Definition:Filtered Algebra

Definition
A filtered algebra is a generalization of the notion of a graded algebra.

A filtered algebra over the field $k$ is an algebra $\left({A, \oplus}\right)$ over $k$ which has an increasing sequence $\left\{{0}\right\} \subset F_0 \subset F_1 \subset \cdots \subset F_i \subset \cdots \subset A$ of substructures of $A$ such that:


 * $\displaystyle A = \bigcup_{i \in \N} F_i$

and that is compatible with the multiplication in the following sense:


 * $\forall m, n \in \N: F_m \cdot F_n \subset F_{n+m}$