Filtered algebra

Filtered algebra generally refers to structures in algebra where there is a filtration, which is a systematic way to impose a structure or hierarchy on the elements of the algebra. More formally, a filtered algebra is an algebra equipped with a filtration, which is an ascending chain of subalgebras indexed by a directed set. This can be useful in various branches of mathematics, particularly in homological algebra, algebraic topology, and representation theory. ### Definitions and Properties 1.