| 朱用文.矩阵逆半群[J].数学研究及应用,2008,28(3):549~557 |
| 矩阵逆半群 |
| Inverse Semigroups of Matrices |
| 投稿时间：2006-07-18 修订日期：2007-03-22 |
| DOI：10.3770/j.issn:1000-341X.2008.03.013 |
| 中文关键词: 矩阵半群 逆半群 格林关系 Clifford半群 半格. |
| 英文关键词:matrix semigroup inverse semigroup Green's relation Clifford semigroup semilattice. |
| 基金项目:国家自然科学基金(No.10571005). |
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| 中文摘要: |
| 讨论矩阵逆半群的一些基本性质, 证明矩阵逆半群的幂等元集是有限布尔格的子半格, 从而证明等秩矩阵逆半群是群, 然后完全确定二级矩阵逆半群的结构：一个二级矩阵逆半群或者同构于二级线性群,或者同构于二级线性群添加一个零元素,或者是交换线性群的有限半格, 或者满足其他一些性质; 对于由某些二级矩阵构成的集合, 我们给出了它们成为矩阵逆半群的充分必要条件. |
| 英文摘要: |
| We discuss some fundamental properties of inverse semigroups of matrices, and prove that the idempotents of such a semigroup constitute a subsemilattice of a finite Boolean lattice, and that the inverse semigroups of some matrices with the same rank are groups. At last, we determine completely the construction of the inverse semigroups of some $2\times 2$ matrices: such a semigroup is isomorphic to a linear group of dimension 2 or a null-adjoined group, or is a finite semilattice of Abelian linear groups of finite dimension, or satisfies some other properties. The necessary and sufficient conditions are given that the sets consisting of some $2\times 2$ matrices become inverse semigroups. |
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