Kernel (mathematics)

This article about kernels in algebra and related fields; for other meanings, see (k,l)-kernel of a digraph and integral transform.

In mathematics, especially algebra, kernels are general constructions which measure the failure of a homomorphism or function to be injective.

In set theory, the kernel of a function f : X → Y is an equivalence relation on X which is defined in terms of f. Specifically:

\ker\left(f\right) = \{\left(x_1,x_2\right) \in X \times X : f\left(x_1\right) = f\left(x_2\right)\}

The function f is injective iff the kernel is the diagonal in $X \times X$ .

This idea generalizes readily to the algebraic setting where f is now a homomorphism. The equivalence relation ker(f) becomes a congruence relation on X (i.e. the equivalence relation is compatible with the algebraic structure).

For many algebraic structures, such as groups, rings, and vector spaces, there is a simpler definition of the kernel that is usually preferred. In these cases the equivalence relation is entirely determined by the equivalence class of the neutral element. In these cases the kernel is defined as the preimage of the neutral element in Y.

\ker\left(f\right) = \{x \in X : f\left(x\right) = 0\}

The congruence relation is now replaced with the notion of a normal subgroup (in the case of groups) or an ideal (in the case of rings). For linear operators between vector spaces, the kernel also goes by the name of null space.

For more on the kernel of a homomorphism, see kernel (algebra).

There exist several notions in category theory which seek to generalize the concept of a kernel in algebra.

In categories with zero morphisms one can define the kernel of a morphism f as the equalizer of f and the parallel zero morphism. For more on this see kernel (category theory).
A kernel pair is a categorical notion which is more closely related to the notion of a congruence relation in algebra. The kernel pair of a morphism f is defined as a pullback of f with itself. In the category of sets this just gives the familiar kernel of a function defined above.
A difference kernel is another name for a binary equalizer. The name comes from preadditive categories where one can define the equalizer of f and g as the kernel of the difference: eq(f,g) = ker(f − g). Difference kernels, however, make sense in arbitrary categories and are often used in conjunction with kernel pairs.

Kernel of a series or integral

The term kernel, in reference to a series, is a synonym for the idea of the generating function. In this same sense, in reference to integrals, it is a synonym for the part of the integrand that defines the integral transform.

Kernel in statistics

A stochastic kernel is the transition function of a (usually discrete) stochastic process.

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Kernel of a series or integral

Kernel in statistics

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