Definition 1

Given a commutative group $G$ written additively and a set $\Delta$, a graduation of type $\Delta$ on $G$ is a family $(G_\lambda)_{\lambda \in \Delta}$ of subgroups of $G$, of which $G$ is the direct sum. The set $G$, with the structure defined by its group law and its graduation, is called a graded (commutative) group of type $\Delta$.

$\Delta$ is called the set of degrees of $G$. An element $x \in G$ is called homogeneous if it belongs to one of the $G_\lambda$, homogeneous of degree $\lambda$ if $x \in G_\lambda$. The element 0 is therefore homogeneous of all degrees; but if $x \neq 0$ is homogeneous, it belongs to only one of the $G_\lambda$; the index $\lambda$ such that $x \in G_\lambda$ is then called the degree of $x$ (or sometimes the weight of $x$) and is sometimes denoted by $\deg(x)$. Every $y \in G$ may be written uniquely as a sum $\sum_\lambda y_\lambda$ of homogeneous elements with $y_\lambda \in G_\lambda$; $y_\lambda$ is called the homogeneous component of degree $\lambda$ (or simply the component of degree $\lambda$) of $y$. When the word "weight" is used instead of "degree", the adjective "homogeneous" is replaced by "isobaric".