filtered algebra
Definition 1.
A filtered algebra over the field $k$ is an algebra $(A,\cdot )$ over $k$ which is endowed with a filtration^{} $\mathcal{F}={\{{F}_{i}\}}_{i\in \mathbb{N}}$ by subspaces^{}, compatible^{} with the multiplication^{} in the following sense
$$\forall m,n\in \mathbb{N},{F}_{m}\cdot {F}_{n}\subset {F}_{n+m}.$$ 
A special case of filtered algebra is a graded algebra^{}. In general there is the following construction that produces a graded algebra out of a filtered algebra.
Definition 2.
Let $(A,\cdot ,\mathcal{F})$ be a filtered algebra then the associated http://planetmath.org/node/3071graded algebra $\mathcal{G}(A)$ is defined as follows:

•
As a vector space^{}
$$\mathcal{G}(A)=\underset{n\in \mathbb{N}}{\oplus}{G}_{n},$$ where,
$${G}_{0}={F}_{0},\text{and}\forall n0,{G}_{n}={F}_{n}/{F}_{n1},$$ 
•
the multiplication is defined by
$$(x+{F}_{n})(y+{F}_{m})=x\cdot y+{F}_{n+m}$$
Theorem 3.
The multiplication is well defined and endows $\mathrm{G}\mathit{}\mathrm{(}A\mathrm{)}$ with the of a graded algebra, with gradation ${\mathrm{\{}{G}_{n}\mathrm{\}}}_{n\mathrm{\in}\mathrm{N}}$. Furthermore if $A$ is associative then so is $\mathrm{G}\mathit{}\mathrm{(}A\mathrm{)}$.
An example of a filtered algebra is the Clifford algebra^{} $\mathrm{Cliff}(V,q)$ of a vector space $V$ endowed with a quadratic form^{} $q$. The associated graded algebra is $\bigwedge V$, the exterior algebra^{} of $V$.
As algebras $A$ and $\mathcal{G}(A)$ are distinct (with the exception of the trivial case that $A$ is graded) but as vector spaces they are isomorphic.
Theorem 4.
The underlying vector spaces of $A$ and $\mathrm{G}\mathit{}\mathrm{(}A\mathrm{)}$ are isomorphic.
Title  filtered algebra 

Canonical name  FilteredAlgebra 
Date of creation  20130322 13:23:55 
Last modified on  20130322 13:23:55 
Owner  Dr_Absentius (537) 
Last modified by  Dr_Absentius (537) 
Numerical id  11 
Author  Dr_Absentius (537) 
Entry type  Definition 
Classification  msc 08A99 