Past exam of the mathematics course of the University of Oxford / 2021 / Prelims / Mathematics I / 5 / b (Recovering two-sided identities)

Suppose $y\ast y=y$. Choose $z$ with $y\ast z=e$, 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,Thus the only idempotent element of a semigroup is $e$. This argument uses no cancellation and does not assume a left identity element.

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.

For any $x$, choose $z$ with $x\ast z=e$. By ii, $z\ast x=e$ as well. Associativity and the right identity element property now giveThus $e$ is also a left identity element. The given binary operation is closed on $X$ and associative, $e$ is a two-sided identity element, and each $x$ has the two-sided inverse element $z$. All group axioms hold, so $(X,\ast )$ is a group. This is the one-sided group axioms criterion.