Kernel of Magma Homomorphism is Submagma
Jump to navigation
Jump to search
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.
$\blacksquare$