A realization theorem for sets of distances

Let H be an atomic monoid. The set of distances Delta(H) of H is the set of all d is an element of N with the following property: there are irreducible elements u(1),..., u(k), v(1) ..., Vk+d such that u(1)....u(k) = v(1) .... v(k+d) but u(1)... u(k) cannot be written as a product of l irreducible elements for any l is an element of N with k < l < k + d. It is well-known (and easy to show) that, if Delta(H) is nonempty, then mm Delta(H) = gcd Delta(H). In this paper we show conversely that for every finite nonempty set Delta subset of N with min Delta = gcd Delta there is a finitely generated Krull monoid H such that Delta(H) = Delta. (C) 2017 Elsevier Inc. All rights reserved.