[[K-algebra]]
# Filtered algebra

A **filtered algebra** is a certain generalization of a [[graded algebra]].
A filtered algebra $A$ over a field $\mathbb{K}$ is an [[algebra]] with an increasing sequence $F_{i}A \leq_{\Vect_{\mathbb{K}}} F_{i+1}A$ of [[Vector subspace|subspaces]] such that $A = \bigcup_{i = 1}^\infty F_{i}A$ and #m/def/falg 
$$
\begin{align*}
F_{i} A \cdot F_{j} A \sube F_{i+j} A
\end{align*}
$$

## Remarks

- Every filtered algebra has a linearly isomorphic [[Associated graded algebra]].
- Often one requires $1 \in F_{0}A$ if $A$ is a [[K-monoid]].


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#state/tidy | #lang/en | #SemBr