Criteria for existence of semigroup homomorphisms and projective rank functions

Abstract

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 necessary 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 $g(p) = w(t_1,\,\dots\,,\,t_n).$ 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, $x^d = y^d \implies x = y$ 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.