Kernel of a congruence

Definition
Let $$\mathcal{V}$$ be a variety of algebras with zero and $$A$$ be an algebra in $$\mathcal{V}$$. Then, a nonempty subset $$S$$ of $$A$$ is termed the kernel of a congruence if it satisfies the following equivalent conditions:


 * 1) There exists a congruence on $$A$$ such that $$S$$ is the congruence class of the zero element.
 * 2) There exists a surjective homomorphism of algebras $$\varphi:A \to B$$ such that $$S$$ is the inverse image under $$\varphi$$ of the zero element of $$B$$.
 * 3) There exists a homomorphism (not necessarily surjective) of algebras $$\varphi:A \to B$$ such that $$S$$ is the inverse image under $$\varphi$$ of the zero element of $$B$$.