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Irreducible component information


In algebraic geometry, an irreducible algebraic set or irreducible variety is an algebraic set that cannot be written as the union of two proper algebraic subsets. An irreducible component of an algebraic set is an algebraic subset that is irreducible and maximal (for set inclusion) for this property. For example, the set of solutions of the equation xy = 0 is not irreducible, and its irreducible components are the two lines of equations x = 0 and y = 0.

It is a fundamental theorem of classical algebraic geometry that every algebraic set may be written in a unique way as a finite union of irreducible components.

These concepts can be reformulated in purely topological terms, using the Zariski topology, for which the closed sets are the algebraic subsets: A topological space is irreducible if it is not the union of two proper closed subsets, and an irreducible component is a maximal subspace (necessarily closed) that is irreducible for the induced topology. Although these concepts may be considered for every topological space, this is rarely done outside algebraic geometry, since most common topological spaces are Hausdorff spaces, and, in a Hausdorff space, the irreducible components are the singletons.

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Irreducible component

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two proper algebraic subsets. An irreducible component of an algebraic set is an algebraic subset that is irreducible and maximal (for set inclusion) for...

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Component

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Component ingredient, in a culinary dish Composition (disambiguation) Decomposition (disambiguation) Giant component Identity component Irreducible component...

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Irreducible polynomial

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In mathematics, an irreducible polynomial is, roughly speaking, a polynomial that cannot be factored into the product of two non-constant polynomials...

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Irreducible complexity

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Irreducible complexity (IC) is the argument that certain biological systems with multiple interacting parts would not function if one of the parts were...

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Hyperconnected space

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In the mathematical field of topology, a hyperconnected space or irreducible space is a topological space X that cannot be written as the union of two...

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Projective variety

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are in general position). If mi denotes the multiplicity of an irreducible component Zi in the intersection (i.e., intersection multiplicity), then the...

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Flat morphism

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for every irreducible component T of f−1(Z), there is an irreducible component T′ of f−1(Z′) containing T. For every irreducible component T of X, the...

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Morphism of algebraic varieties

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∣ Z  an irreducible component of  f − 1 ( f ( x ) )  containing  x } . {\displaystyle e(x)=\max\{\dim Z\mid Z{\text{ an irreducible component of }}f^{-1}(f(x)){\text{...

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Emergence

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direct causal action of a high-level system on its components; qualities produced this way are irreducible to the system's constituent parts. The whole is...

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Dimension of a scheme

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V_{1}\subsetneq \cdots \subsetneq V_{\ell }\subset X.} An irreducible subset of X is an irreducible component of X if and only if the codimension of it in X is...

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Hypersurface

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follows: hypersurfaces are exactly the algebraic sets whose all irreducible components have dimension n – 1. A real hypersurface is a hypersurface that...

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Representation theory

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representation has a subrepresentation, but only has one non-trivial irreducible component. For example, the additive group ( R , + ) {\displaystyle (\mathbb...

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Absolute irreducibility

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absolutely irreducible if it is irreducible over the complex field. For example, x 2 + y 2 − 1 {\displaystyle x^{2}+y^{2}-1} is absolutely irreducible, but...

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Primary decomposition

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algebraic set may be uniquely decomposed into a finite union of irreducible components. It has a straightforward extension to modules stating that every...

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Algebraic geometry

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is unique. Thus its elements are called the irreducible components of the algebraic set. An irreducible algebraic set is also called a variety. It turns...

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Markov chain

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transition per state is either not irreducible or not aperiodic, hence cannot be ergodic. Some authors call any irreducible, positive recurrent Markov chains...

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Ideal quotient

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One geometric application of the ideal quotient is removing an irreducible component of an affine scheme. For example, let I = ( x y z ) , J = ( x y...

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Intersection theory

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dim A + dim B − dim X, then A · B is a linear combination of the irreducible components of A ∩ B, with coefficients the intersection multiplicities. At...

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Factorization of polynomials over finite fields

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factorization of a polynomial consists of decomposing it into a product of irreducible factors. This decomposition is theoretically possible and is unique for...

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Isotypic component

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decomposed into irreducible submodules V = ⨁ i = 1 N V i {\displaystyle V=\bigoplus _{i=1}^{N}V_{i}} . Each finite-dimensional irreducible representation...

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Representation theory of finite groups

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group G {\displaystyle G} are called disjoint, if they have no irreducible component in common, i.e. if ⟨ V 1 , V 2 ⟩ G = 0. {\displaystyle \langle V_{1}...

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