Exercise 13, § 2

Suppose that, for all $x \in E$, $\gamma_x$ is surjective. Show that if, for an element $a \in E$, $\gamma_a$ is a bijective mapping, $\gamma_x$ is a bijective mapping for all $x \in E$ (use Exercise 12(b)); this holds in particular if there exist two elements $a, b$ of $E$ such that $ab = b$ (use Exercise 12(a)); $E$ is then a left semi-group whose residue $R$ is empty; moreover, for every idempotent $u$, every element of $Eu$ is invertible in $Eu$.