Filtered algebra

A filtered algebra is a certain generalization of a graded algebra. A filtered algebra $𝐴$ over a field $𝕂$ is an algebra with an increasing sequence $𝐹 𝑖 𝐴 \leq 𝖵 𝖾 𝖼 𝗍 𝕂 𝐹 𝑖 + 1 𝐴$ of subspaces such that $𝐴 = ⋃ \infty 𝑖 = 1 𝐹 𝑖 𝐴$ and #m/def/ralg

𝐹 𝑖 𝐴 \cdot 𝐹 𝑗 𝐴 \subseteq 𝐹 𝑖 + 𝑗 𝐴

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.

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