Graded (mathematics)

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 $X$ is said to be $I$-graded for an index set $I$ if it has a gradation or grading, i.e. a decomposition into a direct sum $X = \bigoplus_ X_i$ 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 (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 if the index set is a direct product of sets; the pairs may be called "bidegrees" (e.g. see Spectral sequence).
* A $I$-graded vector space or graded linear space is thus a vector space with a decomposition into a direct sum $V = \bigoplus_ V_i$ 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 additive abelian groups $R_i$ such that $R_i R_j \subseteq R_$, with $i$ taken from some monoid, usually $\N$ or $\mathbb$, or semigroup (for a ring without identity).
** The associated graded ring of a commutative ring $R$ with respect to a proper ideal $I$ is $\operatorname_I R = \bigoplus_ I^n/I^$.
* A graded module is left module $M$ over a graded ring that is a direct sum $\bigoplus_ M_i$ of modules satisfying $R_i M_j \subseteq M_$.
** The associated graded module of an $R$-module $M$ with respect to a proper ideal $I$ is $\operatorname_I M = \bigoplus_ I^n M/ I^ M$.
** A differential graded module, differential graded $\mathbb$-module or DG-module is a graded module $M$ with a differential $d \colon M \to M \colon M_i \to M_$ making $M$ a chain complex, i.e. $d \circ d = 0$ .
* A graded algebra is an algebra $A$ over a ring $R$ that is graded as a ring; if $R$ is graded we also require $A_i R_j \subseteq A_ \supseteq R_iA_j$.
** The graded Leibniz rule for a map $d\colon A \to A$ on a graded algebra $A$ specifies that $d(a \cdot b) = (da) \cdot b + (-1)^a \cdot (db)$.
** A 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 on a graded algebra ''A'' is a homogeneous linear map of grade ''d'' = , ''D'', on ''A'' such that $D(ab) = D(a)b + \varepsilon^aD(b), \varepsilon = \pm 1$ acting on homogeneous elements of ''A''.
** A graded derivation is a sum of homogeneous derivations with the same $\varepsilon$.
** A DGA is an augmented DG-algebra, or differential graded augmented algebra, (see Differential graded algebra).
** A superalgebra is a $\mathbb_2$-graded algebra.
*** A graded-commutative superalgebra satisfies the "supercommutative" law $yx = (-1)^xy.$ 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 is a Lie algebra that 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 $\Z_2$-gradation.
** A differential graded Lie algebra is a graded vector space over a field of characteristic zero together with a bilinear map , colon L_i \otimes L_j \to L_ and a differential $d\colon L_i \to L_$ satisfying ,y= (-1)^ ,x 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 $BW(F)$ classifying finite-dimensional graded central division algebras over the field ''F''.
* An $\mathcal$-graded category for a category $\mathcal$ is a category $\mathcal$ together with a functor $F\colon \mathcal \rightarrow \mathcal$.
** A differential graded category or DG category is a category whose morphism sets form differential graded $\mathbb$-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 $P$ 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 $x \implies \rho(y) = \rho(x)+1$ .

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Linear algebra
Differential operators