Delta Sets of Numerical Monoids Are Eventually Periodic

Let M be a numerical monoid (i.e., an additive submonoid of N0) with minimal generating set 〈n1, . . . , nt〉. For m ∈ M , if m = Pt i=1 xini, then Pt i=1 xi is called a factorization length of m. We denote by L(m) = {m1, . . . , mk} (where mi < mi+1 for each 1 ≤ i < k) the set of all possible factorization lengths of m. The Delta set of m is defined by ∆(m) = {mi+1 −mi | 1 ≤ i < k } and the Delta set of M by ∆(M) = ∪0 6=m∈M∆(m). In this paper, we expand on the study of ∆(M) begun in [2] and [3] by showing that the delta sets of a numerical monoid are eventually periodic. More specifically, we show for all x ≥ 2kn2nk in M that ∆(x) = ∆(x + n1nk). Let M be a commutative cancellative monoid with set M• of nonunits and A(M) of irreducible elements. We assume thatM is atomic (i.e., every nonunit can be written as a product of irreducible elements). Problems involving the factorization properties of elements inM into irreducible elements have been a frequent topic in the mathematical literature over the past 20 years (see [6] and the references cited therein). Most of this work entails a study of the length set of an element x ∈ M which is defined as L(x) = {l | ∃ a1, . . . , al ∈ A(M) such that x = a1 · · · al} . If L(x) = {l1, . . . , lj} with l1 < l2 < . . . < lj , then define the delta set of x as the set of consecutive differences of lengths, ∆(x) = {li+1 − li | 1 ≤ i < j}. The delta set of M is defined as ∆(M) = ⋃