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Order (mathematics)

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	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 cardinality of a group or period of an element Order, or degree of a polynomial Degree of a polynomial When a polynomial is expressed as a sum or difference of term s , the exponent of the term with the highest exponent is the degree of the polynomial.... Order, or dimension of a matrix Matrix (mathematics) In mathematics, a matrix is a rectangular array of numbers, as shown at the right. In addition to a number of elementary, entrywise operations such as matrix addition a key notion is matrix multiplication.... Order (ring theory) Order (ring theory) In mathematics, an order in the sense of ring theory is a subring of a ring that satisfies the conditions#R is a ring which is a finite-dimensional algebra over the rational number field ... , an algebraic structure Ordered group Ordered group In abstract algebra, an ordered group is a group equipped with a partial order "=" which 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.... Ordered field Ordered field In mathematics, an ordered field is a field together with a total ordering of its elements that agrees in a certain sense with the field operations.... In arithmetic Order (number theory) Order (number theory) In number theory, the order of an element is the smallest integer such that Note that the order is only defined when , i.e. a and n are coprimes.... , the multiplicative order Multiplicative order In number theory, given an integer a and a positive integer n with greatest common divisor = 1, the multiplicative order of a modulo n is the smallest positive integer k withThe order of a is usually written ordn a, or On.... in modular arithmetic Orders of magnitude, a class of scale or magnitude of any amount Order of operations Order of operations In algebra and computer programming, when a number or expression is both preceded and followed by an operator such as minus or multiplication, a rule is needed to specify which operator should be applied first; this rule is known as a precedence rule, or more informally order of operation.... In analysis Orders of approximation Orders of approximation Orders of approximation have been used not only in science, engineering, and other quantitative disciplines to make approximations with various degrees of precision but also more generally, and more loosely, to indicate relative precision outside these disciplines in the form of "first level", "second level" and so on, "approximations".... in Big O notation Order of convergence, a measurement of convergence Order, or order of highest derivative, of a differential equation Ordered list, a tuple Tuple In mathematics, a tuple is a sequence of a specific number of values, called the components of the tuple. These components can be any kind of mathematical objects, where each component of a tuple is a value of a specified type.... or sequence Sequence In mathematics, a sequence is an ordered list of objects . Like a Set , it contains Element , and the number of terms is called the length of the sequence.... Z-order (curve) Z-order (curve) Z-order, or Morton-order, first proposed in 1966 by G. M. Morton, is a space-filling curve which is often used in computer science: Due to its good locality-preserving behaviour it is used in data structures for mapping multidimensional data to one dimension.... , a space-filling curve In combinatorics Order in the Josephus permutation Weak order of permutations Weak order of permutations In mathematics, the set of permutations on n items can be given the structure of a partial order, called the weak order of permutations. The weak order of permutations forms a lattice .... Ordered selections and partitions Twelvefold way In combinatorics, a branch of mathematics, the twelvefold way provides a unified framework for counting permutations, combinations and Partition of a set.... of the twelvefold way in combinatorics Ordered set, a permutation Permutation In several fields of mathematics the term permutation is used with different but closely related meanings. They all relate to the notion of mapping the element s of a set to other elements of the same set, i.e., exchanging elements of a set.... , bijection Bijection In mathematics, a bijection, or a bijective function is a function f from a set X to a set Y with the property that, for every y in Y, there is exactly one x in X such that f = y.... or cyclic order Cyclic order In combinatorics mathematics, a cyclic order on a set X with n elements is an arrangement of X as on a clock face, for an n-hour clock.... Unordered subset Subset In mathematics, especially in set theory, a Set A is a subset of a set B if A is "contained" inside B. Notice that A and B may coincide.... or combination In fractals Complexor Complexor In 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 Pythagoras tree The Pythagoras tree is a plane fractal constructed from squares. It is named after Pythagoras because each triple of touching squares encloses a right triangle, in a configuration traditionally used to depict the Pythagorean theorem.... Order of extension in Lakes of Wada Lakes of Wada In mathematics, the lakes of Wada are three disjoint set connected set open sets of the plane with the counterintuitive property that they all have the same boundary .... Order of Rényi dimensions Fractal dimension In 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.... In geometry Long-range aperiodic order, in pinwheel tiling Pinwheel tiling The pinwheel tiling is an aperiodic tiling proposed by John H. Conway and Charles Radin.It is constructed with a right triangle which appears in infinitely many orientations.... for instance In graphs Graph order, the number of nodes in a graph Graph (mathematics) In mathematics a graph is an abstract representation of a set of objects where some pairs of the objects are connected by links. The interconnected objects are represented by mathematical abstractions called vertices, and the links that connect some pairs of vertices are called edges.... Ordered pair Ordered pair In mathematics, an ordered pair is a collection of two distinguishable objects, one being the first coordinate system , and the other being the second coordinate .... , including undirected and directed graphs Ordered triple Tuple In mathematics, a tuple is a sequence of a specific number of values, called the components of the tuple. These components can be any kind of mathematical objects, where each component of a tuple is a value of a specified type.... , or mixed graph Glossary of graph theory Glossary of graph theory Graph theory is a growing area in mathematical research, and has a large specialized vocabulary. Some authors use the same word with different meanings.... In other mathematical theories Order in Ramsey theory Ramsey theory Ramsey theory, named for Frank P. Ramsey, is a branch of mathematics that studies the conditions under which order must appear. Problems in Ramsey theory typically ask a question of the form: how many elements of some structure must there be to guarantee that a particular property will hold?... , uniform structures in consequence to critical set cardinality Order theory Order theory Order theory is a branch of mathematics that studies various kinds of binary relations that capture the intuitive notion of ordering, providing a framework for saying when one thing is "less than" or "precedes" another.... , which studies various binary relations known as orders Dense order Dense order 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 < y, there is a z in X such that x < z < y.... of rational and real numbers Order topology 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.... , a topology of Total order Total order In mathematics and set theory, a total order, linear order, simple order, or ordering is a binary relation on some Set X.... for totally ordered sets Partially ordered set Partially ordered set In mathematics, especially order theory, a partially ordered set formalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a Set .... , or poset Total order Total order In mathematics and set theory, a total order, linear order, simple order, or ordering is a binary relation on some Set X.... , a generalization of the complete ordered field of real numbers Glossary of order theory 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 theory, lattice , and domain theory. Note that there is a structured list of order topics available as well.... List of order theory topics List of order topics This is a list of order topics.An alphabetical list of many notions of order theory can be found in the order theory glossary. See also inequality, extreme value, optimization , domain theory.... 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 tonality music, and then mostly developed in connection with atonal music; the concepts of set theory are very general and can be applied to tonal and atonal... encompasses ordered pitch and pitch classes (Musical set theory) Type theory encompasses: First-order logic First-order logic First-order logic is a formal deductive system used in mathematics, philosophy, linguistics, and computer science. It goes by many names, including: first-order predicate calculus , the lower predicate calculus, the language of first-order logic or predicate logic.... , Second-order logic 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.... and Higher-order logic Higher-order logic In mathematics, higher-order logic is distinguished from first-order logic in a number of ways.One of these is the type of Free variables and bound variables appearing in quantifications; in first-order logic, roughly speaking, it is forbidden to quantify over Predicate s....