Definition:Kernel of Magma Homomorphism

Definition
Let $\struct {S, \circ}$ be a magma.

Let $\struct {T, *}$ be a magma with an identity element $e$.

Let $\phi: \struct {S, \circ} \to \struct {T, *}$ be a magma homomorphism.

The kernel of $\phi$ is the subset of the domain of $\phi$ defined as:
 * $\map \ker \phi = \set {x \in S: \map \phi x = e}$

That is, $\map \ker \phi$ is the subset of $S$ that maps to the identity of $T$.