Kernel of Magma Homomorphism is Submagma

Theorem

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

Let $\struct {T, \circ}$ be an algebraic structure with an identity $e$.

Let $\phi: S \to T$ be a homomorphism.

Then the kernel of $\phi$ is a submagma of $\struct {S, *}$.

That is:

$\struct {\map {\phi^{-1} } e, *}$ is a submagma of $\struct {S, *}$

where $\map {\phi^{-1} } e$ denote the preimage of $e$.

Proof

Let $x, y \in \map {\phi^{-1} } e$.

By the definition of a magma, $S$ is closed under $*$.

That is:

$\forall x, y \in S: x * y \in S$

Hence:

$x * y \in \Dom \phi$

It is to be shown that:

$x * y \in \map {\phi^{-1} } e$

Thus:

\(\ds x, y\)

\(\in\)

\(\ds \map {\phi^{-1} } e\)

by hypothesis

\(\ds \leadstoandfrom \ \ \)

\(\ds \map \phi x\)

\(=\)

\(\ds e\)

Definition of Kernel of Magma Homomorphism

\(\, \ds \land \, \)

\(\ds \map \phi y\)

\(=\)

\(\ds e\)

\(\ds \leadstoandfrom \ \ \)

\(\ds \map \phi x \circ \map \phi y\)

\(=\)

\(\ds e\)

Definition of Identity Element

\(\ds \leadstoandfrom \ \ \)

\(\ds \map \phi {x * y}\)

\(=\)

\(\ds e\)

Definition of Homomorphism (Abstract Algebra)

\(\ds \leadstoandfrom \ \ \)

\(\ds x*y\)

\(\in\)

\(\ds \map {\phi^{-1} } e\)

Definition of Preimage of Element under Mapping

Hence the result.

$\blacksquare$