Structure of commutative cancellative subarchimedean semigroups

A commutative semigroup $S$ is subarchimedean if there is an element $z\in S$ such that for every $a\in S$ there exist a positive integer $n$ and $x\in S$ such that $z^n=ax$. Such a semigroup is archimedean if this holds for all ${z\in S}$. A commutative cancellative idempotent-free archimedean semigroup is an $\frak{N}$-semigroup. We study the structure of semigroups in the title as related to $\frak{N}$-semigroups.