带数乘的矩阵半群  

Matrix Semigroups with Multiplication by Scalars

作　　者：朱用文[1] 郭爱丽[1] 

机构地区：[1]烟台大学数学与信息科学学院,烟台264005

出　　处：《数学学报（中文版）》2010年第6期1181-1186,共6页Acta Mathematica Sinica：Chinese Series

基　　金：国家自然科学基金资助项目(10571005)

摘　　要：通过一个半群在另外一个半群上的作用,我们定义带数乘的半群或G-半群.将这个概念运用到矩阵半群便得到G-矩阵半群的概念.如果S是一个域F上的矩阵半群而G是F的一个乘法子半群,那么我们可以自然地定义G在S上的作用就是通常的数乘运算,从而可以定义矩阵半群S的伴随半群[S].我们研究一个矩阵半群与其伴随半群之间的关系,得到如下的结论:在一定的条件之下,S是正则的(单的、纯正的、逆的、Clifford)半群(群)的充分必要条件是[S]也是如此.最后,我们刻画[S]的幂零性和幂幺性,给出了一些充分必要条件.By means of an action of a semigroup on another,we define a new concept called a semigroup with multiplication by scalars or a G-semigroup.This concept and some fundamental results about it are applied to matrix semigroups so that the G-semigroup of matrices are defined.If S is a matrix semigroup over a field F and G is a multiplicative subsemigroup of F,then we define an action of G on S naturally by the usual multiplication by the scalars which induces a new matrix semigroup which we call the adjoint semigroup of S and denote by[S].We investigate the relationship between a matrix semigroup and its adjoint semigroup,obtain the following results： under certain conditions,S is a regular（simple,orthodox,inverse,Clifford） semigroup （group） if and only if so is[S].At last,we characterize the nilpotency and unipotency of the adjoint semigroup of a matrix semigroup and give some necessary and sufficient conditions.

关 键 词：矩阵半群 数乘 G-半群 

分 类 号：O152.7[理学—数学]