V$. In particular we get $s | V$. Writing $V = s V''$ and $U = s U'$ for some $V'',U' \in M$, the assumption $\varphi(U) | \varphi(V)$ implies $\varphi(U') | \varphi(V'')$ by cancellability, and then $U' | V''$ by the induction assumption. It follows that $U|V$ which proves the claim.

The lemma has the following consequence.

The morphism $\varphi : M \to N$ is injective. If $G_M$, $G_N$ denotes the group of fractions of $M,N$, then $\varphi$ can be extended to $\widetilde{\varphi} : G_M \into G_N$.

Let $U,V \in M$ with $\varphi(U) = \varphi(V)$. By the lemma we get $U|V$ and $V | U$. This implies $\ell(U) = \ell(V)$ hence $U = V$. Composing $\varphi : M \to N$ with the natural morphism $N \into G_N$ yields a monoid morphism $M \to G_N$. Since $G_N$ is a group this morphism factors through the morphism $M \to G_M$ and this provides $\widetilde{\varphi} : G_M \to G_N$. Let $g \in \Ker \widetilde{\varphi}$. Since $g \in G_M$ there exists $a,b \in M$ with $g = ab^{-1}$ hence $\varphi(a) = \varphi(b)$, $a=b$ and $g = 1$.

We consider the following extra assumption on $\varphi$. We assume that, for all $m \in M$ and 