Kernel of a congruence
From Groupprops
Definition
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
.