# Graded structure

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=⨁$_{i∈I}$X$_i${\textstyle 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 $\mathbb {N}$ or $\mathbb {Z}$, and may be required to have extra structure depending on the type of $X$.
** Grading by $\mathbb {Z} _{2}$ (i.e. $\mathbb {Z} /2\mathbb {Z}$) is also important; see e.g. signed set (the $\mathbb {Z} _{2}$-graded sets).
** The trivial ($\mathbb {Z}$- or $\mathbb {N}$-) 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=⨁$_{i∈I}$V$_i${\textstyle V=\bigoplus _{i\in I}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 $R_{i}R_{j}\subseteq R_{i+j}$, with $i$ taken from some monoid, usually $\mathbb {N}$ or $\mathbb {Z}$, or semigroup (for a ring without identity).
** The associated graded ring of a commutative ring $R$ with respect to a proper ideal $I$ is gr$_I$⁡R=⨁$_{n∈N}$I$^n$/I$^{n+1}${\textstyle \operatorname {gr} _{I}R=\bigoplus _{n\in \mathbb {N} }I^{n}/I^{n+1}}.
* A graded module is left module $M$ over a graded ring that is a direct sum ⨁$_{i∈I}$M$_i${\textstyle \bigoplus _{i\in I}M_{i}} of modules satisfying $R_{i}M_{j}\subseteq M_{i+j}$.
** The associated graded module of an $R$-module $M$ with respect to a proper ideal $I$ is gr$_I$⁡M=⨁$_{n∈N}$I$^n$M/I$^{n+1}$M{\textstyle \operatorname {gr} _{I}M=\bigoplus _{n\in \mathbb {N} }I^{n}M/I^{n+1}M}.
** A 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 is an algebra $A$ over a ring $R$ that is graded as a ring; if $R$ is graded we also require $A_{i}R_{j}\subseteq A_{i+j}\supseteq R_{i}A_{j}$.
** The graded Leibniz rule for a map $d\colon A\to A$ on a graded algebra $A$ specifies that $d(a\cdot b)=(da)\cdot b+(-1)^{|a|}a\cdot (db)$.
** 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 $D(ab)=D(a)b+\varepsilon ^{|a||D|}aD(b),\varepsilon =\pm 1$ 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 $\mathbb {Z} _{2}$-graded algebra.
*** A 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 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 $\mathbb {Z} _{2}$-gradation.
** A differential graded Lie algebra is a graded vector space over a field of characteristic zero together with a bilinear map $[\ ,]\colon L_{i}\otimes L_{j}\to L_{i+j}$ 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" 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 ${\mathcal {A}}$-graded category for a category ${\mathcal {A}}$ is a category ${\mathcal {C}}$ together with a functor $F\colon {\mathcal {C}}\rightarrow {\mathcal {A}}$.
** A differential graded category or DG category is a category whose morphism sets form differential graded $\mathbb {Z}$-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 $\rho \colon P\to \mathbb {N}$ compatible with the ordering (i.e. $\rho (x)<\rho (y)\implies x<y$) such that $y$ covers $x\implies \rho (y)=\rho (x)+1$ .