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# 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* $\ker(f)$ of $f$ is defined to be

$\ker(f):=\bigcup_{{b\in B}}K(b)\times K(b).$ |

It is easy to see that $\ker(f)=\{(x,y)\in A\times A\mid f(x)=f(y)\}$. 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. $\ker(f)$ is reflexive: for any $a\in A$, $a\in K(f(a))$, so that $(a,a)\in K(f(a))^{2}\subseteq\ker(f)$

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

3. $\ker(f)$ is transitive: if $(a_{1},a_{2}),(a_{2},a_{3})\in\ker(f)$, then $f(a_{1})=f(a_{2})=f(a_{3})$, so $(a_{1},a_{3})\in\ker(f)$.

We write $a_{1}\equiv a_{2}\;\;(\mathop{{\rm mod}}\ker(f))$ to denote $(a_{1},a_{2})\in\ker(f)$.

In fact, $\ker(f)$ is a congruence relation: for any $n$-ary operator symbol $\omega\in O$, suppose $c_{1},\ldots,c_{n}$ and $d_{1},\ldots,d_{n}$ are two sets of elements in $A$ with $c_{i}\equiv d_{i}\mod\ker(f)$. Then

$f(\omega_{A}(c_{1},\ldots,c_{n})=\omega_{B}(f(c_{1}),\ldots,f(c_{n}))=\omega_{% B}(f(d_{1}),\ldots,f(d_{n}))=f(\omega_{A}(d_{1},\ldots,d_{n})),$ |

so $\omega_{A}(c_{1},\ldots,c_{n})\equiv\omega_{A}(d_{1},\ldots,d_{n})\;\;(\mathop%
{{\rm mod}}\ker(f))$. For this reason, $\ker(f)$ 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 in model theory.

## Mathematics Subject Classification

08A05*no label found*

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