The realization problem for delta sets of numerical semigroups

Abstract

The delta set of a numerical semigroup S, denoted ∆(S), is a factorization invariant that measures the complexity of the sets of lengths of elements in S. We study the following problem: Which finite sets occur as the delta set of a numerical semigroup S? It is known that min ∆(S) = gcd ∆(S) is a necessary condition. For any two-element set {d, td} we produce a semigroup S with this delta set. We then show that for t ≥ 2, the set {d, td} occurs as the delta set of some numerical semigroup of embedding dimension three if and only if t = 2.

Factoring in the Chicken McNugget Monoid

“…Notice that in the expressions of 36, 42, 45, 48, 54, and 63, only the first two generators appear (hence, these are acting like factorizations in the numerical monoid 2, 3 and the tame degree of this monoid is 3; see Example 20). Thus, the tame degrees of 18,24,27,30,36,38,42,45,48,54,58, and 63 are all 3. Table 3: Factorizations of elements necessary to compute the tame degree.…”

“…Intuitively, the delta set records the "gaps" in (or "missing") factorization lengths. There is a wealth of recent work concerning the computation of the delta set of a numerical monoid [5,7,10,13,15,16,18,22]. For numerical monoids with three generators, the computation of the delta set is tightly related to Euclid's extended greatest common divisor algorithm [23,24].…”

Factorization theory in commutative monoids

“…It is an open problem whether there is a numerical monoid H with prescribed sets of distances (see [22]). F. Gotti proved that there is a primary BF-submonoid H of (Q ≥0 ), +) such that every finite nonempty set L ⊂ N ≥2 occurs as a set of lengths of H (see [59,Theorem 3.6], and compare with Remark 5.7).…”

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

“…We use the same notation as in A1, and assert that L(U 14] which implies that [6,14] ∈ L(G). Suppose that k ≥ 9, and that the assertion holds for all…”

“…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.…”