On Numerical Experiments with Symmetric Semigroups Generated by Three Elements and Their Generalization

TL;DR

This paper explains and generalizes numerical experiments on symmetric semigroups generated by three elements, providing formulas for their Frobenius numbers and analyzing their symmetry properties.

Contribution

It introduces a generalized class of symmetric numerical semigroups and computes their universal Frobenius numbers, extending prior experimental observations.

Findings

Derived formulas for Frobenius numbers of generalized semigroups

Identified conditions for symmetry and nonsymmetry in these semigroups

Reduced minimal generating sets for specific parameter values

Abstract

We give a simple explanation of numerical experiments of V. Arnold with two sequences of symmetric numerical semigroups, S(4,6+4k,87-4k) and S(9,3+9k,85-9k) generated by three elements. We present a generalization of these sequences by numerical semigroups S(r_1^2,r_1r_2+r_1^2k,r_3-r_1^2k), k\in{\mathbb Z}, r_1,r_2,r_3\in{\mathbb Z}^+, r_1\geq 2 and \gcd(r_1,r_2)=\gcd(r_1,r_3)=1, and calculate their universal Frobenius number Phi(r_1,r_2,r_3) for the wide range of k providing semigroups be symmetric. We show that this kind of semigroups admit also nonsymmetric representatives. We describe the reduction of the minimal generating sets of these semigroups up to {r_1^2,r_3-r_1^2k} for sporadic values of k and find these values by solving the quadratic Diophantine equation.