Classes of complete intersection numerical semigroups

Abstract

We consider several classes of complete intersection numerical semigroups, aris- ing from many different contexts like algebraic geometry, commutative algebra, coding theory and factorization theory. In particular, we determine all the logical implications among these classes and provide examples. Most of these classes are shown to be well-behaved with respect to the operation of gluing.Comment: 13 page

Isolated factorizations and their applications in simplicial affine semigroups

“…Recall that numerical semigroups are Cohen-Macaulay. Therefore, the previous result yields that if S is an α-rectangular numerical semigroup for n, then it is free for an arrangement of its minimal generators starting by n. This result was proven in [6] when n = m(S). Now we study the set of isolated factorizations of α-rectangular semigroups.…”

“…Our definition generalizes that of [6]. The characterizations obtained in [6] for numerical semigroups can be easily generalized to the current setting; Proposition 4.4 and Proposition 4.9 generalize [6, Proposition 2.6] and [6, Theorem 3.3], respectively.…”

“…Remark 4.2. In [6] the authors define a α-rectangular numerical semigroup as a numerical semigroup S minimally generated by {n 1 < · · · < n e } such that…”

“…Since then many families of free numerical semigroups have been studied; see, for instance, [7,6] and the references given therein. One of these families is that of α-rectangular numerical semigroups [6]. In this paper we give a wider definition of α-rectangularity and generalize it to the context of simplicial affine semigroups.…”

numericalsgps, a GAP package for numerical semigroups

“…The functions implemented by Sammartano are mainly focused on deriving properties of the semigroup algebra k[ [S]] and its associated graded algebra from properties of the numerical semigroup S. He offers procedures to determine purity and M-purity of S ( [13]), Buchbaum ( [21]), Gorenstein ( [22]) and complete intersection ( [24]) property for the graded algebra; some special shapes of the Apéry sets (α, β and γ -rectangular, see [23]); and the type sequence of a numerical semigroup ( [7]). …”