When exactly is the preimage of the group of units the group of units?

Question

Let $M$ and $N$ be monoids. Denote by $M^\times$ and $N^\times$ the respective groups of units. Let $f:M\to N$ is a homomorphism of monoids.

Is there a necessary and sufficient condition for the preimage $f^{-1}(N^\times)$ to be equal to $M^\times$?

Even in the easiest examples, I don't seem to find anything promising. For instance, in the case of the inclusion of multiplicative monoids $\mathbf{Z}\to \mathbf{Q}$, the preimage of $3$; a unit in $\mathbf{Q}$, is a non-unit in $\mathbf{Z}$.

Answer 1

First of all $f(M^\times) \subseteq N^\times$ and thus $M^\times \subseteq f^{-1}(N^\times)$. Thus a necessary and sufficient condition for the preimage $f^{-1}(N^\times)$ to be equal to $M^\times$ is that, for every element $n \in N^\times$, the preimage $f^{-1}(n)$ is a subset of $M^\times$. This is a trivial and certainly disappointing answer, but I am afraid you can't go beyond that.