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Frobenius Number Of Almost Symmetric Numerical Generalized Almost Arithmetic Semigroups

Marcel Morales, Nguyen Thi Dung

TL;DR

The article extends the study of numerical generalized almost arithmetic semigroups (AAG-semigroups) generated by $a, ha+d, \dots, ha+kd, c$, with embedding dimension $k+2$, by deriving the Pseudo-Frobenius set under a standing assumption and classifying symmetric and almost symmetric cases. It leverages a Gröbner-basis framework to translate semigroup properties into monomial computations, establishing bijections between the monomial Apéry/PF sets and their classical counterparts and enabling explicit descriptions of $PF(S)$ and ${\rm Ap}(S,a_0)$. The work then provides complete characterizations of symmetric and almost symmetric AAG-semigroups, including explicit parameterizations of $(a,d,c)$ and closed-form Frobenius-number formulas, with a quadratic avenue for computing $F(S)$ by solving equations in an auxiliary parameter. An algorithm is proposed to decide almost symmetry and to determine the type $t(S)$ and Frobenius number, grounded in the PF decomposition and Euclidean-type computations. Overall, the results generalize prior findings (e.g., Garcia-Marco, Ramirez Alfonsín, Rodseth) to broader AAG settings and offer practical, computable criteria and formulas for Frobenius-number problems in this family of numerical semigroups.

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{é}ry set and a Gr{ö}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{í}a-Marco, J. L. Ram{í}rez Alfons{í}n and O. J. R{ø}dseth; we also find a quadratic formula for its Frobenius number that generalizes some results of J.C. Rosales, and P.A. Garc{í}a-S{á}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.

Frobenius Number Of Almost Symmetric Numerical Generalized Almost Arithmetic Semigroups

TL;DR

The article extends the study of numerical generalized almost arithmetic semigroups (AAG-semigroups) generated by , with embedding dimension , by deriving the Pseudo-Frobenius set under a standing assumption and classifying symmetric and almost symmetric cases. It leverages a Gröbner-basis framework to translate semigroup properties into monomial computations, establishing bijections between the monomial Apéry/PF sets and their classical counterparts and enabling explicit descriptions of and . The work then provides complete characterizations of symmetric and almost symmetric AAG-semigroups, including explicit parameterizations of and closed-form Frobenius-number formulas, with a quadratic avenue for computing by solving equations in an auxiliary parameter. An algorithm is proposed to decide almost symmetry and to determine the type and Frobenius number, grounded in the PF decomposition and Euclidean-type computations. Overall, the results generalize prior findings (e.g., Garcia-Marco, Ramirez Alfonsín, Rodseth) to broader AAG settings and offer practical, computable criteria and formulas for Frobenius-number problems in this family of numerical semigroups.

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{é}ry set and a Gr{ö}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{í}a-Marco, J. L. Ram{í}rez Alfons{í}n and O. J. R{ø}dseth; we also find a quadratic formula for its Frobenius number that generalizes some results of J.C. Rosales, and P.A. Garc{í}a-S{á}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.
Paper Structure (11 sections, 20 theorems, 39 equations)

This paper contains 11 sections, 20 theorems, 39 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Numerical generalized almost arithmetic semigroups (AAG-semigroups)
  4. Gröbner basis of AAG-semigroups
  5. Pseudo Frobenius set of AAG-semigroups
  6. Symmetric AAG-semigroups
  7. Almost Symmetric AAG-semigroups, $d\in \mathbb{Z}^*$
  8. Almost symmetric AAG-semigroups with $\widetilde{\rm Frob(S) }\in\widetilde{PF(S)_\varepsilon }$ with $\varepsilon \in \{1,2\},{\rm card}(\widetilde{PF(S)_\varepsilon })\geq 2$
  9. Almost symmetric AAG-semigroups with $\widetilde{PF(S)_2}=\{\widetilde{\rm Frob(S)}\}$
  10. Almost symmetric AAG-semigroups with $\widetilde{\rm PF(S)_1}=\{\widetilde{\rm Frob(S)}\}$
  11. Formula for Frobenius number of Almost Symmetric AAG-semigroups

Key Result

Lemma 2.2

N Suppose $t(S)>1$. Let $PF(S)=\{f_1,\dots, f_{t(S)-1}, F(S)\}$ with $f_1<\dots<f_{t(S)-1}$. Then $S$ is almost symmetric if and only if $f_i+ f_{t(S)-i}= F(S)$ for $i=1,\dots,t(S)-1$.

Theorems & Definitions (37)

  • Definition 2.1
  • Lemma 2.2
  • Corollary 2.3
  • Lemma 2.4
  • proof
  • Remark 3.1
  • Lemma 3.2
  • Corollary 3.3
  • Lemma 3.4
  • Lemma 3.5
  • ...and 27 more