kernel of a homomorphism between algebraic systems
It is easy to see that . Since it is a subset of , it is relation on . Furthermore, it is an equivalence relation on : 11In general, is a partition of a set iff is an equivalence relation on .
We write to denote .
In fact, is a congruence relation: for any -ary operator symbol , suppose and are two sets of elements in with . Then
so . For this reason, is also called the congruence induced by .
Example. If are groups and is a group homomorphism. Then the kernel of , 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 is the identity of ).
Remark. The above can be generalized. See the analog (http://planetmath.org/KernelOfAHomomorphismIsACongruence) in model theory.
|Title||kernel of a homomorphism between algebraic systems|
|Date of creation||2013-03-22 16:26:20|
|Last modified on||2013-03-22 16:26:20|
|Last modified by||CWoo (3771)|
|Defines||congruence induced by a homomorphism|