Let $M,N$ be cancellative monoids with identity $\epsilon$ and suppose that $k\colon M\rightarrow N$ is a function such that

- $k(\epsilon)=\epsilon$
- for all $a,b\in M$, there exists $v\in N$ such that $k(ab)=k(a)v$

What are maps satisfying the two points above called? The $v$ in the second condition is unique by the cancellation law. Suppose both monoids have a trivial group of invertible elements and the ordering: $a \leq b$ iff there exists $c$ such that $ac=b$. In this case, the second condition above is equivalent to "k is monotone".

When $M$ has non-trivial group of invertible elements, $\leq$ is no longer a partial ordering and must be modified: redefine so that $a < ab$ for all $a\in M$ and non-invertible $b$. Going to the trouble of working with this ordering merely to have the convenience of using the short name "monotone, identity-preserving" to describe such a $k$ is an obstacle I'd like to overcome.

There is a simple theorem statement here relating such maps and right $M$-actions, and I'm looking for a reference that studies such objects and hopefully includes this theorem and others. I tried looking in Clifford and Preston, no luck.