| 裴惠生,卢凤梅.一类线性变换半群的格林关系[J].数学研究及应用,2009,29(5):931~944 |
| 一类线性变换半群的格林关系 |
| Green's Relations on a Kind of Semigroups of Linear Transformations |
| 投稿时间：2007-05-18 修订日期：2007-11-22 |
| DOI：10.3770/j.issn:1000-341X.2009.05.021 |
| 中文关键词: 线性空间 线性变换 半群 格林关系 正则元. |
| 英文关键词:linear spaces linear transformations semigroups Green's equivalence regular semigroups. |
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| 中文摘要: |
| 设$V$是数域$F$上的有限维线性空间.$L(V)$表示$V$上所有线性变换, 关于变换的复合, 作成的半群. 假设 $V=V_1 \oplus V_2 \oplus\cdots\oplus V_m$ 是$V$的一个直和分解, 这里$V_1,\,V_2,\,\cdots,V_m$ 是$V$的具有相同维数的子空间.. 称线性变换$f\in L(V)$是直和保持的, 如果对于每个 $i\ (1\leq i\leq m)$, 存在$j\ (1\leq j\leq m)$ 使得 $f(V_i)\subseteq |
| 英文摘要: |
| Let $V$ be a linear space over a field $F$ with finite dimension, $L(V)$ the semigroup, under composition, of all linear transformations from $V$ into itself. Suppose that $V=V_1 \oplus V_2 \oplus\cdots\oplus V_m$ is a direct sum decomposition of $V$, where $V_1,V_2,\ldots,V_m$ are subspaces of $V$ with the same dimension. A linear transformation $f\in L(V)$ is said to be sum-preserving, if for each $i\ (1\leq i\leq m)$, there exists some $j\ (1\leq j\leq m)$ such that $f(V_i)\subseteq V_j$. It is easy to verify that all sum-preserving linear transformations form a subsemigroup of $L(V)$ which is denoted by $L^{\oplus}(V)$. In this paper, we first describe Green's relations on the semigroup $L^{\oplus}(V)$. Then we consider the regularity of elements and give a condition for an element in $L^{\oplus}(V)$ to be regular. Finally, Green's equivalences for regular elements are also characterized. |
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