论文标题
生成热带基质单体的集合，演示和生长
Generating sets, presentations, and growth of tropical matrix monoids
论文作者
论文摘要
我们为$ n \ times n $上三角形矩阵的单型属于$ n \ times n $ times n $ times n $ times n $ niangular矩阵的属于最小和不繁殖的产生集。我们表明，$ n \ times n $矩阵在热带整数上，$ m_n（\ mathbb {z} _ \ mathrm {max}）$，仅在$ n \ leq 2 $时才有限地生成，并且仅在$ n \ leq 2 $时才生成，并且仅在$ n = $ n = $ n = $ n = $ n = $ n = $ n = $ n = $ n = $ n = $ n = $ n = $ n = $ n = $ n = $ n = $ n = 1 $中。当$ n \ leq 3 $时，明确构建了最小和非繁殖生成集。然后，我们为热带整数上的$ n \ times n $上三角矩阵构建演示文稿，$ ut_n（\ mathbb {z} _ \ mathrm {max}）$，表明它是所有$ n \ in \ nathbb {n} $的$ n \ in \ in \ n} $的$ n \。最后，我们基于在双能半肌肉上的$ n \ times n $矩阵的有限生成的子群的生长函数的多项式程度上建立上限，并表明这些边界对于热带半级是锋利的。
We construct minimal and irredundant generating sets for a family of submonoids of the monoid of $n \times n$ upper triangular matrices over a commutative semiring. We show that the monoid of $n \times n$ matrices over the tropical integers, $M_n(\mathbb{Z}_\mathrm{max})$, is finitely generated if and only if $n \leq 2$, and finitely presented if and only if $n = 1$. Minimal and irredundant generating sets are explicitly constructed when $n \leq 3$. We then construct a presentation for the monoid of $n \times n$ upper triangular matrices over the tropical integers, $UT_n(\mathbb{Z}_\mathrm{max})$, demonstrating that it is finitely presented for all $n \in \mathbb{N}$. Finally, we establish upper bounds on the polynomial degree of the growth function of finitely generated subsemigroups of the monoid of $n \times n$ matrices over a bipotent semiring and show that these bounds are sharp for the tropical semiring.
