Principal Matrices of Numerical Semigroups

TL;DR

This paper investigates the properties and structure of principal matrices associated with numerical semigroups, establishing rank bounds and characterizations for specific cases, especially in embedding dimensions 4 and 5.

Contribution

It provides new structure theorems for pseudo principal matrices and characterizes semigroups via their principal matrices in low embedding dimensions.

Findings

Principal matrices have rank at least n/2.

Characterization of semigroups in embedding dimensions 4 and 5.

Sufficient conditions for pseudo principal matrices to be principal.

Abstract

Principal matrices of a numerical semigroup of embedding dimension n are special types of $n \times n$ matrices over integers of rank $\leq n - 1$. We show that such matrices and even the pseudo principal matrices of size n must have rank $\geq \frac{n}{2}$ regardless of the embedding dimension. We give structure theorems for pseudo principal matrices for which at least one $n - 1 \times n - 1$ principal minor vanish and thereby characterize the semigroups in embedding dimensions $4$ and $5$ in terms of their principal matrices. When the pseudo principal matrix is of rank $n - 1$, we give a sufficient condition for it to be principal.