kernel of a homomorphism between algebraic systems

Let f:(A,O)(B,O) be a homomorphismPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath between two algebraic systems A and B (with O as the operator set). Each element bB corresponds to a subset K(b):=f-1(b) in A. Then {K(b)bB} forms a partition of A. The kernel ker(f) of f is defined to be


It is easy to see that ker(f)={(x,y)A×Af(x)=f(y)}. Since it is a subset of A×A, it is relation on A. Furthermore, it is an equivalence relationMathworldPlanetmath on A: 11In general, {Ni} is a partition of a set A iff Ni2 is an equivalence relation on A.

  1. 1.

    ker(f) is reflexiveMathworldPlanetmathPlanetmathPlanetmathPlanetmath: for any aA, aK(f(a)), so that (a,a)K(f(a))2ker(f)

  2. 2.

    ker(f) is symmetricPlanetmathPlanetmathPlanetmath: if (a1,a2)ker(f), then f(a1)=f(a2), so that (a2,a1)ker(f)

  3. 3.

    ker(f) is transitiveMathworldPlanetmathPlanetmathPlanetmathPlanetmath: if (a1,a2),(a2,a3)ker(f), then f(a1)=f(a2)=f(a3), so (a1,a3)ker(f).

We write a1a2(modker(f)) to denote (a1,a2)ker(f).

In fact, ker(f) is a congruence relationPlanetmathPlanetmath: for any n-ary operator symbol ωO, suppose c1,,cn and d1,,dn are two sets of elements in A with cidimodker(f). Then


so ωA(c1,,cn)ωA(d1,,dn)(modker(f)). For this reason, ker(f) is also called the congruencePlanetmathPlanetmath induced by f.

Example. If A,B are groups and f:AB is a group homomorphism. Then the kernel of f, using the definition above is just the union of the square of the cosets of


the traditional definition of the kernel of a group homomorphism (where e is the identityPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath of B).

Remark. The above can be generalized. See the analog ( in model theoryMathworldPlanetmath.

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