# Epimorphism Preserves Commutativity

## Contents

## Theorem

Let $\left({S, \circ}\right)$ and $\left({T, *}\right)$ be algebraic structures.

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

Let $\circ$ be a commutative operation.

Then $*$ is also a commutative operation.

## Proof

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:

\(\displaystyle u * v\) | \(=\) | \(\displaystyle \phi \left({x}\right) * \phi \left({y}\right)\) | $\phi$ is a surjection | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \phi \left({x \circ y}\right)\) | Morphism Property | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \phi \left({y \circ x}\right)\) | Commutativity of $\circ$ | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \phi \left({y}\right) * \phi {\left({x}\right)}\) | Morphism Property | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle v * u\) | by definition as above |

$\blacksquare$

## Warning

Note that this result is applied to epimorphisms.

For a general homomorphism which is not surjective, nothing definite can be said about the behaviour of the elements of its codomain which are not part of its image.

## Also see

## Sources

- 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): $\S 12$: Theorem $12.2$