Order (mathematics)

Encyclopedia

**Order**in mathematics may refer to:

## In algebra

- Order (group theory)Order (group theory)In group theory, a branch of mathematics, the term order is used in two closely related senses:* The order of a group is its cardinality, i.e., the number of its elements....

, the cardinality of a group or period of an element - Order, or degree of a polynomialDegree of a polynomialThe degree of a polynomial represents the highest degree of a polynominal's terms , should the polynomial be expressed in canonical form . The degree of an individual term is the sum of the exponents acting on the term's variables...
- Order, or dimension of a matrix
- Order (ring theory), an algebraic structure
- Ordered groupOrdered groupIn abstract algebra, a partially-ordered group is a group 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 then a+g ≤ b+g and g+a ≤ g+b.An element x of G is called positive element if 0 ≤ x...
- Ordered fieldOrdered fieldIn mathematics, an ordered field is a field together with a total ordering of its elements that is compatible with the field operations. Historically, the axiomatization of an ordered field was abstracted gradually from the real numbers, by mathematicians including David Hilbert, Otto Hölder and...

## In arithmetic

- Order (number theory), the multiplicative orderMultiplicative orderIn number theory, given an integer a and a positive integer n with gcd = 1, the multiplicative order of a modulo n is the smallest positive integer k withThe order of a modulo n is usually written ordn, or On.- Example :To determine the multiplicative order of 4 modulo 7, we compute 42 = 16 ≡ 2 ...

in modular arithmetic - Orders of magnitude, a class of scale or magnitude of any amount
- Order of operationsOrder of operationsIn mathematics and computer programming, the order of operations is a rule used to clarify unambiguously which procedures should be performed first in a given mathematical expression....

## In analysis

- Orders of approximationOrders of approximationIn science, engineering, and other quantitative disciplines, orders of approximation refer to formal or informal terms for how precise an approximation is, and to indicate progressively more refined approximations: in increasing order of precision, a zeroth order approximation, a first order...

in Big O notation - Order of convergence, a measurement of convergence
- Order (differential equation), or order of highest derivative, of a differential equation
- Order of an entire functionEntire functionIn complex analysis, an entire function, also called an integral function, is a complex-valued function that is holomorphic over the whole complex plane...
- Ordered list, a tupleTupleIn mathematics and computer science, a tuple is an ordered list of elements. In set theory, an n-tuple is a sequence of n elements, where n is a positive integer. There is also one 0-tuple, an empty sequence. An n-tuple is defined inductively using the construction of an ordered pair...

or sequenceSequenceIn mathematics, a sequence is an ordered list of objects . Like a set, it contains members , and the number of terms is called the length of the sequence. Unlike a set, order matters, and exactly the same elements can appear multiple times at different positions in the sequence... - Z-order (curve)Z-order (curve)In mathematical analysis and computer science, Z-order, Morton order, or Morton code is a space-filling curve which maps multidimensional data to one dimension while preserving locality of the data points. It was introduced in 1966 by G. M. Morton...

, a space-filling curve - NURBS order, a number one greater than the degree of the polynomial representation of a non-uniform rational B-spline

## In combinatorics

- Order in the Josephus permutation
- Weak order of permutations
- Ordered selections and partitions of the twelvefold way in combinatorics
- Ordered set, a permutationPermutationIn mathematics, the notion of permutation is used with several slightly different meanings, all related to the act of permuting objects or values. Informally, a permutation of a set of objects is an arrangement of those objects into a particular order...

, bijectionBijectionA bijection is a function giving an exact pairing of the elements of two sets. A bijection from the set X to the set Y has an inverse function from Y to X. If X and Y are finite sets, then the existence of a bijection means they have the same number of elements...

or cyclic order - Unordered subsetSubsetIn mathematics, especially in set theory, a set A is a subset of a set B if A is "contained" inside B. A and B may coincide. The relationship of one set being a subset of another is called inclusion or sometimes containment...

or combination

## In statistics

- MeanMeanIn statistics, mean has two related meanings:* the arithmetic mean .* the expected value of a random variable, which is also called the population mean....

, medianMedianIn probability theory and statistics, a median is described as the numerical value separating the higher half of a sample, a population, or a probability distribution, from the lower half. The median of a finite list of numbers can be found by arranging all the observations from lowest value to...

, quantiles are first-order statistics - VarianceVarianceIn probability theory and statistics, the variance is a measure of how far a set of numbers is spread out. It is one of several descriptors of a probability distribution, describing how far the numbers lie from the mean . In particular, the variance is one of the moments of a distribution...

, correlationCorrelationIn statistics, dependence refers to any statistical relationship between two random variables or two sets of data. Correlation refers to any of a broad class of statistical relationships involving dependence....

, power spectrum are second-order statistics - SkewnessSkewnessIn probability theory and statistics, skewness is a measure of the asymmetry of the probability distribution of a real-valued random variable. The skewness value can be positive or negative, or even undefined...

