Exercise 11, § 2

E is called a left semi-group if every element $x$ of E is left cancellable.

(a) If $u$ is an idempotent ($§ 1$, no. 4), then $ux = x$ for all $x \in E$ (use Exercise 9(a)); $u$ is identity element for the law induced on $Eu$.

(b) If $u$ and $v$ are two distinct idempotents of E, then $Eu \cap Ev = \varnothing$ and the sets $Eu$ and $Ev$ (with the laws induced by that on E) are isomorphic.

(c) Let R be the complement of the union of the sets $Eu$, where $u$ runs through the set of idempotents of E. Show that $ER \subset R$; it follows that R is a stable subset of E. If R is non-empty, then $aR \neq R$ for all $a \in R$ (use Exercise 9(a) to prove that, in the contrary case, R would contain an idempotent); in particular, R is then infinite. R is called the residue of the left semi-group E.

(d) If R is non-empty and there exists at least one idempotent $u$ in E (that is $E \neq R$), for all $x \in REu, xEu \neq Eu$; in particular, no element of $REu$ is invertible in $Eu$ and $REu$ is an infinite set (use Exercise 8).

(e) If E has an identity element $e$, $e$ is the only idempotent in E and R is empty (note that $E = Ee$).

(f) If there exists a right cancellable element $a$ in E (in particular if the given law on E is commutative), either E has an identity element or $E = R$ (note that if there exists an idempotent $u$, then $xa = xua$ for all $x \in E$). For example, if E is the set of integers $\geq 1$, with addition, E is a commutative semi-group such that $E = R$.

(g) If there exists $a \in E$ such that $\delta_a$ is surjective, either E has an identity element or $E = R$ (examine separately the case where $a \in R$ and the case where $a \in Eu$ for an idempotent $u$).