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 $\struct {A_k, \oplus}$ 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:


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

Also see

 * Definition:Graded Algebra