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 $X = \bigoplus_{i \in I} X_i$ 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 (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 $V = \bigoplus_{i \in I} V_i$ 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 $$\N$$ 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 $\operatorname{gr}_I R = \bigoplus_{n \in \N} I^n/I^{n+1}$.
 * A graded module is left module $$M$$ over a graded ring that is a direct sum $\bigoplus_{i \in I} M_i$ 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 $\operatorname{gr}_I M = \bigoplus_{n \in \N} I^n M/ I^{n+1} M$.
 * 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 $$A_i R_j \subseteq A_{i+j} \supseteq R_iA_j$$.
 * The graded Leibniz rule for a map $$d\colon A \to A$$ on a graded algebra $$A$$ specifies that $$d(a \cdot b) = (da) \cdot b + (-1)^{|a|}a \cdot (db)$$.
 * 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 $$\Z_2$$-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 $$\rho\colon P \to \N$$ compatible with the ordering (i.e. $$\rho(x) < \rho(y) \implies x < y$$) such that $$y$$ covers $$x \implies \rho(y) = \rho(x)+1$$.