Criteria for existence of semigroup homomorphisms and projective rank functions

TL;DR

This paper establishes criteria for when homomorphisms between semigroups exist, based on relations and cancellative properties, and applies these results to characterize integer-valued rank functions on modules.

Contribution

It provides sufficient conditions for the existence of semigroup homomorphisms and introduces an elementary criterion for rank functions on finitely generated projective modules.

Findings

Identifies necessary and sufficient conditions for homomorphism existence.

Shows that certain algebraic properties of target semigroups guarantee homomorphism extension.

Provides a criterion for the existence of integer-valued rank functions on modules.

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 <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 $g(p) = w(t_1,\,\dots\,,\,t_n).$ 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.