Equivalence relation expressing that two elements have the same image under a function
In set theory , the kernel of a function
f
{\displaystyle f}
(or equivalence kernel [ 1] ) may be taken to be either
the equivalence relation on the function's domain that roughly expresses the idea of "equivalent as far as the function
f
{\displaystyle f}
can tell",[ 2] or
the corresponding partition of the domain.
An unrelated notion is that of the kernel of a non-empty family of sets
B
,
{\displaystyle {\mathcal {B}},}
which by definition is the intersection of all its elements:
ker
B
=
⋂
B
∈
B
B
.
{\displaystyle \ker {\mathcal {B}}~=~\bigcap _{B\in {\mathcal {B}}}\,B.}
This definition is used in the theory of filters to classify them as being free or principal .
^ Mac Lane, Saunders ; Birkhoff, Garrett (1999), Algebra , Chelsea Publishing Company , p. 33, ISBN 0821816462 .
^ Bergman, Clifford (2011), Universal Algebra: Fundamentals and Selected Topics , Pure and Applied Mathematics, vol. 301, CRC Press , pp. 14–16, ISBN 9781439851296 .