Criteria for existence of semigroup homomorphisms and projective rank functions

Let $P,$ $S,$ and $T$ be semigroups, $f:P\to S$ and $g:P\to T$ semigroup homomorphisms, and $X$ a generating set for $S$ (possibly infinite). Clearly, a <i>necessary</i> condition for there to exist a homomorphism $S\to T$ making a commuting triangle with $f$ and $g$ is that for every relation $f(p)=w(x_1,\dots ,x_n)$ holding in $S$, with $p\in P,$ $w$ a semigroup word, and $x_1,\dots ,x_n\in X,$ there exist $t_1,\dots ,t_n\in T$ satisfying $g(p)=w(t_1,\dots ,t_n)\mathrm{.}$ Under what assumptions will that also be sufficient? We show that one such family of assumptions is that (i) every element of $S$ is a divisor some element of $f(P),$ (ii) $T$ is right and left cancellative, (iii) $T$ is power-cancellative, i.e, $x^d=y^d⟹x=y$ for $d>0,$ and (iv) a certain technical condition which, in particular, holds if $T$ admits a semigroup ordering with the order-type of the natural numbers. As an application, we obtain an elementary criterion for the existence of an integer-valued rank function on finitely generated projective modules over a ring.