Isomorphism Preserves Associativity

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 isomorphism.

Then $\circ$ is associative iff $*$ is associative.

Proof 1
Let $\left({S, \circ}\right)$ be an algebraic structure in which $\circ$ is associative.

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

As an isomorphism is surjective, it follows that:


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

So:

As $\phi$ is an isomorphism, it follows from Inverse Isomorphism that $\phi^{-1}$ is also a isomorphism.

Thus the result for $\phi$ can be applied to $\phi^{-1}$.

Proof 2
We have that an isomorphism is a homomorphism which is also a bijection.

By definition, an epimorphism is a homomorphism which is also a surjection.

That is, an isomorphism is an epimorphism which is also an injection.

Thus Epimorphism Preserves Associativity can be applied.