In mathematics, a filtration is an indexed set Si of subobjects of a given algebraic structure S, with the index i running over some index set I that is a totally ordered set, subject to the condition that if i ≤ j in I then Si ⊆ Sj. The concept dual to a filtration is called a cofiltration.
Sometimes, as in a filtered algebra, there is instead the requirement that the be subobjects with respect to certain operations (say, vector addition), but with respect to other operations (say, multiplication), they instead satisfy, where here the index set is the natural numbers; this is by analogy with a graded algebra.
Sometimes, filtrations are supposed to satisfy the additional requirement that the union of the be the whole, or (in more general cases, when the notion of union does not make sense) that the canonical homomorphism from the direct limit of the to is an isomorphism. Whether this requirement is assumed or not usually depends on the author of the text and is often explicitly stated. We are not going to impose this requirement in this article.
There is also the notion of a descending filtration, which is required to satisfy in lieu of (and, occasionally, instead of ). Again, it depends on the context how exactly the word "filtration" is to be understood. Descending filtrations are not to be confused with cofiltrations (which consist of quotient objects rather than subobjects).
Filtrations are widely used in abstract algebra, homological algebra (where they are related in an important way to spectral sequences), and in measure theory and probability theory for nested sequences of σ-algebras. In functional analysis and numerical analysis, other terminology is usually used, such as scale of spaces or nested spaces.