morphism

Functions in applications are often functions of several variables, or multivariate functions: the values they take depend on a number of different factors. From a mathematical point of view all the variables must be made explicit in order to have a functional relationship - no 'hidden' factors are allowed. Then again, from the mathematical point of view, there is no qualitative difference between functions of one and of several variables. A function of three real variables is just a function that a ...

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Function mathematics, Function mathematics - Introduction, Function mathematics - Functions of more than one variable, Function mathematics - History, Function mathematics - Formal definition, Function mathematics - Domains codomains and ranges, Function mathematics - Injective surjective and bijective functions, Function mathematics - Images and preimages, Function mathematics - Graph of a function, Function mathematics - Examples of functions, Function mathematics - Properties of functions, Function mathematics - Ambiguous functions, Function mathematics - n-ary function: function of several variables, Function mathematics - Composing functions, Function mathematics - Inverse function, Function mathematics - Restrictions and extensions, Function mathematics - Pointwise operations, Function mathematics - Computable and non-computable functions, Function mathematics - Functions from the categorical viewpoint

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The functions f: X → Y and g: Y → Z can be composed by first applying f to an argument x and then applying g to the result. Thus one obtains a composite function g o f: X → Z defined by (g o f)(x) = g(f(x)) for all x in X. As an example, suppose that an airplane's height at time t is ...

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Function mathematics, Function mathematics - Introduction, Function mathematics - Functions of more than one variable, Function mathematics - History, Function mathematics - Formal definition, Function mathematics - Domains codomains and ranges, Function mathematics - Injective surjective and bijective functions, Function mathematics - Images and preimages, Function mathematics - Graph of a function, Function mathematics - Examples of functions, Function mathematics - Properties of functions, Function mathematics - Ambiguous functions, Function mathematics - n-ary function: function of several variables, Function mathematics - Composing functions, Function mathematics - Inverse function, Function mathematics - Restrictions and extensions, Function mathematics - Pointwise operations, Function mathematics - Computable and non-computable functions, Function mathematics - Functions from the categorical viewpoint

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If M is a metric space with metric d, and ~ is an equivalence relation on M, then we can endow the quotient set M/~ with the following (pseudo)metric. Given two equivalence classes [x] and [y], we define where the infimum is taken over all finite sequences and with [p1] = [x], [qn] = [y], . In general this will only define a pseudometric, i.e. See also:

Metric space, Metric space - History, Metric space - Definition, Metric space - Examples, Metric space - Metric spaces as topological spaces, Metric space - Boundedness and compactness, Metric space - Separation properties and extension of continuous functions, Metric space - Distance between points and sets, Metric space - Equivalence of metric spaces, Metric space - Quotient metric space

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If F and G are covariant functors between the categories C and D, then a natural transformation η from F to G associates to every object X in C a morphism ηX : F(X) → G(X) in D, such that for every morphism f : X → Y in C we have ηY O F(f) = G(f) O ηX. This equation can ...

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Natural transformation, Natural transformation - Definition, Natural transformation - Examples, Natural transformation - A worked example, Natural transformation - Further examples, Natural transformation - Operations with natural transformations, Natural transformation - Functor categories, Natural transformation - Yoneda lemma, Natural transformation - Historical notes

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The action of G on X is called transitive if for any two x, y in X there exists a g in G such that g·x = y; n-transitive if G acts transitively on Xn. sharply n-transitive if G acts regularly on Xn. faithful (or effective) if for any two different g, h ...

