Douglas Hofstadter provides an informal definition: The word "isomorphism" applies when two complex structures can be mapped onto each other, in such a way that to each part of one structure there is a corresponding part in the other structure, where "corresponding" means that the two parts play similar roles in their respective structures. (Gödel, Escher, Bach, p. 49) Formally, an isomorphism is a bijective map f such that both f and its inverse f −1 are homomorphisms, ...

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Isomorphism, Isomorphism - Definition, Isomorphism - Purpose, Isomorphism - Physical analogies, Isomorphism - Practical example, Isomorphism - Two abstract examples, Isomorphism - A relation-preserving isomorphism, Isomorphism - An operation-preserving isomorphism, Isomorphism - Applications

Read more here: » Isomorphism: Encyclopedia II - Isomorphism - Definition

Because abstract algebra studies sets with operations that generate interesting structure or properties on the set, the most interesting functions are those which preserve the operations. These functions are known as homomorphisms. For example, consider the natural numbers with addition as the operation. A function which preserves addition should have this property: f(a + b) = f(a) + f(b). Note that f(x) = 3x is a homomorphism, since f(a + b< ...

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

Read more here: » Homomorphism: Encyclopedia II - Homomorphism - Informal discussion

Homomorphism is one of the fundamental concepts in abstract algebra. Because abstract algebra studies sets with operations that generate interesting structure or properties on the set, the most interesting functions are those which preserve the operation. For example, consider the natural numbers with addition as the operation. A function which preserves addition should have this property: f(a + b) = f(a) + f(b). Note that f(x) = 3x is a homomorphism, ...

See also:

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

Read more here: » Homomorphism: Encyclopedia II - Homomorphism - Homomorphism for beginners

There may be different notions of connectedness that are intuitively similar, but different as formally defined concepts. We might wish to call a topological space connected if each pair of points in it is joined by a path. However this concept turns out to be different from standard topological connectedness; in particular, there are connected topological spaces for which this property does not hold. Because of this, different terminology is used; ...

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Connectedness, Connectedness - Other notions of connectedness, Connectedness - Connectivity

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Category theory is a mathematical theory that deals in an abstract way with mathematical structures and relationships between them. It is half-jokingly known as "generalized abstract nonsense". Categories appear in most branches of mathematics, in some areas of theoretical computer science and mathematical physics, and have been a unifying notion. Categories were first introduced by Samuel Eilenberg and Saunders Ma ...

Including:

• Category theory - Background
• Category theory - Historical notes
• Category theory - Categories objects and morphisms
• Category theory - Some properties of morphisms
• Category theory - Functors
• Category theory - Natural transformations and isomorphisms
• Category theory - Universal constructions limits and colimits
• Category theory - Equivalent categories
• Category theory - Further concepts and results
• Category theory - Higher-dimensional categories

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In mathematics, especially in algebraic geometry and the theory of complex manifolds, a coherent sheaf F on a locally ringed space X is a sheaf isomorphic with the cokernel of a morphism of OX-modules OXm → OXn. Here OX is the structure sheaf of local rings, given by definition on X. The form of the definition is a global (on X) way of carrying across the idea of a finitely-presented mo ...

Including:

• Coherent sheaf - Coherent cohomology

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In abstract algebra, a branch of mathematics, a monoid is an algebraic structure with a single, associative binary operation and an identity element. In other words, it is a unital semigroup. Monoid - Definition. A monoid is a magma (M,*), i.e. a set M with binary operation * : M × M → M, obeying the following axioms: Associativity: for all a, b, c in M, (a*b)*c = a*(b*c) Identity ...

Including:

• Monoid - Definition
• Monoid - Examples
• Monoid - Properties
• Monoid - Monoid homomorphisms
• Monoid - Relation to category theory

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In category theory, a zero morphism is a special kind of "trivial" morphism. Suppose C is a category, and for any two objects X and Y in C we are given a morphism 0XY : X → Y with the following property: for any two morphism f : R → S and g : U → V we obtain a commutative diagram: Then the morphisms 0XY are c ...

