Example

(1) Given any commutative monoid $\Delta$ (with identity element 0) and a commutative group $G$, a graduation $(G_\lambda)_{\lambda \in \Delta}$ is defined on $G$ by taking $G_0 = G$ and $G_\lambda = \{0\}$ for $\lambda \neq 0$; this graduation is called trivial.

(2) Let $\Delta, \Delta'$ be two sets and $\rho$ a mapping of $\Delta$ into $\Delta'$. Let $(G_\lambda)_{\lambda \in \Delta}$ be a graduation of type $\Delta$ on a commutative group $G$; for $\mu \in \Delta'$, let $G'_\mu$ be the sum of the $G_\lambda$ such that $\rho(\lambda) = \mu$; clearly $(G'_\mu)_{\mu \in \Delta'}$ is a graduation of type $\Delta'$ on $G$, said to be derived from $(G_\lambda)$ by means of the mapping $\rho$.

When $\Delta$ is a commutative group written additively and $\rho$ the mapping $\lambda \mapsto -\lambda$ of $\Delta$ onto itself, $(G'_\mu)$ is called the opposite graduation of $(G_\lambda)$.

(3) If $\Delta = \Delta_1 \times \Delta_2$ is a product of two sets, a graduation of type $\Delta$ is called a bigraduation of types $\Delta_1, \Delta_2$. For all $\lambda \in \Delta_1$, let $G'_\lambda = \bigoplus_{\mu \in \Delta_2} G_{\lambda \mu}$ and, for all $\mu \in \Delta_2$, let $G''_\mu = \bigoplus_{\lambda \in \Delta_1} G_{\lambda \mu}$; clearly $(G'_\lambda)_{\lambda \in \Delta_1}$ is a graduation of type $\Delta_1$ and $(G''_\mu)_{\mu \in \Delta_2}$ a graduation of type $\Delta_2$ on $G$; these graduations are called the partial graduations derived from the bigraduation $(G_{\lambda \mu})$. Note that $G_{\lambda \mu} = G'_\lambda \cap G''_\mu$; conversely, if $(G'_\lambda)_{\lambda \in \Delta_1}$ and $(G''_\mu)_{\mu \in \Delta_2}$ are two graduations on $G$ such that $G$ is the direct sum of the $G_{\lambda \mu} = G'_\lambda \cap G''_\mu$, these subgroups form a bigraduation of types $\Delta_1, \Delta_2$ on $G$, of which $(G'_\lambda)$ and $(G''_\mu)$ are the partial graduations. We leave to the reader the task of generalizing this to the case where $\Delta$ is a finite product of sets.

(4) Let $\Delta_0$ be a commutative monoid written additively, with identity element denoted by 0; let $I$ be any set and $\Delta_0^{(I)} = \Delta$ denote the submonoid of the product $\Delta_0^I$ consisting of the families $(\lambda_i)_{i \in I}$ of finite support. Let $\rho : \Delta \to \Delta_0$ be the surjective (codiagonal) homomorphism of $\Delta$ into $\Delta_0$ defined by $\rho((\lambda_i)) = \sum_{i \in I} \lambda_i$. From every graduation of type $\Delta$ a graduation of type $\Delta_0$ is derived by means of $\rho$ (Example 2); it is called the total graduation associated with the given "multigraduation" of type $\Delta$.

The definitions and examples of this no. extend immediately to the case where $G$ is a group which is not necessarily commutative; it is simply necessary to replace everywhere the notion of direct sum by that of "restricted sum" (§ 1, no. 6, Remark). Note that in this case the $G_\lambda$ are normal subgroups of $G$ and that for $\lambda \neq \mu$ every element of $G_\lambda$ is permutable with every element of $G_\mu$.