Show that the operation $*$ is commutative is a structural property.
Give a careful proof that the indicated property of a binary structure $\langle S,* \rangle$ is indeed a structural property.
I've started this problem as:
Let $\langle S,* \rangle$ be isomorphic to $\langle T,\Box\rangle$. Also let $f:S \to T$.
This means that for $a,b \in S$, then $f(a*b)=f(a)\Box f(b)$ and $f(a*b)=f(b*a)=f(b)\Box f(a)$ and therefore $f(a)\Box f(b)=f(b) \Box f(a)$. This means that an operation $*$ is commutative is a structural property. Does this work?
Best Answer
Start like this: Let $\langle S,* \rangle$ be isomorphic to $\langle T, \Box\rangle$. Assume $\langle S,* \rangle$ is commutative. I claim that $\langle T, \Box\rangle$ is commutative. To prove this, let $a,b \in T$. ...continue computation to get... $a\Box b = b \Box a$. Therefore ....