Numerical semigroups from rational matrices II: matricial dimension does not exceed multiplicity

TL;DR

This paper proves that the matricial dimension of a numerical semigroup is at most its multiplicity, providing a tight bound for many cases including symmetric numerical semigroups, advancing understanding of their algebraic structure.

Contribution

It establishes a new upper bound on the matricial dimension of numerical semigroups, improving previous bounds and characterizing cases where the bound is tight.

Findings

Matricial dimension is at most the multiplicity of the semigroup.

The bound is tight for many numerical semigroups, including all symmetric ones.

The result significantly refines the understanding of the structure of numerical semigroups.

Abstract

We continue our study of exponent semigroups of rational matrices. Our main result is that the matricial dimension of a numerical semigroup is at most its multiplicity (the least generator), greatly improving upon the previous upper bound (the conductor). For many numerical semigroups, including all symmetric numerical semigroups, our upper bound is tight.