Editing Kernel (algebra) (section)

== Universal algebra ==
All the above cases may be unified and generalized in [[universal algebra]]. Let ''A'' and ''B'' be [[algebraic structure]]s of a given type and let ''f'' be a homomorphism of that type from ''A'' to ''B''. Then the ''kernel'' of ''f'' is the subset of the [[direct product]] {{nowrap|''A'' × ''A''}} consisting of all those [[ordered pair]]s of elements of ''A'' whose components are both mapped by ''f'' to the same element in ''B''.<ref name=":3">{{Cite book |last1=Burris |last2=Sankappanavar |first1=Stanley |first2=H.P. |title=A Course in Universal Algebra |publisher=S. Burris and H.P. Sankappanavar |isbn=978-0-9880552-0-9 |edition=Millennium |publication-date=2012}}</ref><ref name=":4">{{Cite book |last1=McKenzie |first1=Ralph |title=Algebras, lattices, varieties |last2=McNulty |first2=George F. |last3=Taylor |first3=W. |date=1987 |publisher=Wadsworth & Brooks/Cole Advanced Books & Software |isbn=978-0-534-07651-1 |series=The Wadsworth & Brooks/Cole mathematics series |location=Monterey, Calif}}</ref> The kernel is usually denoted {{nowrap|ker ''f''}} (or a variation). In symbols:
: <math>\operatorname{ker} f = \left\{\left(a, b\right) \in A \times A : f(a) = f\left(b\right)\right\}\mbox{.}</math>

The homomorphism ''f'' is injective if and only if its kernel is exactly the diagonal set {{nowrap|{{mset|(''a'', ''a'') : ''a'' &isin; ''A''}}}}, which is always at least contained inside the kernel.<ref name=":3" /><ref name=":4" />

It is easy to see that ker ''f'' is an [[equivalence relation]] on ''A'', and in fact a [[congruence relation]].
Thus, it makes sense to speak of the [[quotient (universal algebra)|quotient algebra]] {{nowrap|''A'' / (ker ''f'')}}.
The [[isomorphism theorem#General|first isomorphism theorem]] in general universal algebra states that this quotient algebra is naturally isomorphic to the image of ''f'' (which is a [[subalgebra]] of ''B'').<ref name=":3" /><ref name=":4" />

Note that the definition of kernel here (as in the monoid example) doesn't depend on the algebraic structure; it is a purely [[set (mathematics)|set]]-theoretic concept.
For more on this general concept, outside of abstract algebra, see [[kernel of a function]].