半群 H_((n,m)) 的独立子半群  

Isolated Subsemigroups of the Semigroup H_((n,m))

作　　者：袁月 赵平 YUAN Yue;ZHAO Ping(School of Mathematics Science,Guizhou Normal University,Guiyang 550025,China)

机构地区：[1]贵州师范大学数学科学学院,贵阳550025

出　　处：《西南大学学报（自然科学版）》2021年第6期74-81,共8页Journal of Southwest University(Natural Science Edition)

基　　金：国家自然科学基金项目(11461014);贵州师范大学2019年博士科研启动项目(GZNUD[2019]13).

摘　　要：设T_(n)是有限集X n={1,2,…,n}上的全变换半群.对1≤m≤n-1,记X m={1,2,…,m}且X_(n-m)=X n\X m.令T(n,m)={α∈T_(n):X mα=X m}H_((n,m))={α∈T(n,m):X_(n-m)αX_(n-m)}则H_((n,m))和T(n,m)都是全变换半群T_(n)的子半群,且H(n,m)T(n,m).设T是半群S的子半群,如果对任意α∈S,n∈N+,由αn∈T可推出α∈T,则称T为S的独立子半群.考虑半群H_((n,m))的独立子半群T,由于独立子半群T可表示为一些包含幂等元的子集的并集,通过分析T的幂等元集E(T)与半群H_((n,m))中元素的关系,根据其定义及半群的封闭性进行构造,对幂等元及幂等元的生成元作运算,发现:若T包含H_((n,m))(n-2)中的某些幂等元,则可推出奇异变换半群Sing_((n,m))必被包含于T的结论;若T包含H_((n,m))的顶端G_((n,m))的某些元素,可推出G_((n,m))必被包含于T的结论.由此,对E(T)分情况讨论,通过所得结论推出独立子半群的结构特征,进而获得H_((n,m))的独立子半群的完全分类.Let T_(n) be a full transformation semigroup on the finite set X n={1,2,…,n}.For 1≤m≤n-1,let X m={1,2,…,m}and X_(n-m)=X n\X m.Let T(n,m)={α∈T_(n):X mα=X m}H_((n,m))={α∈T(n,m):X_(n-m)αX_(n-m)}then both H_((n,m))and T(n,m)are subsemigroups of the full transformation semigroup T_(n) and H_((n,m))T(n,m).Let T be a subsemigroup of the semigroup S,and ifαn∈T impliesα∈T for allα∈S and n∈N+,then T is an isolated subsemigroup of the semigroup S.In this paper,we consider the isolated subsemigroup T of the semigroup H_((n,m)).As the isolated subsemigroup T can be expressed as a union of subsets containing idempotents,we can,by analyzing the relationship between the set E(T)of the idempotents of T and the elements in the semigroup H_((n,m)),construct it according to its definition and the closure of the semigroup and calculate the idempotents and the generation elements of the idempotents.It is found that if T contains some idempotents in H_((n,m))(n-2),then it can be concluded that the singular transformation semigroup Sing_((n,m))must be included in T,and that G_((n,m))must be included in T if T contains some elements at the top G_((n,m))of the semigroup H_((n,m)).The structural characteristics of the isolated subsemigroup T are deduced by discussing different cases of E(T),and then the complete classification of the isolated subsemigroup of H_((n,m))is obtained.

关 键 词：全变换半群 幂等元 独立子半群 

分 类 号：O152.7[理学—数学]