morphism

A homomorphism is a map from one algebraic structure to another of the same type that preserves all the relevant structure; i.e. properties like identity elements, inverse elements, and binary operations. N.B. Some authors use the word homomorphism in a larger context than that of algebra. Some take it to mean any kind of structure preserving map (such as continuous maps in topology), or even a more abstract kind of map—what we term a morphism—used in category theory. This article only treats the algebraic context. For more gene ...

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Homomorphism, Homomorphism - Informal discussion, Homomorphism - Formal definition, Homomorphism - Types of homomorphisms, Homomorphism - Kernel of a homomorphism

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The above terms are used in an analogous fashion in category theory, however, the definitions in category theory are more subtle; see the article on morphism for more details. Note that in the larger context of structure preserving maps, it is generally insufficient to define an isomorphism as a bijective morphism. One must also require that the inverse is a morphism of the same type. In the algebraic setting (at least within the context of universal algebra) this extra condition is automatically satisfied. ...

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Homomorphism, Homomorphism - Homomorphism for beginners, Homomorphism - Homomorphism for mathematicians, Homomorphism - Types of homomorphisms, Homomorphism - Kernel of a homomorphism

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In topology and related areas of mathematics a continuous function is a morphism between topological spaces; that is, a mapping which preserves the topological structure. Intuitively, a function is continuous if it maps nearby points to nearby points. For metric spaces, nearness is measured in terms of distance, leading to the ε-δ definition used in real analysis. For more general topological spaces, nearness is measured less directly in terms of open sets, leading to the definition below. If a top ...

Including:

• Continuous function topology - Definitions
• Continuous function topology - Open and closed set definition
• Continuous function topology - Neighborhood definition
• Continuous function topology - Closure and interior operator definition
• Continuous function topology - Closeness relation definition
• Continuous function topology - Useful properties of continuous maps
• Continuous function topology - Other notes

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The above terms are used in an analogous fashion in category theory, however, the definitions in category theory are more subtle; see the article on morphism for more details. Note that in the larger context of structure preserving maps, it is generally insufficient to define an isomorphism as a bijective morphism. One must also require that the inverse is a morphism of the same type. In the algebraic setting (at least within the context of universal algebra) this extra condition is automatically satisfied. ...

See also:

Homomorphism, Homomorphism - Informal discussion, Homomorphism - Formal definition, Homomorphism - Types of homomorphisms, Homomorphism - Kernel of a homomorphism

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A real vector bundle is given by the following data: topological spaces X (the "base space") and E (the "total space") a continuous map Ï€ : E → X (the "projection") for every x in X, the structure of a real vector space on the fiber π−1({x}) satisfying the following compatibility condition: for every point in X there is an open neighborhood U, a natural number n, and a homeomorphism φ : U × See also:

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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In the category of sets the image of a morphism is the inclusion from the ordinary image to Y. In many concrete categories such as groups, abelian groups and (left- or right) modules, the image of a morphism is the image of the correspondent morphism in the category of sets. In any normal category with a zero object and kernels and cokernels for every morphism, the image of a morphism f can be expressed as follows: im f = ker coker f See also:

Image category theory, Image category theory - Examples

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From one point of view, a groupoid is simply a category in which every morphism is an isomorphism (that is, invertible). To be explicit, a groupoid G is: A set G0 of objects; For each pair of objects x and y in G0, a set G(x,y) of morphisms (or arrows) from x to y — we write f : x → y to indicate that f is an elemen ...

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Groupoid, Groupoid - Definitions, Groupoid - Examples, Groupoid - Relation to groups, Groupoid - Covariance in special relativity, Groupoid - Lie groupoids and Lie algebroids

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Consider a group G acting on a set X. The orbit of a point x in X is the set of elements of X to which x can be moved by the elements of G. The orbit of x is denoted by Gx: The defining properties of a group guarantee that the set of orbits of X under the action of G form a partition of X. The associated equivalence relation is defined by saying x ~ y iff there exists a g in G with g·x< ...

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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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To rephrase this definition in a way that will work in any category C that has sufficient structure, we note that we can write the objects and morphisms involved in the definition above in a diagram which we will call (G), for "gluing": Here the first map is the product of the restriction maps resU,Ui,:F(U)→F(Ui) and each pair of arrows represents the two restrictions resUi,Ui∩Uj:USee also:

Gluing axiom, Gluing axiom - Removing restrictions on C, Gluing axiom - Sheaves on a basis of open sets, Gluing axiom - The logic of C, Gluing axiom - Sheafification, Gluing axiom - Other gluing axioms

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Let C be a category. In order to define a kernel in the general category-theoretical sense, C needs to have zero morphisms. In that case, if f : X → Y is an arbitrary morphism in C, then a kernel of f is an equaliser of f and the zero morphism from X to Y. In symbols: ker(f) = eq(f, 0XY) To be more explicit, the following universal property can be used. A kernel of f is any morphism k : See also:

Kernel category theory, Kernel category theory - Definition, Kernel category theory - Examples, Kernel category theory - Relation to other categorical concepts, Kernel category theory - Relationship to algebraic kernels

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The prototypical example is the ring Z of all integers. Every field is an integral domain. Conversely, every Artinian integral domain is a field. In particular, the only finite integral domains are the finite fields. Rings of polynomials are integral domains if the coefficients come from an integral domain. For instance, the ring Z[X] of all polynomials in one variable with integer coefficients is an integral domain; so is the ring R[X,Y] of all polynomi ...

