In analysis, one consider an integral operator *T* which transforms a function *f* into a function *T**f* given by the integral formula

Unrelated to this, if *f* is any function in any context, then the *kernel* of *f* is a certain equivalence relation on the domain of *f* which is defined in terms of *f*.
For more on this in general, see Kernel (function).

This notion is used heavily in abstract algebra.
But in the case of Mal'cev algebras, it can be replaced by a simpler definition; the *kernel* of a homomorphism *f* is the preimage under *f* of the zero element of the codomain.
For more on this, see Kernel (algebra).

Finally, for this last notion of kernel is generalised in a certain sense in category theory; the *kernel* of a morphism *f* is the difference kernel of *f* and the corresponding zero morphism (if this exists).
For more on this, see Kernel (category theory).