Detecting divisibility in uniquely divisible $\mathfrak{N}$-semigroups via homomorphisms to $\mathbb{R}_{>0}$

An $\mathfrak{N}$-semigroup is an archimedean, idempotent-free, cancellative and commutative semigroup. For example, the additive semigroup $\mathbb{R}_{>0}$ of strictly positive real numbers is a $\mathfrak{N}$-semigroup. Hewitt-Zuckerman showed that every $\mathfrak{N}$-semigroup admits a semigrou