, bispectrumBispectrumIn mathematics, in the area of statistical analysis, the bispectrum is a statistic used to search for nonlinear interactions. The Fourier transform of the second-order cumulant, i.e., the autocorrelation function, is the traditional power spectrum...

, kurtosisKurtosisIn probability theory and statistics, kurtosis is any measure of the "peakedness" of the probability distribution of a real-valued random variable...

are examples of higher-order statisticsHigher-order statisticsHigher-order statistics are descriptive measures of, among other things, qualities of probability distributions and sample distributions, and are, themselves, extensions of first- and second-order measures to higher orders. Skewness and kurtosis are examples of this...

## In fractals

- ComplexorComplexorIn chaos theory a complexor is mathematically equivalent to a chaotic attractor. The word was coined by Marcial Losada , derived from the words "complex order"....

, or complex order in fractals - Orders of construction in the Pythagoras tree
- Order of extension in Lakes of WadaLakes of WadaIn mathematics, the lakes of Wada are three disjoint connected open sets of the plane with the counterintuitive property that they all have the same boundary....
- Order of Rényi dimensionsFractal dimensionIn fractal geometry, the fractal dimension, D, is a statistical quantity that gives an indication of how completely a fractal appears to fill space, as one zooms down to finer and finer scales. There are many specific definitions of fractal dimension. The most important theoretical fractal...

## In graphs

- Graph order, the number of nodes in a graph
- Ordered pairOrdered pairIn mathematics, an ordered pair is a pair of mathematical objects. In the ordered pair , the object a is called the first entry, and the object b the second entry of the pair...

, including undirected and directed graphs - Ordered tripleTupleIn mathematics and computer science, a tuple is an ordered list of elements. In set theory, an n-tuple is a sequence of n elements, where n is a positive integer. There is also one 0-tuple, an empty sequence. An n-tuple is defined inductively using the construction of an ordered pair...

, or mixed graph- Glossary of graph theoryGlossary of graph theoryGraph theory is a growing area in mathematical research, and has a large specialized vocabulary. Some authors use the same word with different meanings. Some authors use different words to mean the same thing. This page attempts to keep up with current usage....

- Glossary of graph theory

## Order theory

- Order theoryOrder theoryOrder theory is a branch of mathematics which investigates our 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 introduces the field and gives some basic definitions...

, which studies various binary relations known as orders - Order (journal)Order (journal)Order is a quarterly peer-reviewed academic journal on order theory and its applications, published by Springer Science+Business Media. It was founded in 1984 by University of Calgary mathematics professor Ivan Rival; as of 2010, its editor in chief is Dwight Duffus, the Goodrich C...

, an academic journal on order theory - Partial order, often called just "order" in order theory texts; a transitive antisymmetric relation
- Total orderTotal orderIn set theory, a total order, linear order, simple order, or ordering is a binary relation on some set X. The relation is transitive, antisymmetric, and total...

, a partial order that is also total, in that either the relation or its inverse holds between any unequal elements - Dense orderDense orderIn mathematics, a partial order ≤ on a set X is said to be dense if, for all x and y in X for which x In mathematics, a partial order ≤ on a set X is said to be dense if, for all x and y in X for which x...

, a total order where between any unequal pair of elements there is always an intervening element in the order - Order topologyOrder topologyIn 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...

, a topology of total order for totally ordered sets - Ordinal numbers, which are assigned to sets based on their set-theoretic order
- Glossary of order theoryGlossary of order theoryThis 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...
- List of order theory topics

## In other mathematical theories

- Order in Ramsey theoryRamsey theoryRamsey theory, named after the British mathematician and philosopher Frank P. Ramsey, is a branch of mathematics that studies the conditions under which order must appear...

, uniform structures in consequence to critical set cardinality - Set theory (music)Set theory (music)Musical set theory provides concepts for categorizing musical objects and describing their relationships. Many of the notions were first elaborated by Howard Hanson in connection with tonal music, and then mostly developed in connection with atonal music by theorists such as Allen Forte , drawing...

encompasses ordered pitch and pitch classes (Musical set theory) - Type theoryType theoryIn mathematics, logic and computer science, type theory is any of several formal systems that can serve as alternatives to naive set theory, or the study of such formalisms in general...

encompasses: zeroth-orderZeroth-order logicZeroth-order logic is first-order logic without quantifiers. A finitely axiomatizable zeroth-order logic is isomorphic to a propositional logic. Zeroth-order logic with axiom schema is a more expressive system than propositional logic...

, first-orderFirst-order logicFirst-order logic is a formal logical system used in mathematics, philosophy, linguistics, and computer science. It goes by many names, including: first-order predicate calculus, the lower predicate calculus, quantification theory, and predicate logic...

, second-orderSecond-order logicIn 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....

and higher-order logicHigher-order logicIn mathematics and logic, a higher-order logic is a form of predicate logic that is distinguished from first-order logic by additional quantifiers and a stronger semantics...