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Group action, Group action - Definition, Group action - Examples, Group action - Types of actions, Group action - Orbits and stabilizers, Group action - Morphisms and isomorphisms between G-sets, Group action - Continuous group actions, Group action - Strongly continuous group action and smooth vector, Group action - Generalizations

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If A and B are associative algebras over the same field K, an algebra homomorphism h: A → B is a K-linear map which is also multiplicative in the sense that h(xy) = h(x) h(y) for all x, y in A. With this notion of morphism, the class of all associative algebras over K becomes a category. Take for example the algebra A of all real-valued continuous functions R → R, and B = RSee also:

Associative algebra, Associative algebra - Definition, Associative algebra - Examples, Associative algebra - Algebra homomorphisms, Associative algebra - Index-free notation, Associative algebra - Generalizations, Associative algebra - Coalgebras, Associative algebra - Representations, Associative algebra - Motivation for a Hopf algebra, Associative algebra - Motivation for a Lie algebra

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In the above definition of an associative algebra, the definition of associativity was made with regard to all of the elements of A. It is sometimes more convenient to have a definition of associativity that does not need to refer to the elements of A. This can be done as follows. An algebra is defined as a map M (multiplication) on a vector space A: An associative algebra is an algebra w ...

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Associative algebra, Associative algebra - Definition, Associative algebra - Examples, Associative algebra - Algebra homomorphisms, Associative algebra - Index-free notation, Associative algebra - Generalizations, Associative algebra - Coalgebras, Associative algebra - Representations, Associative algebra - Motivation for a Hopf algebra, Associative algebra - Motivation for a Lie algebra

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Given a vector bundle π : E → X and an open subset U of X, we can consider sections of π on U, i.e. continuous functions s : U → E with πs = idU. Essentially, a section assigns to every point of U a vector from the attached vector space, in a continuous manner. As an example, sections of the tangent bundle of a di ...

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Vector bundle, Vector bundle - Definition and first consequences, Vector bundle - Vector bundle morphisms, Vector bundle - Sections and locally free sheaves, Vector bundle - Operations on vector bundles, Vector bundle - Variants and generalizations

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A morphism from the vector bundle π1 : E1 → X1 to the vector bundle π2 : E2 → X2 is given by a pair of continuous maps f : E1 → E2 and g : X1 → X2 such that gπ1 = π2f for every x in X1, the map π1−1({x}) → π2−1({g(x)}) induced by f ...

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Vector bundle, Vector bundle - Definition and first consequences, Vector bundle - Vector bundle morphisms, Vector bundle - Sections and locally free sheaves, Vector bundle - Operations on vector bundles, Vector bundle - Variants and generalizations

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If η : F → G and ε : G → H are natural transformations between functors F,G:C → D, then we can compose them to get a natural transformation εη : F → H. This is done componentwise: (εη)X = εXηX. This "vertical composition" of natural transformation is associative and has an identity, and allows one to consider the collection of all functor ...

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Natural transformation, Natural transformation - Definition, Natural transformation - Examples, Natural transformation - A worked example, Natural transformation - Further examples, Natural transformation - Operations with natural transformations, Natural transformation - Functor categories, Natural transformation - Yoneda lemma, Natural transformation - Historical notes

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Main article: Yoneda lemma If X is an object of the category C, then the assignment Y MorC(X, Y) defines a covariant functor FX : C → Set. This functor is called representable. The natural transformations from a representable functor to an arbitrary functor F : C → Set are completely known and eas ...

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Natural transformation, Natural transformation - Definition, Natural transformation - Examples, Natural transformation - A worked example, Natural transformation - Further examples, Natural transformation - Operations with natural transformations, Natural transformation - Functor categories, Natural transformation - Yoneda lemma, Natural transformation - Historical notes

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(More can be found at list of functions.) The relation wght between persons in the United States and their weights at a particular time. The relation between nations and their capitals, if we exclude those nations that maintain multiple capitals [1]. The relation sqr between natural numbers n and their squares n2. The relation ln between positive real numbers x and their natural logarithms ln(x). Note that the relation between real numb ...