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Integral domain, Integral domain - Examples, Integral domain - Divisibility prime and irreducible elements, Integral domain - Field of fractions, Integral domain - Characteristic and homomorphisms

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Not all categories have initial or terminal objects, as will be seen below. Directly from the definition, one can show however that if an initial object exists, then it is unique up to a unique isomorphism. The same is true for terminal objects. The automorphism group of an initial (or terminal) object I is trivial. Aut(I) = Hom(I,I) = { idI }. ...

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Initial object, Initial object - Properties, Initial object - Examples

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Given two variables X and Y, one constructs the polynomial ring R[X], and then, on top of it, the ring (R[X])[Y]. This ring is considered the polynomial ring in the two variables R[X,Y]. For example, the polynomial P(X,Y) = X2Y2 + 4XY2 + 5X3 â ...

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Polynomial ring, Polynomial ring - Definition of a polynomial, Polynomial ring - The polynomial ring R[X], Polynomial ring - The polynomial ring in several variables, Polynomial ring - Equivalent definition, Polynomial ring - Properties, Polynomial ring - Some uses of polynomial rings

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As mentioned above, lattices can be characterized both as posets and as algebraic structures. Both approaches and their relationship are explained below. Lattice order - Lattices as posets. Consider a poset (L, ≤). L is a lattice if for all elements x and y of L, the set {x, y} has both a least upper bound (join, or supremum) and a greatest l ...

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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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A morphism of ringed spaces is simply a morphism of sheaves. Explicitly, a morphism from (X, OX) to (Y, OY) is given by the following data: a continuous map f : X → Y a family of ring homomorphisms φV : OY(V) → OX(f -1(V)) for every open set V of Y which commute with the restriction maps. That is, if V1 ⊂ VSee also:

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

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An associative algebra A over a field K is defined to be a vector space over K together with a K-bilinear multiplication A x A → A (where the image of (x,y) is written as xy) such that the associative law holds: (x y) z = x (y z) for all x, y and z in A. The bilinearity of the multiplication can be expressed as (x + y) z = x z + y z  
...

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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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Although Lie algebras are often studied in their own right, historically they arose as a means to study Lie groups. Given a Lie group, a Lie algebra can be associated to it either by endowing the tangent space to the identity with the differential of the adjoint map, or by considering the left-invariant vector fields as mentioned in the examples. This association is functorial, meaning that homomorphisms of Lie groups lift to homomorphisms of Lie algebras, and various properties are satisfied by this lifting: it commutes with composition, it maps subgroups, kernels, quotients and cokernels of Lie groups to subalgebras, kernels, ...

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Lie algebra, Lie algebra - Definition, Lie algebra - Examples, Lie algebra - Homomorphisms subalgebras and ideals, Lie algebra - Relation to Lie groups, Lie algebra - Classification of Lie algebras, Lie algebra - Category theoretic definition

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The exact definition of an automorphism depends on the type of "mathematical object" in question and what, precisely, constitutes an "isomorphism" of that object. The most general setting in which these words have meaning is an abstract branch of mathematics called category theory. Category theory deals with abstract objects and morphisms between those objects. In category theory, an automorphism is an endomorphism (i.e. a morphism from an object to itself) which is also an isomorph ...

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Automorphism, Automorphism - Definition, Automorphism - Automorphism group, Automorphism - Examples, Automorphism - Inner and outer automorphisms, Automorphism - Reference

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A precise definition is required for the purposes of mathematics. A function is a binary relation, f, with the property that for an element x there is no more than one element y such that x is related to y. This uniquely determined element y is denoted f(x). Because two definitions of binary relation are in use, there are actually two definitions of function, in ...

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Function mathematics, Function mathematics - Mathematical definition of a function, Function mathematics - First definition, Function mathematics - Second definition, Function mathematics - History of the concept, Function mathematics - Functions in other fields, Function mathematics - Domain codomain argument image, Function mathematics - Graph of a function, Function mathematics - Specifying a function, Function mathematics - Functions with multiple inputs and outputs, Function mathematics - Functions of two or more variables, Function mathematics - Functions with output in a product set, Function mathematics - Binary operations, Function mathematics - Argument order and lambda notation, Function mathematics - Examples of functions, Function mathematics - Image of a set, Function mathematics - Range of a function, Function mathematics - Preimage of a set, Function mathematics - Composing functions, Function mathematics - Inverse function, Function mathematics - Set of all functions, Function mathematics - Is a function more than its graph?, Function mathematics - Partial functions and multi-functions, Function mathematics - Classes of functions, Function mathematics - Injective surjective bijective, Function mathematics - Other properties, Function mathematics - Restrictions and extensions, Function mathematics - Pointwise operations, Function mathematics - Computable and non-computable functions, Function mathematics - Lambda calculus, Function mathematics - Functions in category theory

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Natural transformation - A worked example. Statements like "Every group is naturally isomorphic to its opposite group" abound in modern mathematics. We will now give the precise meaning of this statement as well as its proof. Consider the category Grp of all groups with group homomorphisms as morphisms. If (G,*) is a group, we define its opposite group (Gop,*op) as follows: Gop is the same set as G, and the operation *< ...

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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 modern idea of a mathematical function was introduced by Leibniz, and the associated notation y = f(x) was invented by Leonhard Euler, in the 18th century. But the intuitive idea of a function as any rule or procedure that assigns an output to each given input proved to be naive. Joseph Fourier, for example, claimed that every function had a Fourier series, something no mathematician would claim today. The concept of a function was not put on a rigorous basis u ...

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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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