Definition:Kernel of Magma Homomorphism

Definition
Let $\left({S, \circ}\right)$ be a magma.

Let $\left({T, *}\right)$ be a magma with an identity element $e$.

Let $\phi: \left({S, \circ}\right) \to \left({T, *}\right)$ be a magma homomorphism.

The kernel of $\phi$ is the subset of the domain of $\phi$ defined as:
 * $\ker \left({\phi}\right) = \left\{{x \in S: \phi \left({x}\right) = e}\right\}$

That is, $\ker \left({\phi}\right)$ is the subset of $S$ that maps to the identity of $T$.