Kernel of a congruence

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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.