Exercise 9, § 2

Under a multiplicative law on a set E let $a$ be a left cancellable element of E.

(a) If there exists an element $u$ such that $au = a$, show that $ux = x$ for all $x \in E$; in particular, if $xu = x$ for all $x \in E$, $u$ is identity element.

(b) If there exist $u \in E$ such that $au = a$ and $b \in E$ such that $ab = u$, show that $ba = u$ (form $aba$); in particular, if there exists an identity element $e$ and an element $b$ such that $ab = e$, $b$ is the inverse of $a$.