kernel of a homomorphism between algebraic systems
Let $f:(A,O)\to (B,O)$ be a homomorphism^{} between two algebraic systems $A$ and $B$ (with $O$ as the operator set). Each element $b\in B$ corresponds to a subset $K(b):={f}^{1}(b)$ in $A$. Then $\{K(b)\mid b\in B\}$ forms a partition of $A$. The kernel $\mathrm{ker}(f)$ of $f$ is defined to be
$$\mathrm{ker}(f):=\bigcup _{b\in B}K(b)\times K(b).$$ 
It is easy to see that $\mathrm{ker}(f)=\{(x,y)\in A\times A\mid f(x)=f(y)\}$. Since it is a subset of $A\times A$, it is relation on $A$. Furthermore, it is an equivalence relation^{} on $A$: ^{1}^{1}In general, $\{{N}_{i}\}$ is a partition of a set $A$ iff $\bigcup {N}_{i}^{2}$ is an equivalence relation on $A$.

1.
$\mathrm{ker}(f)$ is reflexive^{}: for any $a\in A$, $a\in K(f(a))$, so that $(a,a)\in K{(f(a))}^{2}\subseteq \mathrm{ker}(f)$

2.
$\mathrm{ker}(f)$ is symmetric^{}: if $({a}_{1},{a}_{2})\in \mathrm{ker}(f)$, then $f({a}_{1})=f({a}_{2})$, so that $({a}_{2},{a}_{1})\in \mathrm{ker}(f)$

3.
$\mathrm{ker}(f)$ is transitive^{}: if $({a}_{1},{a}_{2}),({a}_{2},{a}_{3})\in \mathrm{ker}(f)$, then $f({a}_{1})=f({a}_{2})=f({a}_{3})$, so $({a}_{1},{a}_{3})\in \mathrm{ker}(f)$.
We write ${a}_{1}\equiv {a}_{2}\phantom{\rule{veryverythickmathspace}{0ex}}(mod\mathrm{ker}(f))$ to denote $({a}_{1},{a}_{2})\in \mathrm{ker}(f)$.
In fact, $\mathrm{ker}(f)$ is a congruence relation^{}: for any $n$ary operator symbol $\omega \in O$, suppose ${c}_{1},\mathrm{\dots},{c}_{n}$ and ${d}_{1},\mathrm{\dots},{d}_{n}$ are two sets of elements in $A$ with ${c}_{i}\equiv {d}_{i}mod\mathrm{ker}(f)$. Then
$$f({\omega}_{A}({c}_{1},\mathrm{\dots},{c}_{n})={\omega}_{B}(f({c}_{1}),\mathrm{\dots},f({c}_{n}))={\omega}_{B}(f({d}_{1}),\mathrm{\dots},f({d}_{n}))=f({\omega}_{A}({d}_{1},\mathrm{\dots},{d}_{n})),$$ 
so ${\omega}_{A}({c}_{1},\mathrm{\dots},{c}_{n})\equiv {\omega}_{A}({d}_{1},\mathrm{\dots},{d}_{n})\phantom{\rule{veryverythickmathspace}{0ex}}(mod\mathrm{ker}(f))$. For this reason, $\mathrm{ker}(f)$ is also called the congruence^{} induced by $f$.
Example. If $A,B$ are groups and $f:A\to B$ is a group homomorphism. Then the kernel of $f$, using the definition above is just the union of the square of the cosets of
$$N=\{x\mid f(x)=e\},$$ 
the traditional definition of the kernel of a group homomorphism (where $e$ is the identity^{} of $B$).
Remark. The above can be generalized. See the analog (http://planetmath.org/KernelOfAHomomorphismIsACongruence) in model theory^{}.
Title  kernel of a homomorphism between algebraic systems 

Canonical name  KernelOfAHomomorphismBetweenAlgebraicSystems 
Date of creation  20130322 16:26:20 
Last modified on  20130322 16:26:20 
Owner  CWoo (3771) 
Last modified by  CWoo (3771) 
Numerical id  11 
Author  CWoo (3771) 
Entry type  Definition 
Classification  msc 08A05 
Synonym  induced congruence 
Related topic  KernelOfAHomomorphismIsACongruence 
Related topic  KernelPair 
Defines  congruence induced by a homomorphism 