This question comes from my attempt to solve Exercise 17(b) of Bourbaki, Algebra, Chapter 1, §2.
Let $E$ be a commutative monoid (written multiplicatively) and $S$ a submonoid of $E$. Define on $E\times S$ an equivalence relation by $(a,s)\sim(b,t):\Leftrightarrow$ "$\exists u\in S$ such that $atu=bsu$". Denote the set $(E\times S)/\sim$ by $\overline{E}$ and the class of $(a,s)$ by $a/s$. For any $a\in E$, let $\epsilon(a)=a/e$.
So far so good. The Bourbaki exercise is actually more general (replacing commutativity by weaker assumptions), but, even in this special case, I can't do the next step:
Show that there exists on $\overline{E}$ one and only one monoid structure such that $\epsilon$ is a monoid homomorphism and such that, for all $s\in S$, $\epsilon(s)$ is invertible.
My problem is with the "only one" part. Given a monoid structure $\otimes$ on $\overline{E}$ with those properties, I see no reason why $(a/s)\otimes(b/t)=(ab)/(st)$ or even $(s/e)\otimes(e/s)=e/e$. I couldn't find a counterexample in the case $E=\mathbb{N}$, nor did the search give me any intuition on why the assertion should be true.
This commutative case is treated in the text of Bourbaki's Algebra, but there no mention is made of "only one".
I'm glad for anything that gets me started.