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一类线性变换半群的格林关系
引用本文: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.
作者姓名:裴惠生  卢凤梅
作者单位:信阳师范学院数学系, 河南 信阳 464000;安阳工业学院理学部, 河南 安阳 454900
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 = V1 V2 ... Vm is a direct sum decomposition of V, where V1,V2,..., Vm are subspaces of V with the same dimension. A linear transformation f ∈ L(V) is said to be sum-preserving, if for each i (1 ≤ i ≤ m), there exists some j (1 ≤ j ≤ m) such that f(Vi) Vj. It is easy to verify that all sum-preserving linear transformations form a subsemigroup of L(V) which is denoted by L (V). In this paper, we first describe Green's relations on the semigroup L (V). Then we consider the regularity of elements and give a condition for an element in L (V) to be regular. Finally, Green's equivalences for regular elements are also characterized.

关 键 词:线性变换  变换半群  格林关系  直和分解  格林等价  空间域  有限维  虚拟机
收稿时间:2007/5/18 0:00:00
修稿时间:2007/11/22 0:00:00

Green's Relations on a Kind of Semigroups of Linear Transformations
PEI Hui Sheng and LU Feng Mei.Green''s Relations on a Kind of Semigroups of Linear Transformations[J].Journal of Mathematical Research with Applications,2009,29(5):931-944.
Authors:PEI Hui Sheng and LU Feng Mei
Institution:Department of Mathematics, Xinyang Normal University, Henan 464000, China;Department of Science, Anyang Polytechnic College, Henan 454900, China
Abstract: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.
Keywords:linear spaces  linear transformations  semigroups  Green's equivalence  regular semigroups  
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