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 $A$ : ¹¹In 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 $a_{1}\equiv a_{2}\pmod{\ker(f)}$ 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 $\omega_{A}(c_{1},\ldots,c_{n})\equiv\omega_{A}(d_{1},\ldots,d_{n})\pmod{\ker(f)}$ . 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 (http://planetmath.org/KernelOfAHomomorphismIsACongruence) in model theory.

Title	kernel of a homomorphism between algebraic systems
Canonical name	KernelOfAHomomorphismBetweenAlgebraicSystems
Date of creation	2013-03-22 16:26:20
Last modified on	2013-03-22 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