Maximal denumerant of a numerical semigroup with embedding dimension less than four

Abstract

Given a numerical semigroup S = a1, a2, . . . , at and s ∈ S, we consider the factorization s = c1a1 +c2a2 +· · ·+ctat where ci ≥ 0. Such a factorization is maximal if c1 +c2 +· · ·+ct is a maximum over all such factorizations of s. We show that the number of maximal factorizations, varying over the elements in S, is always bounded. Thus, we define dmax(S) to be the maximum number of maximal factorizations of elements in S. We study maximal factorizations in depth when S has embedding dimension less than four, and establish formulas for dmax(S) in this case.The next example demonstrates how we can easily compute the maximal denumerant of all basic semigroups with a fixed multiplicity.Example 4.5. Let S = 7, a 2 , a 3 be a basic semigroup with multiplicity a 1 = 7. Using Proposition 4.3, we carry out the following steps:

Classes of complete intersection numerical semigroups

“…In connection to this, the number of maximal representations of elements in a semigroup has been investigated recently (cf. [12], [13]). Now we give the criteria for the remaining classes.…”

When the associated graded ring of a semigroup ring is Complete Intersection

“…. This kind of representations and, in particular, the number of maximal representations of elements of S have been studied in [4]. We say that Ap(S) is of unique maximal expression if every ω ∈ Ap(S) has a unique maximal representation.…”

“…Since Ap(S) is γ-rectangular, by Theorem 2.22, m = ν i=2 (γ i + 1); thus in the above chain we have all equalities and, therefore, 4 ). (3) S = 5, 6, 9 : here Ap(S) is not γ-rectangular and gr m (R) is Cohen-Macaulay (as can be checked using [6, Proposition 5.1]); S is symmetric, but not M-pure (since 9 M 18).…”

The realization problem for delta sets of numerical semigroups

“…There are several factorization invariants of numerical semigroups and related commutative monoids that have been studied extensively in the recent literature, for example the maximal denumerant [8], the catenary and tame degree [3,10,25], and the ω-invariant [1,16]. In this paper we focus on another invariant, the delta set [4,6,7,9,11,12,13,17,19].…”