A realization theorem for sets of distances

Abstract

Let H be an atomic monoid. The set of distances Δ(H) of H is the set of all d∈N with the following property: there are irreducible elements u1,…,uk,v1…,vk+d such that u1⋅…⋅uk=v1⋅…⋅vk+d but u1⋅…⋅uk cannot be written as a product of ℓ irreducible elements for any ℓ∈N with k<ℓ<k+d. It is well-known (and easy to show) that, if Δ(H) is nonempty, then min⁡Δ(H)=gcd⁡Δ(H). In this paper we show conversely that for every finite nonempty set Δ⊂N with min⁡Δ=gcd⁡Δ there is a finitely generated Krull monoid H such that Δ(H)=Δ.