Solution
= Solution
{c}
Suppose $y*y=y$. Choose $z$ with $y*z=e$, using the given right <inverse element> property. Multiply the idempotent relation on the right by $z$. By <associativity> and the given right <identity element> property,
$$
e=y*z=(y*y)*z=y*(y*z)=y*e=y.
$$
Thus \b[the only <idempotent element of a semigroup> is $e$]. This argument uses no cancellation and does not assume a left <identity element>.