First the terms as I understand them:
The congruence is defined as equivalence that preserves structure (the operations) of algebra.
Homomorphism is a map
f: A->Bbetween two algebraic structures of the same type, that preserves structure. This means that it doesn't matter if we map the result of operation, or map the operands and then perform operation. For example
f(a+b) = f(a)+f(b); a,b \in A.
By this definition: Kernel of homomorphism is a set of elements from
Awhich are mapped to neutral element of
B. For example if the map is from
(R^2,*) -> (R,*)(2D vectors to real number) defined as
f(x) = |x|(length of vector), all vectors that would map to
1(which is neutral in
(R,*)) would make up the kernel. Therefore the kernel is set of all unit vectors.
From other definition: Generally, the kernel is a congruence relation.
Regarding this, I have following questions:
- Is there anything wrong with the definitions or examples above?
- How can it be, that generally the kernel of map is a congruence, but kernel of homomorphism is a set of elements from
A? Shouldn't be these 2 distinct? What is the relationship between them?
- How would a quotient algebra defined as
kerFis kernel of homomorphism look like in example from third definition? How is it different from quotient algebra where
kerFis general kernel from fourth definition?