Kernel of a congruence
Let be a variety of algebras with zero and be an algebra in . Then, a nonempty subset of is termed the kernel of a congruence if it satisfies the following equivalent conditions:
- There exists a congruence on such that is the congruence class of the zero element.
- There exists a surjective homomorphism of algebras such that is the inverse image under of the zero element of .
- There exists a homomorphism (not necessarily surjective) of algebras such that is the inverse image under of the zero element of .