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An Entity of Type : yago:MathematicalRelation113783581, within Data Space : dbpedia.org associated with source document(s)  In mathematics, the term “graded” has a number of meanings, mostly related: In abstract algebra, it refers to a family of concepts: * An algebraic structure is said to be -graded for an index set if it has a gradation or grading, i.e. a decomposition into a direct sum of structures; the elements of are said to be “homogeneous of degree i”. * The index set is most commonly or , and may be required to have extra structure depending on the type of . * Grading by (i.e. ) is also important; see e.g. signed set (the -graded sets). * The trivial (- or -) gradation has for and a suitable trivial structure . * An algebraic structure is said to be if the index set is a direct product of sets; the pairs may be called “bidegrees” (e.g. see spectral sequence). * A -graded vector space o

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• Graded (mathematics)
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• In mathematics, the term “graded” has a number of meanings, mostly related: In abstract algebra, it refers to a family of concepts: * An algebraic structure is said to be -graded for an index set if it has a gradation or grading, i.e. a decomposition into a direct sum of structures; the elements of are said to be “homogeneous of degree i”. * The index set is most commonly or , and may be required to have extra structure depending on the type of . * Grading by (i.e. ) is also important; see e.g. signed set (the -graded sets). * The trivial (- or -) gradation has for and a suitable trivial structure . * An algebraic structure is said to be if the index set is a direct product of sets; the pairs may be called “bidegrees” (e.g. see spectral sequence). * A -graded vector space o
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• In mathematics, the term “graded” has a number of meanings, mostly related: In abstract algebra, it refers to a family of concepts: * An algebraic structure is said to be -graded for an index set if it has a gradation or grading, i.e. a decomposition into a direct sum of structures; the elements of are said to be “homogeneous of degree i”. * The index set is most commonly or , and may be required to have extra structure depending on the type of . * Grading by (i.e. ) is also important; see e.g. signed set (the -graded sets). * The trivial (- or -) gradation has for and a suitable trivial structure . * An algebraic structure is said to be if the index set is a direct product of sets; the pairs may be called “bidegrees” (e.g. see spectral sequence). * A -graded vector space or graded linear space is thus a vector space with a decomposition into a direct sum of spaces. * A graded linear map is a map between graded vector spaces respecting their gradations. * A graded ring is a ring that is a direct sum of abelian groups such that , with taken from some monoid, usually or , or semigroup (for a ring without identity). * The associated graded ring of a commutative ring with respect to a proper ideal is . * A graded module is left module over a graded ring which is a direct sum of modules satisfying . * The associated graded module of an -module with respect to a proper ideal is . * A differential graded module, differential graded -module or DG-module is a graded module with a differential making a chain complex, i.e. . * A graded algebra is an algebra over a ring that is graded as a ring; if is graded we also require . * The graded Leibniz rule for a map on a graded algebra specifies that . * A differential graded algebra, DG-algebra or DGAlgebra is a graded algebra which is a differential graded module whose differential obeys the graded Leibniz rule. * A homogeneous derivation on a graded algebra A is a homogeneous linear map of grade d = |D| on A such that acting on homogeneous elements of A. * A graded derivation is a sum of homogeneous derivations with the same . * A DGA is an augmented DG-algebra, or , (see differential graded algebra). * A superalgebra is a -graded algebra. * A graded-commutative superalgebra satisfies the “supercommutative” law for homogeneous x,y, where represents the “parity” of , i.e. 0 or 1 depending on the component in which it lies. * CDGA may refer to the category of augmented differential graded commutative algebras. * A graded Lie algebra is a Lie algebra which is graded as a vector space by a gradation compatible with its Lie bracket. * A graded Lie superalgebra is a graded Lie algebra with the requirement for anticommutativity of its Lie bracket relaxed. * A supergraded Lie superalgebra is a graded Lie superalgebra with an additional super -gradation. * A differential graded Lie algebra is a graded vector space over a field of characteristic zero together with a bilinear map and a differential satisfying for any homogeneous elements x, y in L, the “graded Jacobi identity” and the graded Leibniz rule. * The Graded Brauer group is a synonym for the Brauer–Wall group classifying finite-dimensional graded central division algebras over the field F. * An -graded category for a category is a category together with a functor . * A differential graded category or DG category is a category whose morphism sets form differential graded -modules. * Graded manifold – extension of the manifold concept based on ideas coming from supersymmetry and supercommutative algebra, including sections on * Graded function * Graded vector fields * Graded exterior forms * Graded differential geometry * Graded differential calculus In other areas of mathematics: * Functionally graded elements are used in finite element analysis. * A graded poset is a poset with a rank function compatible with the ordering (i.e. ) such that covers .
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