Is only the commutative identity equivalent to the commutative identity?

Question

Let our signature be that of a single binary operation $+$. Suppose I have an equational identity $E$ such that $E$ is equivalent to the commutative identity $x+y=y+x$. In other words, $E$ implies and is implied by the commutative identity. Must $E$ simply be an alphabetic variant of the commutative identity? That is, must $E$ be of the form $v_1+v_2=v_2+v_1$ for distinct variables $v_1$ and $v_2$?

Answer 1

Yes. Note that the two sides of $E$ must be equal when interpreted in the free commutative monoid on the variables, so in particular each variable must occur the same number of times on each side. If there are three or more occurrences of variables on each side of $E$, then $E$ does not imply commutativity, since it does not imply any equation about sums of only two elements. (Explicitly, consider a structure $\{a,b,a+b,b+a,\infty\}$ where any sum besides $a+b$ and $b+a$ is defined to be $\infty$; this will satisfy $E$ but not commutativity.) So there are at most two occurrences of variables on each side of $E$, and then it is clear that $E$ must have the form $v_1+v_2=v_2+v_1$ in order to be equivalent to commutativity.