\begin{definition}[Definition:Filtered Algebra]
A '''filtered algebra''' is a generalization of the notion of a graded algebra.
A '''filtered algebra''' over the field $k$ is an algebra $\struct {A_k, \oplus}$ over $k$ which has an increasing sequence $\set 0 \subset F_0 \subset F_1 \subset \cdots \subset F_i \subset \cdots \subset A$ of substructures of $A$ such that:
:$\ds A = \bigcup_{i \mathop \in \N} F_i$
and that is compatible with the multiplication in the following sense:  
:$\forall m, n \in \N: F_m \cdot F_n \subset F_{n + m}$
\end{definition}