Exercise 17, § 2

Let $E$ be a monoid and $S$ a stable subset of $E$ with the following properties:

(α) For all $s \in S$ and all $a \in E$, there exist $t \in S$ and $b \in E$ such that $sb = at$.

(β) For all $a, b \in E$ and all $s \in S$ such that $sa = sb$, there exist $t \in S$ such that $at = bt$.

(a) In $E \times S$ let $(a, s) \sim (b, t)$ denote the relation: "there exist $s'$ and $t'$ in $S$ such that $tt' = ss'$ and $as' = bt'$."

Show that $\sim$ is an equivalence relation. We write $\overline{E} = E \times S / \sim$, denote by $as^{-1}$ the equivalence class of $(a, s)$ (mod $\sim$) and by $\varepsilon : E \to \overline{E}$ the mapping $a \mapsto ae^{-1}$.

(b) Show that there exists on $\overline{E}$ one and only one monoid structure such that $\varepsilon$ is a unital homomorphism and such that, for all $s \in S$, $\varepsilon(s)$ is invertible.

(c) Show that $(\overline{E}, \varepsilon)$ has the universal property described in Theorem 1 (no. 4).

(d) Show that $\varepsilon$ is injective if and only if the elements of $S$ are cancellable.