# Graded structure > Mediated Wiki article. Canonical URL: https://mediated.wiki/source/Graded_structure > Markdown URL: https://mediated.wiki/source/Graded_structure.md > Source: https://en.wikipedia.org/wiki/Graded_structure > Source revision: 1235547315 > License: Creative Commons Attribution-ShareAlike 4.0 International (https://creativecommons.org/licenses/by-sa/4.0/) For other uses, see [graded (disambiguation)](/source/Graded_(disambiguation)). In [mathematics](/source/Mathematics), the term "**graded**" has a number of meanings, mostly related: In [abstract algebra](/source/Abstract_algebra), it refers to a family of concepts: - An [algebraic structure](/source/Algebraic_structure) X is said to be I-**graded** for an [index set](/source/Index_set) I if it has a **gradation** or **grading**, i.e. a decomposition into a [direct sum](/source/Direct_sum) X = \bigoplus_{i \in I} X_i of structures; the elements of X_i are said to be "**homogeneous** of **degree** *i*. - The index set I is most commonly \N or \Z, and may be required to have extra structure depending on the type of X. - Grading by \Z_2 (i.e. \Z/2\Z) is also important; see e.g. [signed set](/source/Signed_set) (the \Z_2-graded sets). - The **trivial** (\Z- or \N-) gradation has X_0 = X, X_i = 0 for i \neq 0 and a suitable trivial structure 0. - An algebraic structure is said to be [doubly graded](/source/Doubly_graded) if the index set is a direct product of sets; the pairs may be called "**bidegrees**" (e.g. see [Spectral sequence](/source/Spectral_sequence)). - A I-[graded vector space](/source/Graded_vector_space) or **graded linear space** is thus a [vector space](/source/Vector_space) with a decomposition into a [direct sum](/source/Direct_sum_of_vector_spaces) V = \bigoplus_{i \in I} V_i of spaces. - A [graded linear map](/source/Graded_linear_map) is a map between graded vector spaces respecting their gradations. - A [graded ring](/source/Graded_ring) is a [ring](/source/Ring_(mathematics)) that is a [direct sum](/source/Direct_sum) of additive [abelian groups](/source/Abelian_group) R_i such that R_i R_j \subseteq R_{i+j}, with i taken from some [monoid](/source/Monoid), usually \N or \mathbb{Z}, or [semigroup](/source/Semigroup) (for a [ring without identity](/source/Ring_without_identity)). - The [associated graded ring](/source/Associated_graded_ring) of a [commutative ring](/source/Commutative_ring) R with respect to a proper [ideal](/source/Ideal_(ring_theory)) I is \operatorname{gr}_I R = \bigoplus_{n \in \N} I^n/I^{n+1}. - A [graded module](/source/Graded_module) is left [module](/source/Module_(mathematics)) M over a graded ring that is a [direct sum](/source/Direct_sum_of_modules) \bigoplus_{i \in I} M_i of modules satisfying R_i M_j \subseteq M_{i+j}. - The [associated graded module](/source/Associated_graded_module) of an R-module M with respect to a proper ideal I is \operatorname{gr}_I M = \bigoplus_{n \in \N} I^n M/ I^{n+1} M. - A [differential graded module](/source/Differential_graded_module), **differential graded \mathbb{Z}-module** or **DG-module** is a graded module M with a **differential** d \colon M \to M \colon M_i \to M_{i+1} making M a **chain complex**, i.e. d \circ d = 0 . - A [graded algebra](/source/Graded_algebra) is an [algebra](/source/Algebra_over_a_ring) A over a ring R that is graded as a ring; if R is graded we also require A_i R_j \subseteq A_{i+j} \supseteq R_iA_j. - The graded [Leibniz rule](/source/Product_rule) for a map d\colon A \to A on a graded algebra A specifies that d(a \cdot b) = (da) \cdot b + (-1)^{|a|}a \cdot (db). - A [differential graded algebra](/source/Differential_graded_algebra), **DG-algebra** or **DGAlgebra** is a graded algebra that is a differential graded module whose differential obeys the graded