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Function mathematics, Function mathematics - Introduction, Function mathematics - Functions of more than one variable, Function mathematics - History, Function mathematics - Formal definition, Function mathematics - Domains codomains and ranges, Function mathematics - Injective surjective and bijective functions, Function mathematics - Images and preimages, Function mathematics - Graph of a function, Function mathematics - Examples of functions, Function mathematics - Properties of functions, Function mathematics - Ambiguous functions, Function mathematics - n-ary function: function of several variables, Function mathematics - Composing functions, Function mathematics - Inverse function, Function mathematics - Restrictions and extensions, Function mathematics - Pointwise operations, Function mathematics - Computable and non-computable functions, Function mathematics - Functions from the categorical viewpoint

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The graph of a function f is the set of all ordered pairs(x, f(x)), for all x in the domain X. There are theorems formulated or proved most easily in terms of the graph, such as the closed graph theorem. If X and Y are real lines, then this definition coincides with the familiar sense of "graph" as a picture of the function, with the ordered pairs plotted as Cartesian coordinates. Note that a binary relation on the two sets X and Y could be identified with an ordere ...

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Function mathematics, Function mathematics - Introduction, Function mathematics - Functions of more than one variable, Function mathematics - History, Function mathematics - Formal definition, Function mathematics - Domains codomains and ranges, Function mathematics - Injective surjective and bijective functions, Function mathematics - Images and preimages, Function mathematics - Graph of a function, Function mathematics - Examples of functions, Function mathematics - Properties of functions, Function mathematics - Ambiguous functions, Function mathematics - n-ary function: function of several variables, Function mathematics - Composing functions, Function mathematics - Inverse function, Function mathematics - Restrictions and extensions, Function mathematics - Pointwise operations, Function mathematics - Computable and non-computable functions, Function mathematics - Functions from the categorical viewpoint

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The appropriate notion of a morphism between two lattices can easily be derived from the algebraic definition above: given two lattices (L, , ) and (M, , ), a homomorphism of lattices is a function f : L → M with the properties that f(xy) = f(x) f(y), and f(xy) = f(x) f(y). Thus f is a homomorphism of the two underlying semilattices. If the lattices are ...

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Lattice order, Lattice order - Formal definition, Lattice order - Lattices as posets, Lattice order - Lattices as algebraic structures, Lattice order - Connection between the two definitions, Lattice order - Examples, Lattice order - Morphisms of lattices, Lattice order - Properties of lattices, Lattice order - Completeness, Lattice order - Distributivity, Lattice order - Modularity, Lattice order - Continuity and algebraicity, Lattice order - Complements and pseudo-complements, Lattice order - Free lattices, Lattice order - Important lattice-theoretic notions

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The definitions above already introduced the simple condition of being a bounded lattice. A number of other important properties, many of which lead to interesting special classes of lattices, will be introduced below. Lattice order - Completeness. A highly relevant class of lattices are the complete lattices. A lattice is complete if all of its subsets have both a join and a meet, which should be contrasted to the above definition of a lattice where one only requires the existence of all (non-empty) finite joins and me ...

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Lattice order, Lattice order - Formal definition, Lattice order - Lattices as posets, Lattice order - Lattices as algebraic structures, Lattice order - Connection between the two definitions, Lattice order - Examples, Lattice order - Morphisms of lattices, Lattice order - Properties of lattices, Lattice order - Completeness, Lattice order - Distributivity, Lattice order - Modularity, Lattice order - Continuity and algebraicity, Lattice order - Complements and pseudo-complements, Lattice order - Free lattices, Lattice order - Important lattice-theoretic notions

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Using the standard definition of universal algebra, a free lattice over a generating set S is a lattice L together with a function i:S→L, such that any function f from S to the underlying set of some lattice M can be factored uniquely through a lattice homomorphism f° from L to M. Stated differently, for every element s of S we find that f(s) = f°(i(s)) and that f° is the only lattice homomorphism w ...

