Let Σ be a fixed signaturePlanetmathPlanetmathPlanetmath, and 𝔄 and 𝔅 be two structuresMathworldPlanetmath for Σ. Given a homomorphismPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath f:𝔄𝔅, the kernel of f is the relationMathworldPlanetmathPlanetmath ker(f) on A defined by


So defined, the kernel of f is a congruenceMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath on 𝔄. If Σ has a constant symbol 0, then the kernel of f is often defined to be the preimageMathworldPlanetmath of 0𝔅 under f. Under this definition, if {0𝔅} is a substructure of 𝔅, then the kernel of f is a substructure of 𝔄.

Title kernel
Canonical name Kernel1
Date of creation 2013-03-22 13:46:34
Last modified on 2013-03-22 13:46:34
Owner almann (2526)
Last modified by almann (2526)
Numerical id 11
Author almann (2526)
Entry type Definition
Classification msc 03C05
Classification msc 03C07
Related topic Kernel
Related topic KernelOfAGroupHomomorphism
Related topic KernelOfALinearTransformation