I asked this on Math Stack Exchange, but it didn't get a single answer. So, I am now asking it here. Let our signature be that of a single binary operation $+$. I define the constant identity to be $x+y=z+w$. The commutative identity is, of course, the well-known identity $x+y=y+x$. I wonder, is there an identity strictly between those two, meaning, is there a single identity $E$ such that the constant identity implies $E$, but not conversely, and $E$ implies the commutative identity, but not conversely?



Is there an identity between the commutative identity and the constant identity?
Yes:
$$(x + x) + y = y + x$$
The constant identity implies this because both sides are $+$es. This does not imply the constant identity because it is true about any set with an operation that is commutative, associative, and idempotent (meaning $x + x = x$ for all $x$), the smallest nontrivial example is $\{x,y\}$ where $x + x = x + y = y + x = x$ and $y + y = y$.


This implies commutativity. $(x + x) + (x + x) = (x + x) + x = x + x$, so $x + x$ is idempotent. $(x+x)+y=((x+x)+(x+x))+y=y+(x+x)$, so $x + x$ commutes with everything. $(x + x) + (y + y) = (y + y) + x = x + y$, and $(x + x) + (y + y) = (y + y) + (x + x)$ because $x+x$ commutes with everything, so $x + y = y + x$, so $+$ is commutative.


This is not implied by commutativity, because (for example) addition of natural numbers is commutative but does not satisfy this identity.