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Lattice order, Lattice order - Formal definition, Lattice order - Lattices as posets, Lattice order - Lattices as algebraic structures, Lattice order - Connection between the two definitions, Lattice order - Examples, Lattice order - Morphisms of lattices, Lattice order - Properties of lattices, Lattice order - Completeness, Lattice order - Distributivity, Lattice order - Modularity, Lattice order - Continuity and algebraicity, Lattice order - Complements and pseudo-complements, Lattice order - Free lattices, Lattice order - Important lattice-theoretic notions

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One often considers continuous group actions: the group G is a topological group, X is a topological space, and the map G × X → X is continuous with respect to the product topology of G × X. The space X is also called a G-space in this case. This is indeed a generalization, since every group can be considered a topological group by using the discrete topology. All the concepts introduced above still work in this context, however we define ...

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Group action, Group action - Definition, Group action - Examples, Group action - Types of actions, Group action - Orbits and stabilizers, Group action - Morphisms and isomorphisms between G-sets, Group action - Continuous group actions, Group action - Strongly continuous group action and smooth vector, Group action - Generalizations

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If X and Y are two G-sets, we define a morphism from X to Y to be a function f : X → Y such that f(g.x) = g.f(x) for all g in G and all x in X. Morphisms of G-sets are also called equivariant maps or G-maps. If such a function f is bijective, then its inverse is also a morphism, and we call f an isomorphism and the two G-sets X and Y are called iso ...

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Group action, Group action - Definition, Group action - Examples, Group action - Types of actions, Group action - Orbits and stabilizers, Group action - Morphisms and isomorphisms between G-sets, Group action - Continuous group actions, Group action - Strongly continuous group action and smooth vector, Group action - Generalizations

Read more here: » Group action: Encyclopedia II - Group action - Morphisms and isomorphisms between G-sets

If G is a group and X is a set, then a (left) group action of G on X is a binary function (where the image of and is written as ) which satisfies the following two axioms: for all and for every (e denotes the identity elemen ...

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Group action, Group action - Definition, Group action - Examples, Group action - Types of actions, Group action - Orbits and stabilizers, Group action - Morphisms and isomorphisms between G-sets, Group action - Continuous group actions, Group action - Strongly continuous group action and smooth vector, Group action - Generalizations

Read more here: » Group action: Encyclopedia II - Group action - Definition

The action of G on X is called transitive if for any two x, y in X there exists an g in G such that g·x = y; n-transitive if G acts transitively on Xn. sharply n-transitive if G acts regularly on Xn. faithful (or effective) if for any two different g, h ...

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Group action, Group action - Definition, Group action - Examples, Group action - Types of actions, Group action - Orbits and stabilizers, Group action - Morphisms and isomorphisms between G-sets, Group action - Continuous group actions, Group action - Strongly continuous group action and smooth vector, Group action - Generalizations

Read more here: » Group action: Encyclopedia II - Group action - Types of actions

In the following, let L be a lattice. We define some order-theoretic notions that are of particular importance in lattice theory. An element x of L is called join-irreducible iff x = a v b implies x = a or x = b for any a, b in L, if L has a 0, x is sometimes required to be different from 0. When the first condition is generalized to arbitrary joins Vai, x i ...

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Lattice order, Lattice order - Formal definition, Lattice order - Lattices as posets, Lattice order - Lattices as algebraic structures, Lattice order - Connection between the two definitions, Lattice order - Examples, Lattice order - Morphisms of lattices, Lattice order - Properties of lattices, Lattice order - Completeness, Lattice order - Distributivity, Lattice order - Modularity, Lattice order - Continuity and algebraicity, Lattice order - Complements and pseudo-complements, Lattice order - Free lattices, Lattice order - Important lattice-theoretic notions

Read more here: » Lattice order: Encyclopedia II - Lattice order - Important lattice-theoretic notions

Formally, a ringed space is a topological space X together with a sheaf of commutative rings OX on X. The sheaf OX is called the structure sheaf of X. A locally ringed space is a ringed space (X, OX) such that all stalks of OX are local rings (i.e. they have unique maximal ideals). Note that it is not required that OX(U) be a local ring for every open set U — in fact, ...

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Ringed space, Ringed space - Definition, Ringed space - Examples, Ringed space - Morphisms, Ringed space - Tangent spaces, Ringed space - O_X modules

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