Leibniz rule. - A [homogeneous derivation](/source/Homogeneous_derivation) on a graded algebra *A* is a homogeneous linear map of grade *d* = |*D*| on *A* such that D(ab) = D(a)b + \varepsilon^{|a||D|}aD(b), \varepsilon = \pm 1 acting on homogeneous elements of *A*. - A [graded derivation](/source/Derivation_(abstract_algebra)#Graded_derivations) is a sum of homogeneous derivations with the same \varepsilon. - A **DGA** is an augmented DG-algebra, or **[differential graded augmented algebra](/source/Differential_graded_augmented_algebra)**, (see [Differential graded algebra](/source/Differential_graded_algebra)). - A [superalgebra](/source/Superalgebra) is a \mathbb{Z}_2-graded algebra. - A [graded-commutative](/source/Graded-commutative) superalgebra satisfies the "supercommutative" law yx = (-1)^{|x| |y|}xy. for homogeneous *x*,*y*, where |a| represents the "parity" of a, 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](/source/Graded_Lie_algebra) is a [Lie algebra](/source/Lie_algebra) that is graded as a vector space by a gradation compatible with its Lie bracket. - A [graded Lie superalgebra](/source/Graded_Lie_superalgebra) is a graded Lie algebra with the requirement for anticommutativity of its Lie bracket relaxed. - A [supergraded Lie superalgebra](/source/Supergraded_Lie_superalgebra) is a graded Lie superalgebra with an additional super \Z_2-gradation. - A [differential graded Lie algebra](/source/Differential_graded_Lie_algebra) is a graded vector space over a [field](/source/Field_(mathematics)) of [characteristic](/source/Characteristic_(algebra)) zero together with a bilinear map [\ , ]\colon L_i \otimes L_j \to L_{i+j} and a differential d\colon L_i \to L_{i-1} satisfying [x,y] = (-1)^{|x||y|+1}[y,x], for any homogeneous elements *x*, *y* in *L*, the "graded [Jacobi identity](/source/Jacobi_identity)" and the graded Leibniz rule. - The **Graded Brauer group** is a synonym for the [Brauer–Wall group](/source/Brauer%E2%80%93Wall_group) BW(F) classifying finite-dimensional graded central [division algebras](/source/Division_algebra) over the field *F*. - An \mathcal{A}-[graded category](/source/Graded_category) for a [category](/source/Category_(mathematics)) \mathcal{A} is a category \mathcal{C} together with a [functor](/source/Functor) F\colon \mathcal{C} \rightarrow \mathcal{A}. - A [differential graded category](/source/Differential_graded_category) or **DG category** is a category whose morphism sets form differential graded \mathbb{Z}-modules. - [Graded manifold](/source/Graded_manifold) – extension of the [manifold](/source/Manifold) concept based on ideas coming from supersymmetry and [supercommutative algebra](/source/Supercommutative_algebra), including sections on - [Graded function](/source/Graded_manifold#Graded_functions) - [Graded vector fields](/source/Graded_manifold#Graded_vector_fields) - [Graded exterior forms](/source/Graded_manifold#Graded_exterior_forms) - [Graded differential geometry](/source/Graded_manifold#Graded_differential_geometry) - [Graded differential calculus](/source/Graded_manifold#Graded_differential_calculus) In other areas of mathematics: - [Functionally graded elements](/source/Functionally_graded_element) are used in [finite element analysis](/source/Finite_element_analysis). - A [graded poset](/source/Graded_poset) is a [poset](/source/Poset) P with a **rank function** \rho\colon P \to \N compatible with the ordering (i.e. \rho(x) < \rho(y) \implies x < y) such that y [covers](/source/Covering_relation) x \implies \rho(y) = \rho(x)+1 . --- Adapted from the Wikipedia article [Graded structure](https://en.wikipedia.org/wiki/Graded_structure) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Graded_structure?action=history)). 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