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 degreei ".
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 doubly graded if the index set is a direct product of sets; the pairs may be called "bidegrees" (e.g. see Spectral sequence).
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 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 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 differential graded augmented algebra, (see Differential 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 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.