I have to give a 'careful' proof to show that a binary structure $(S,*)$ with the property
for each $ c \in S$ the equation $x*x=c$ has a solution $x \in S$.
Question: I need to prove $x*x=c$ is a structural property note: I tagged this with solution verification, thus looking for exactly that
Suppose (S,*) has the property $x*x=c$ for each $c \in S$ will have solution $x \in S$
Further suppose that there is an isomorphism $\phi$ such that $\phi : (S,*) \to (S',*')$
then for each $c' \in S'$ there is some solution $x' \in S'$ such that $x' *' x' = c'$
Proof:
Because $\phi$ is an isomorphism we know $\forall x \in S, x' \in S'$ that $\phi(x * x) = \phi(x) *' \phi(x) = x' *' x'$
Now, $x*x=c \implies \phi(x*x)=\phi(c) \implies \phi(c) = \phi(x) *' \phi(x)$
and because $\phi(x) *' \phi(x) = x' *' x'$ this implies that $\phi(c) = x' *' x'$
thus $\phi(c) \in S'$ and there must be $c' \in S'$ such that $\phi(c) = c'$ where $x' *' x' =c'$
Therefore we have a structural property.
Best Answer
Let $\phi: (S,*) \rightarrow (S',*')$ be an isomorphism where $(S,*)$ has the property $x*x=c$ for $c,x \in S$. Because, $\phi$ is an isomorphism we know $\forall x \in S, x' \in S'$ that $\phi(x * x) = x' *' x' = \phi(x) *' \phi(x)$. Now, $x*x=c \Longrightarrow \phi(x*x) =\phi(c) \Longrightarrow \phi(c) = \phi(x) *' \phi(x)$. And because, $\phi(x) *' \phi(x) = x' *' x'$ this implies $\phi(c) = x' *' x'$. Thus, because $x' *' x' \in S'$ we know that $\phi(c) \in S'$. Now we know $\phi(c) = x' * ' x'$ is an element in $S'$ such that $\phi(c) $ is equal to that that element. Therefore, $x*x=c$ is a structural property preserved under an isomorphism $\phi$ and we may denote this element in $S'$ as $c'$ such that $\phi(c) = c'$