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

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...

In mathematics, the concept of irreducibility is used in several ways. A polynomial over a field may be an irreducible polynomial if it cannot be factored...

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...

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

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

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...

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

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...

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

point), every irreducible component of the non-empty algebraic set defined by the p i {\displaystyle p_{i}} is an irreducible component of the algebraic...

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

passing through P has an irreducible component that is reduced to the single point P. Therefore, the local ring at this component of the coordinate ring...

be a regular scheme and V, W closed integral subschemes. Then an irreducible component P of V ∩ W := V × X W {\displaystyle V\cap W:=V\times _{X}W} is...

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...

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}...

∣ 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{...

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...

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

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...

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

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...

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

O may be decomposed as the direct sum of irreducible unitary representations of O. Each isotypic component in this decomposition is called a superselection...

scheme with infinitely many irreducible components. (By contrast, a Noetherian scheme has only finitely many irreducible components.) It is also fruitful to...

of degree d 2 {\displaystyle d_{2}} , that does not contain any irreducible component of V; under these hypotheses, the intersection of V and H has dimension...

things) the decomposition of tensor products of irreducible representations of GLn into irreducible components is formulated in terms of certain skew semistandard...

then — under the assumption that V is locally normal at W — each irreducible component of the transform T[W] is of higher dimension than W. Here T is essentially...

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

irreducible representations (i.e., a reducible representation into irreducible representations, in cases where the numbers and types of irreducible components...