What is "filtration"

Filtration \Fil*tra"tion\, n. [Cf. F. filtration.] The act or process of filtering; the mechanical separation of a liquid from the undissolved particles floating in it.

"act or process of filtering," c.1600, perhaps from French filtration (1570s), noun of action from filter "to filter" (see filter (v.)).

n. The act or process of filtering; the mechanical separation of a liquid from the undissolved particles floating in it.

1. n. the process whereby fluids pass through a filter or a filtering medium

2. the act of changing a fluid by passing it through a filter

Filtration is any of various mechanical, physical or biological operations that separate solids from fluids ( liquids or gases) by adding a medium through which only the fluid can pass. The fluid that passes through is called the filtrate. In physical filters oversize solids in the fluid are retained and in biological filters particulates are trapped and ingested and metabolites are retained and removed. However, the separation is not complete; solids will be contaminated with some fluid and filtrate will contain fine particles (depending on the pore size ,filter thickness and biological activity). Filtration occurs both in nature and in engineered systems; there are biologic, geologic, and industrial forms. For example, in animals (including humans), renal filtration removes wastes from the blood, and in water treatment and sewage treatment, undesirable constituents are removed by absorption into a biological film grown on or in the filter medium, as in slow sand filtration.

In mathematics, a filtration F is an indexed set S 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 the index i is the time parameter of some stochastic process, then the filtration can be interpreted as representing all historical but not future information available about the stochastic process, with the algebraic object S gaining in complexity with time. Hence, a process that is adapted to a filtration F, is also called non-anticipating, i.e. one that cannot see into the future.

Sometimes, as in a filtered algebra, there is instead the requirement that the S be subalgebras with respect to some operations (say, vector addition), but not with respect to other operations (say, multiplication), that satisfy S ⋅ S ⊂ S, where 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 S be the whole S, or (in more general cases, when the notion of union does not make sense) that the canonical homomorphism from the direct limit of the S to S is an isomorphism. Whether this requirement is assumed or not usually depends on the author of the text and is often explicitly stated. This article does not impose this requirement.

There is also the notion of a descending filtration, which is required to satisfy S ⊇ S in lieu of S ⊆ S (and, occasionally, ⋂S = 0 instead of ⋃S = S). 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).

The concept dual to a filtration is called a cofiltration.

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.