Past exam of the mathematics course of the University of Oxford / 2021 / Prelims / Mathematics I / 5 / b / ii / Solution

Suppose $x\ast z=e$. Using only associativity and the given right identity element property,Thus $z\ast x$ is an idempotent element of a semigroup. By i it equals $e$, provingNo left identity element or cancellation law is needed for this deduction.