Principal Matrices of Numerical Semigroups

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.