Epimorphism Preserves Commutativity

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Theorem

Let $\phi: \left({S, \circ}\right) \to \left({T, *}\right)$ be an epimorphism.

If $\circ$ is commutative, then so is $*$.

Let $\left({S, \circ}\right)$ be an algebraic structure in which $\circ$ is commutative.

Let $\phi: \left({S, \circ}\right) \to \left({T, *}\right)$ be an epimorphism.

As an epimorphism is surjective, it follows that:

$\forall u, v \in T: \exists x, y \in S: \phi \left({x}\right) = u, \phi \left({y}\right) = v$

So:

$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle u * v$	$=$	$\displaystyle \phi \left({x}\right) * \phi \left({y}\right)$	$\displaystyle $	$\displaystyle $	$\displaystyle $	as $\phi$ is a surjection
$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $	$=$	$\displaystyle \phi \left({x \circ y}\right)$	$\displaystyle $	$\displaystyle $	$\displaystyle $	Morphism Property
$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $	$=$	$\displaystyle \phi \left({y \circ x}\right)$	$\displaystyle $	$\displaystyle $	$\displaystyle $	Commutativity of $\circ$
$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $	$=$	$\displaystyle \phi \left({y}\right) * \phi {\left({x}\right)}$	$\displaystyle $	$\displaystyle $	$\displaystyle $	Morphism Property
$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $	$=$	$\displaystyle v * u$	$\displaystyle $	$\displaystyle $	$\displaystyle $	by definition as above

$\blacksquare$

Note that this result is applied to epimorphisms. For a general homomorphism which is not surjective, we can say nothing definite about the behaviour of the elements of its codomain which are not part of its image.

\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle u * v\)	\(=\)	\(\displaystyle \phi \left({x}\right) * \phi \left({y}\right)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	as $\phi$ is a surjection
\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \phi \left({x \circ y}\right)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Morphism Property
\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \phi \left({y \circ x}\right)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Commutativity of $\circ$
\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \phi \left({y}\right) * \phi {\left({x}\right)}\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Morphism Property
\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle v * u\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	by definition as above