Every homomorphism between (rational) Puiseux monoids is multiplication by a non-negative rational

Let a (rational) Puiseux monoid be a non-trivial submonoid of the non-negative rational numbers under (the usual operation of) addition. It is not difficult to show that, if $$f \colon H \to K$$ is a (monoid) homomorphism between Puiseux monoids, then there exists a rational number $$r \ge 0$$ such that $$f(x) = r x$$ for every $$x \in H$$.

Proof. Set $$H^+ := H \setminus \{0\}$$ and pick $$p, q \in H^+$$ (since $$f$$ maps $$0$$ to $$0$$, we can focus on the positive elements). There then exist positive integers $$a$$, $$b$$, $$c$$, and $$d$$ such that $$p = a/b$$ and $$q = m/n$$. It follows that $$mb f(p) = f(mbp) = f(ma) = f(naq) = na f(q),$$ and hence $$f(p)/p = f(q)/q$$. That is, the function $$H^+ \to \mathbb Q_{\ge 0} \colon x \mapsto f(x)/x$$ is constant. []

Question. The result, along with a slighly more complicated proof, appears as Proposition 3.2(1) in [F. Gotti, Puiseux monoids and transfer homomorphisms, J. Algebra 516 (2018), 95–114], but I feel it must be older. Do you know of any reference that supports my feeling?

Note that the case of numerical monoids (that is, submonoids of $$(\mathbb N, +)$$ with finite complement in $$\mathbb N$$) is covered by Theorem 3.2 in [J.C. Higgins, Representing $$N$$-semigroups, Bull. Austral. Math. Soc. 1 (1969), 115–125], as I learned from Benjamin Steinberg's answer in another thread some time ago.

• I think that, by the term "positive rational semigroup", Sasaki and Tamura mean a (non-empty) subsemigroup of the positive rational numbers under addition. Corollary 4 in their paper says that any homomorphism between positive rational semigroups is in fact an isomorphism (of the form $x \mapsto rx$, where $r$ is a positive rational number). This is not the case with Puiseux monoids, but only because it is missing the zero homomorphism. Commented Feb 13 at 3:47
• So, to recover the result in the OP from Sasaki and Tamura's corollary, I think one should first argue (as trivial as it may be) that a monoid hom $f \colon H \to K$ between Puiseux monoids restricts to a sgrp hom between the "positive cones" of $H$ and $K$, unless $f$ is identically zero. Commented Feb 13 at 3:56