Numerical semigroups II: pseudo-symmetric AA-Semigroups

TL;DR

This paper investigates the structure of AA-semigroups, establishing bounds on their type, characterizing pseudo-symmetry, and providing algorithms and methods for identifying and constructing pseudo-symmetric AA-semigroups.

Contribution

It introduces bounds on the type of AA-semigroups, characterizes pseudo-symmetry, and offers algorithms for their detection and construction.

Findings

Existence of an upper bound for the type of AA-semigroups depending on the number of generators

Polynomial time algorithm to decide pseudo-symmetry of AA-semigroups

Methods to construct pseudo-symmetric AA-semigroups with many generators

Abstract

This paper is a continuation of the paper "Numerical Semigroups: Ap\'ery 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 $a,a+d,a+2d,\dots,a+kd,c$. 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.