or magnitude of any amount

Combinatorics

*Order in the Josephus permutation * Ordered selections and partitions of the twelvefold way in combinatorics *Ordered set, a

bijection In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other ...

cyclic order In mathematics, a cyclic order is a way to arrange a set of objects in a circle. Unlike most structures in order theory, a cyclic order is not modeled as a binary relation, such as "". One does not say that east is "more clockwise" than west. Ins ...

, or

permutation In mathematics, a permutation of a set is, loosely speaking, an arrangement of its members into a sequence or linear order, or if the set is already ordered, a rearrangement of its elements. The word "permutation" also refers to the act or pro ...

*Unordered

subset In mathematics, set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset of ...

or combination * Weak order of permutations

Fractals

* Complexor, or complex order in fractals *Order of extension in

Lakes of Wada In mathematics, the are three disjoint set, disjoint connected set, connected open sets of the plane (geometry), plane or open unit square with the counterintuitive property that they all have the same boundary (topology), boundary. In other wor ...

*Order of

fractal dimension In mathematics, more specifically in fractal geometry, a fractal dimension is a ratio providing a statistical index of complexity comparing how detail in a pattern (strictly speaking, a fractal pattern) changes with the scale at which it is me ...

(Rényi dimensions) *Orders of construction in the Pythagoras tree

Geometry

*Long-range aperiodic order, in

pinwheel tiling In geometry, pinwheel tilings are non-periodic tilings defined by Charles Radin and based on a construction due to John Conway. They are the first known non-periodic tilings to each have the property that their tiles appear in infinitely many or ...

, for instance

Graphs

* Graph order, the number of nodes in a graph *First order and second order

logic of graphs In the mathematical fields of graph theory and finite model theory, the logic of graphs deals with formal specifications of graph properties using sentences of mathematical logic. There are several variations in the types of logical operation that ...

* Topological ordering of directed acyclic graphs * Degeneracy ordering of undirected graphs * Elimination ordering of chordal graphs *Order, the complexity of a structure within a graph: see

haven (graph theory) In graph theory, a haven is a certain type of function on sets of vertices in an undirected graph. If a haven exists, it can be used by an evader to win a pursuit–evasion game on the graph, by consulting the function at each step of the game to ...

and

bramble (graph theory) In graph theory, a bramble for an undirected graph is a family of connected subgraphs of that all touch each other: for every pair of disjoint subgraphs, there must exist an edge in that has one endpoint in each subgraph. The ''order'' of a ...

Logic

In logic, model theory and type theory: *

Zeroth-order logic Zeroth-order logic is first-order logic without variables or quantifiers. Some authors use the phrase "zeroth-order logic" as a synonym for the propositional calculus,. but an alternative definition extends propositional logic by adding constant ...

First-order logic First-order logic—also known as predicate logic, quantificational logic, and first-order predicate calculus—is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic uses quantifie ...

Second-order logic In logic and mathematics, second-order logic is an extension of first-order logic, which itself is an extension of propositional logic. Second-order logic is in turn extended by higher-order logic and type theory. First-order logic quantifies on ...

Higher-order logic mathematics and logic, a higher-order logic is a form of predicate logic that is distinguished from first-order logic by additional quantifiers and, sometimes, stronger semantics. Higher-order logics with their standard semantics are more express ...

Order theory

* ''Order'' (journal), an academic journal on order theory *

Dense order In mathematics, a partial order or total order < on a

set

X

is said to be dense if, for all

x

, a total order wherein between any unequal pair of elements there is always an intervening element in the order *

Glossary of order theory This is a glossary of some terms used in various branches of mathematics that are related to the fields of order, lattice, and domain theory. Note that there is a structured list of order topics available as well. Other helpful resources might ...

Lexicographical order In mathematics, the lexicographic or lexicographical order (also known as lexical order, or dictionary order) is a generalization of the alphabetical order of the dictionaries to sequences of ordered symbols or, more generally, of elements of ...

, an ordering method on sequences analogous to alphabetical order on words * List of order topics, list of order theory topics *

Order theory Order theory is a branch of mathematics that investigates the intuitive notion of order using binary relations. It provides a formal framework for describing statements such as "this is less than that" or "this precedes that". This article intr ...

, study of various binary relations known as orders *

Order topology In mathematics, an order topology is a certain topology that can be defined on any totally ordered set. It is a natural generalization of the topology of the real numbers to arbitrary totally ordered sets. If ''X'' is a totally ordered set, th ...

, a topology of total order for totally ordered sets *

Ordinal numbers In set theory, an ordinal number, or ordinal, is a generalization of ordinal numerals (first, second, th, etc.) aimed to extend enumeration to infinite sets. A finite set can be enumerated by successively labeling each element with the least ...

, numbers assigned to sets based on their set-theoretic order *

Partial order In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binary ...

, often called just "order" in order theory texts, a transitive antisymmetric relation *

, a partial order that is also total, in that either the relation or its inverse holds between any unequal elements

Statistics

Order statistics In statistics, the ''k''th order statistic of a statistical sample is equal to its ''k''th-smallest value. Together with rank statistics, order statistics are among the most fundamental tools in non-parametric statistics and inference. Importa ...

* First-order statistics, e.g.,

arithmetic mean In mathematics and statistics, the arithmetic mean ( ) or arithmetic average, or just the '' mean'' or the ''average'' (when the context is clear), is the sum of a collection of numbers divided by the count of numbers in the collection. The co ...

, median, quantiles * Second-order statistics, e.g.,

correlation In statistics, correlation or dependence is any statistical relationship, whether causal or not, between two random variables or bivariate data. Although in the broadest sense, "correlation" may indicate any type of association, in statistic ...

, power spectrum, variance *

Higher-order statistics In statistics, the term higher-order statistics (HOS) refers to functions which use the third or higher power of a sample, as opposed to more conventional techniques of lower-order statistics, which use constant, linear, and quadratic terms (zero ...

, e.g., bispectrum, kurtosis, skewness {{SIA, mathematics Mathematical terminology