n×n矩阵的极大传递半群（修订与扩展版）
Maximally transitive semigroups of $n\times n$ matrices (revised and extended version)
AI总结：
该研究证明了n×n矩阵的2生成元半群可在对应矩阵空间中稠密，且其在(𝕂ⁿ)ⁿ上拓扑传递，同时给出了相关半群稠密性的自包含证明。
摘要：
设𝕂为ℝ或ℂ，我们证明对每个n≥1，存在一对元素属于𝕂的n×n矩阵，它们生成所有n×n矩阵构成的空间Mₙ(𝕂)的一个稠密子半群。由于单个矩阵的幂永远不稠密，两个生成元是最优的。等价地，存在一个2生成元线性映射半群，其在(𝕂ⁿ)ⁿ上的作用是拓扑传递的，而不存在任何线性映射半群在(𝕂ⁿ)ⁿ⁺¹上的作用是拓扑传递的。事实上，对该半群而言，(𝕂ⁿ)ᵏ中每个轨道闭包（对所有k≥1）都达到线性代数允许的最大规模。在此过程中，我们基于SL₂(ℝ)的闭子半群的简单判据，给出了GL₂⁺(ℝ)和M₂(ℝ)的显式2生成元子半群稠密性的简短自包含证明。
英文摘要：
Let $\mathbb{K}=\mathbb{R}$ or $\mathbb{C}$. We prove that for every $n\ge 1$ there is a pair of $n\times n$ matrices with entries in $\mathbb{K}$ that generates a dense subsemigroup of the space $M_n(\mathbb{K})$ of all $n\times n$ matrices. Since the powers of a single matrix are never dense, two generators is optimal. Equivalently, there is a $2$-generator semigroup of linear maps whose action on $(\mathbb{K}^n)^n$ is topologically transitive, whereas no semigroup of linear maps acts topologically transitively on $(\mathbb{K}^n)^{n+1}$. In fact, for such a semigroup every orbit closure in $(\mathbb{K}^n)^k$, for every $k\ge1$, is as large as linear algebra allows. Along the way we give a short self-contained proof of the density of explicit $2$-generator subsemigroups of $GL_2^+(\mathbb{R})$ and of $M_2(\mathbb{R})$, based on a simple criterion for closed subsemigroups of $SL_2(\mathbb{R})$.
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