论文标题
数值半群的主要矩阵
Principal Matrices of Numerical Semigroups
论文作者
论文摘要
嵌入尺寸n的数值半群的主要矩阵是$ n \ times n $矩阵的特殊类型，而级别的整数$ \ \ leq n -1 $。我们表明，无论嵌入维度如何我们为伪主矩阵提供了结构定理，其中至少有一个$ n -1 \ times n -1 $主要的小次要矩阵消失，从而将半群的特征在于嵌入尺寸的$ 4 $和$ 5 $的主矩阵。当伪主矩阵为等级$ n -1 $时，我们给出了足够的条件，使其成为本金。
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.
