Numerical semigroups II: pseudo-symmetric AA-Semigroups

Ignacio García-Marco (1) , Jorge Ramirez Alfonsin (2) , Oystein J. Rodseth (3)

1 LIP - Laboratoire de l'Informatique du Parallélisme
2 GTA
3 UiB - University of Bergen

This paper is a continuation of the paper " Numerical Semigroups: Apéry Sets and Hilbert Series". We consider the general numerical AA-semigroup, i.e., semigroups consisting of all non-negative integer linear combinations of relatively prime positive integers of the form $a, a+d, a+2d,. .. , a+kd, c$. We first prove that, in contrast to arbitrary numerical semigroups, there exists an upper bound for the type of AA-semigroups that only depends on the number of generators of the semigroup. We then present two characterizations of pseudo-symmetric AA-semigroups. The first one leads to a polynomial time algorithm to decide whether an AA-semigroup is pseudo-symmetric. The second one gives a method to construct pseudo-symmetric AA-semigroups and provides explicit families of pseudo-symmetric semigroups with arbitrarily large number of generators .

Mots clés: Numerical semigroup; Apéry set; Frobenius number; Cohen-Macaulay type; genus; pseudo-symmetry

Domaines: Algèbre commutative [math.AC]

Ignacio García-Marco, Jorge Ramirez Alfonsin, Oystein J. Rodseth. Numerical semigroups II: pseudo-symmetric AA-Semigroups. 2016. ⟨hal-01263035⟩