On Delta Sets of Numerical Monoids

Abstract

Let S be a numerical monoid (i.e. an additive submonoid of ℕ0) with minimal generating set 〈n1,…,nt〉. For m ∈ S, if [Formula: see text], then [Formula: see text] is called a factorization length of m. We denote by [Formula: see text] (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 Δ(m) = {mi+1 - mi|1 ≤ i < k} and the Delta set of S by Δ(S) = ∪m∈SΔ(m). In this paper, we address some basic questions concerning the structure of the set Δ(S). In Sec. 2, we find upper and lower bounds on Δ(S) by finding such bounds on the Delta set of any monoid S where the associated reduced monoid S red is finitely generated. We prove in Sec. 3 that if S = 〈n, n + k, n + 2k,…,n + bk〉, then Δ(S) = {k}. In Sec. 4 we offer some specific constructions which yield for any k and v in ℕ a numerical monoid S with Δ(S) = {k, 2k,…,vk}. Moreover, we show that Delta sets of numerical monoids may contain natural "gaps" by arguing that Δ(〈n, n + 1, n2 - n - 1〉) = {1,2,…,n - 2, 2n - 5}.

On the Delta set of a singular arithmetical congruence monoid

“…Since 2 has multiplicative order 4 modulo 5, we see that max ∆(M 6,30 ) = 4. It is pretty easy to find witnesses to 1, 2, 3 ∈ ∆(M 6,30 ), and so ∆(M 6,30 ) = [1,4]. The local decomposition of M 6,30 is M 6,10 ∩ M 6,15 , both of which have ∆-set equal to [1,3] by Theorem 3.1.…”

“…For instance, the ∆-set of the Hilbert monoid H m (which is a Krull monoid) is equivalent to that of the block monoid on (Z/mZ) × (see [9]), and in general, little can be said about the ∆-set of a block monoid on a finite abelian group unless it is cyclic. The ∆-set of a numerical monoid (an additive submonoid of N 0 ) has been analyzed rigorously in [4] where the authors characterize ∆(S) when S is a numerical monoid with equally spaced generators. In particular, if S = a, a + k, a + 2k, .…”

Systems of Sets of Lengths: Transfer Krull Monoids Versus Weakly Krull Monoids

“…Sets of lengths of numerical monoids have found wide attention in the literature (see, among others, [9,1,14]). As can be seen from Theorem 5.5.3, the structure of their sets of lengths is simpler than the structure of sets of lengths of transfer Krull monoids over finite abelian groups.…”

On delta sets and their realizable subsets in Krull monoids with cyclic class groups

“…A large class of Dedekind domains (whose multiplicative monoids are Krull) with such delta sets are constructed in [9]. Another example of such a monoid is a primitive numerical monoid whose minimal generating set forms an arithmetic sequence (see [3,Theorem 3.9]). By [7,Theorem 1], the sequence of sets {∆(n)} n∈S is eventually periodic, and hence using the periodic bound in that theorem allows one to check for all realizable subsets of {1, 2, 3} in finite time.…”