Anticommutative operation on a set with more than one element is not commutative and has no identity element?

abstract-algebra

If we prove that there is no identity, then it will follow that the operation is not commutative, since we already have a right identity.

Note that by (ii), $xx=r$ for all $x$. Now suppose that $rx=x$. Then: $$(xr)(rx) = xx = r$$ hence by (ii) we conclude that $x=r$.

Therefore, if $X$ has more than one element, $x\neq r$, then $rx\neq x$, hence $r$ is not a two-sided identity, and moreover, $rx\neq x=xr$, so the operation is not commutative.

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