Stupid question, but one occasionally reads such things as "the operation $\ast$ is noncommutative for all $x,y$ such that $x\neq y$" or "$x\ast y$ is commutative iff $x=y$". These statements bother me, because they imply that there is some operation $\cdot$ for which $x=y\not\implies x\cdot y=y\cdot x$ which in turn implies $x\cdot x\neq x\cdot x$.

Is this the result of poor writing, or is there some legitimate reason to call an operation between an element and itself "commutative".