论文标题
矩阵单体的最小生成集
Minimal generating sets for matrix monoids
论文作者
论文摘要
在本文中，我们确定了半序的几种众所周知的矩阵的最小生成集。特别是，我们找到了由$ n \ leq 8 $时的monoids的最小生成集。 $ n \ times n $ boolean矩阵包含身份矩阵（反射布尔矩阵）时，当$ n \ leq 7 $时； $ n \ times n $ boolean矩阵包含置换（hall矩阵）$ n \ leq 8 $时；每个维度的上部和下三角布尔矩阵； $ 2 \ times 2 $矩阵，$ \ mathbb {n} \ cup \ { - \ fty \} $，加上$ \ oplus $由$ x \ oplus y = \ max（x，y）$和乘以$ x \ imimes $ \ otimimes $ \ x \ x \ x \ ytem y y y y y y y y y y y = x + y y = x + y = x + y = x + y = $ 2 \ times 2 $矩阵上的最大值半矩阵由$ t = t + 1 $生成的一致性，其中$ t \ in \ mathbb {n} $; $ 2 \ times 2 $矩阵矩阵及其在\ mathbb {n} $中的$ t = t = t + 1 $生成的一致性及其有限的商。以及$ n \ times n $矩阵上的$ \ mathbb {z} / n \ mathbb {z} $相对于其单位组。
In this paper, we determine minimal generating sets for several well-known monoids of matrices over semirings. In particular, we find minimal generating sets for the monoids consisting of: all $n\times n$ boolean matrices when $n\leq 8$; the $n\times n$ boolean matrices containing the identity matrix (the reflexive boolean matrices) when $n\leq 7$; the $n\times n$ boolean matrices containing a permutation (the Hall matrices) when $n \leq 8$; the upper, and lower, triangular boolean matrices of every dimension; the $2 \times 2$ matrices over the semiring $\mathbb{N} \cup \{-\infty\}$ with addition $\oplus$ defined by $x\oplus y = \max(x, y)$ and multiplication $\otimes$ given by $x\otimes y = x + y$ (the max-plus semiring); the $2\times 2$ matrices over any quotient of the max-plus semiring by the congruence generated by $t = t + 1$ where $t\in \mathbb{N}$; the $2\times 2$ matrices over the min-plus semiring and its finite quotients by the congruences generated by $t = t + 1$ for all $t\in \mathbb{N}$; and the $n \times n$ matrices over $\mathbb{Z} / n\mathbb{Z}$ relative to their group of units.