Including:

• Zero morphism - Examples

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In mathematics, a sheaf F on a topological space X is something that assigns a structure F(U) (such as a set, group, or ring) to each open set U of X. The structures F(U) are compatible with the operations of restricting the open set to smaller subsets and gluing smaller open sets to obtain a bigger one. A presheaf is similar to a sheaf, but it may not be possible to glue. Sheaves enable one to discuss in a refined way what is a local property, as appl ...

Including:

• Sheaf mathematics - Introduction
• Sheaf mathematics - The formal definition
• Sheaf mathematics - Definition of a presheaf
• Sheaf mathematics - The gluing axiom
• Sheaf mathematics - Examples
• Sheaf mathematics - Morphisms of sheaves
• Sheaf mathematics - Stalks of a sheaf at a point and germs of functions
• Sheaf mathematics - The étale space of a sheaf
• Sheaf mathematics - Generalizations
• Sheaf mathematics - History

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Analogy is either the cognitive process of transferring information from a particular subject (the analogue or source) to another particular subject (the target), or a linguistic expression corresponding to such a process. In a narrower sense, analogy is an inference or an argument from a particular to another particular, as opposed to deduction, induction and abduction, where at least one of the premises or the conclusion is general. The word analogy can also refer to the relation between the source and the target themselves, which is often, though not necessarily, a simil ...

Including:

• Analogy - Models and theories of analogy
• Analogy - Identity of relation
• Analogy - Shared abstraction
• Analogy - Special case of induction
• Analogy - Hidden deduction
• Analogy - Shared structure
• Analogy - High-level perception
• Analogy - Applications and types of analogy
• Analogy - Linguistics
• Analogy - Mathematics
• Analogy - Artificial intelligence
• Analogy - Anatomy
• Analogy - Law
• Analogy - Engineering

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Alexander Grothendieck (born March 28, 1928) was one of the most important mathematicians active in the 20th century. He was also one of its most extreme scientific personalities, with achievements over a short span of years that are still scarcely credible in their broad scope and sheer bulk, and an approach that antagonised even close followers. He made major contributions to algebraic geometry, homological algebra, and functional analysis. He was awarded the Fields Medal in 1966, and co-awarded the Crafoord Prize with Pierre Delig ...

Including:

• Alexander Grothendieck - Mathematical achievements
• Alexander Grothendieck - Major mathematical topics from Récoltes et Semailles
• Alexander Grothendieck - Life
• Alexander Grothendieck - Childhood and studies
• Alexander Grothendieck - Politics and retreat from scientific community
• Alexander Grothendieck - Manuscripts written in the 1980s
• Alexander Grothendieck - Disappearance

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In mathematics, connectedness is used to refer to various properties meaning, in some sense, "all one piece". When a mathematical object has such a property, we say it is connected; otherwise it is disconnected. When a disconnected object can be split naturally into connected pieces, each piece is usually called a component (or connected component). Many fields of mathematics include a formally defined property known as connectedness. In each field, the property may be defined differently. How ...

Including:

• Connectedness - Other notions of connectedness
• Connectedness - Connectivity

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In mathematics, categories allow one to formalize notions involving abstract structure and processes which preserve structure. Categories appear in virtually every branch of modern mathematics and are a central unifying notion. The study of categories in their own right is known as category theory. For more extensive motivational background and historical notes, see category theory and the list of category theory topics. Category mathematics - Definition. A category C consists of Including:

• Category mathematics - Definition
• Category mathematics - Examples
• Category mathematics - Types of morphisms
• Category mathematics - Types of categories

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In mathematics, particularly in algebraic geometry, complex analysis and number theory, an abelian variety is a complex torus that can be embedded into projective space. Also it is used for the generalization of this concept studied in algebraic geometry over fields more general than the complex numbers. One-dimensional abelian varieties are elliptic curves. Abelian variety - History and motivation. The success in the early nineteenth century of the theory of elliptic functions in giving a basis for the the ...

