A commutative semigroup S is subarchimedean if there exists z ∈ S such that for any a ∈ S, there are n ＞ 0 and x ∈ S such that zn ＝ ax. A commutative cancellative idempotent-free subarchimedean semigroup is a -semigroup.These semigroups admit Tamura-like representations of the form N(G, I) and N(G,ψ), and their groups of quotients Z(G, I) and Z(G, ψ). We consider categories whose objects are of the form (G, I; N), (G, ψ; N), (G, I; Z), and (G, ψ; Z)with suitable morphisms, and establish functorial relationships among these categories as well as with the categories of -semigroups and non-periodic abelian groups.