kernel of a homomorphism is a congruence

Let Σ be a fixed signaturePlanetmathPlanetmathPlanetmath, and 𝔄 and 𝔅 two structuresMathworldPlanetmath for Σ. If f:𝔄𝔅 is a homomorphismPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath, then ker(f) is a congruenceMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath on 𝔄.


If F is an n-ary function symbol of Σ, and f(ai)=f(ai), then

f(F𝔄(a1,,an)) =F𝔅(f(a1),,f(an))
Title kernel of a homomorphism is a congruence
Canonical name KernelOfAHomomorphismIsACongruence
Date of creation 2013-03-22 13:48:03
Last modified on 2013-03-22 13:48:03
Owner almann (2526)
Last modified by almann (2526)
Numerical id 8
Author almann (2526)
Entry type Theorem
Classification msc 03C05
Classification msc 03C07
Related topic KernelOfAHomomorphismBetweenAlgebraicSystems