Including:

• Abelian variety - History and motivation
• Abelian variety - Analytic theory
• Abelian variety - Definition
• Abelian variety - Riemann conditions
• Abelian variety - The Jacobian of an algebraic curve
• Abelian variety - Abelian functions
• Abelian variety - Algebraic definition
• Abelian variety - Structure of the group of points
• Abelian variety - Polarization and dual abelian variety
• Abelian variety - Dual abelian variety
• Abelian variety - Polarizations
• Abelian variety - Polarizations over the complex numbers
• Abelian variety - Abelian scheme

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Isomorphism - A relation-preserving isomorphism. For example, if one object consists of a set X with an ordering ≤ and the other object consists of a set Y with an ordering then an isomorphism from X to Y is a bijective function f : X → Y such that iff u ≤ v. Such an isomorphism is called an order isomorphism. Isom ...

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Isomorphism, Isomorphism - Definition, Isomorphism - Purpose, Isomorphism - Physical analogies, Isomorphism - Practical example, Isomorphism - Two abstract examples, Isomorphism - A relation-preserving isomorphism, Isomorphism - An operation-preserving isomorphism, Isomorphism - Applications

Read more here: » Isomorphism: Encyclopedia II - Isomorphism - Two abstract examples

Group isomorphism is where the objects in question are groups. Similarly, if the objects are fields, it is called a field isomorphism. In Analysis, the Legendre transform maps hard differential equations into easier algebraic equations. In universal algebra, one can provide a general definition of isomorphism that covers these and many other cases. For a more general definition, see category theory. In graph theory, an isomorphism between two graphs G and H is a bijective map f from the ...

See also:

Isomorphism, Isomorphism - Definition, Isomorphism - Purpose, Isomorphism - Physical analogies, Isomorphism - Practical example, Isomorphism - Two abstract examples, Isomorphism - A relation-preserving isomorphism, Isomorphism - An operation-preserving isomorphism, Isomorphism - Applications

Read more here: » Isomorphism: Encyclopedia II - Isomorphism - Applications

The following is an example of an isomorphism from ordinary algebra. Consider the logarithm function: For any fixed base b, the logarithm function logb maps from the positive real numbers onto the real numbers ; formally: This mapping is one-to-one and onto, that is, it is a bijection from the domain to the codomain of the logarithm function. In addition to being an isomorphism of sets, the logarithm function also preserves certain operations. Specifically, consider the group of positive real numbers under ordinary multiplication. The logarithm function o ...

See also:

Isomorphism, Isomorphism - Definition, Isomorphism - Purpose, Isomorphism - Physical analogies, Isomorphism - Practical example, Isomorphism - Two abstract examples, Isomorphism - A relation-preserving isomorphism, Isomorphism - An operation-preserving isomorphism, Isomorphism - Applications

Read more here: » Isomorphism: Encyclopedia II - Isomorphism - Practical example

Main article: kernel (algebra) Any homomorphism f : X → Y defines an equivalence relation ~ on X by a ~ b iff f(a) = f(b). The relation ~ is called the kernel of f. It is a congruence relation on X. The quotient set X/~ can then be given an object-structure in a natural way, e.g., [x] * [y] = [x * y]. In that case the image of X in Y under the homomorphism f is neces ...

See also:

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

Read more here: » Homomorphism: Encyclopedia II - Homomorphism - Kernel of a homomorphism

In abstract algebra, a homomorphism is a map from one algebraic structure to another of the same type that preserves all the relevant structure. 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. ...

See also:

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

Read more here: » Homomorphism: Encyclopedia II - Homomorphism - Homomorphism for mathematicians

In mathematics, a vector bundle is a geometrical construct where to every point of a topological space (or manifold, or algebraic variety) we attach a vector space in a compatible way, so that all those vector spaces, "glued together", form another topological space (or manifold or variety). A typical example is the tangent bundle of a differentiable manifold: to every point of the manifold we attach the tangent space of the manifold at that point. Or consider a smooth curve in R2, and attach to every point of the curve the line normal to the curve at that ...

Including:

• 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

Read more here: » Vector bundle: Encyclopedia - Vector bundle