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 $B$.