Delta sets for nonsymmetric numerical semigroups with embedding dimension three

Abstract

We present a fast algorithm to compute the Delta set of a nonsymmetric numerical semigroup with embedding dimension three. We also characterize the sets of integers that are the Delta set of a numerical semigroup of this kind.

Delta sets for symmetric numerical semigroups with embedding dimension three

“…As we mentioned above, this was not needed in the nonsymmetric case, since in that setting the Delta sets of the Betti elements of the numerical semigroup are singletons. Then, we prove our main result, which yields an algorithm that works in the same way as in the non-symmetric case; in particular, notice that Example 20 provided in this paper gives the same Delta set as in [10,Example 38], because it is obtained from the same δ 1 and δ 2 .…”

“…The minimum was known to be the greatest common divisor of the Delta set since [13]; however, the elements in the interval determined by this minimum and maximum element of the Delta set are not known in general. Some realization results were given in [4], while in [10] the sets that can be realized as the Delta set of a non-symmetric numerical semigroup with embedding dimension three are completely characterized. In that paper, the authors present a procedure that strongly reduces the time needed to compute the Delta sets of non-symmetric embedding dimension three numerical semigroups.…”

“…Unfortunately, from our reasoning, it emerged that there are no relations between these two cases; we think that it is not possible to find a common way to deduce both settings together. This paper, even though is complementary to [10], is self-contained, and therefore can be read separately from [10]. After some preliminary results, we explain how we can choose the elements δ 1 and δ 2 needed to start the algorithm.…”

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.…”

“…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]. We now summarize the main results in [ if r = 0, 6, 9, 15, where n = 20q + r for q, r ∈ N 0 and r < 20.…”