Frobenius Number Of Almost Symmetric Numerical Generalized Almost Arithmetic Semigroups

TL;DR

This paper extends the understanding of Frobenius numbers for a class of numerical semigroups called generalized almost arithmetic semigroups, providing formulas, characterizations, and algorithms for their properties.

Contribution

It offers a quadratic formula for the Frobenius number and characterizes the semigroup's symmetry properties under certain conditions.

Findings

Derived a quadratic formula for the Frobenius number.

Provided a complete description of symmetric and almost symmetric semigroups.

Developed an algorithm to determine symmetry type and Frobenius number.

Abstract

Let a, k, h, c be positive integers and d a non zero integer. Recall that a numerical generalized almost arithmetic semigroup S is a semigroup minimally generated by relatively prime positive integers a, ha + d, ha + 2d, . . . , ha + kd, c, that is its embedding dimension is k + 2. In a previous work, the authors described the Ap{\'e}ry set and a Gr{\"o}bner basis of the ideal defining S under one technical assumption, the complete version will be published in a forthcoming paper. In this paper we continue with this assumption and we describe the Pseudo Frobenius set. As a consequence we give a complete description of S when it is symmetric or almost symmetric as well as generalize and extend the previous results of Ignacio Garc{\'i}a-Marco, J. L. Ram{\'i}rez Alfons{\'i}n and O. J. R{{\o}}dseth; we also find a quadratic formula for its Frobenius number that generalizes some results of J.C. Rosales, and P.A. Garc{\'i}a-S{\'a}nchez. Moreover, for given numbers a, d, k, h, c, a simple algorithm allows us to determine if S is almost symmetric or not and furthermore to find its type and Frobenius number.