半群D_(k)T_(n)的秩和平方幂等元秩  

On the Rank and the Quasi-Idempotent Rank of the Semigroup D_(k)T_(n)

作　　者：张梁松 杨平平 罗永贵 ZHANG Liang-song;YANG Ping-ping;LUO Yong-gui(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,贵州贵阳550025

出　　处：《数学的实践与认识》2023年第7期220-227,共8页Mathematics in Practice and Theory

基　　金：贵州师范大学学术基金项目(黔师新苗[2021]B08号)。

摘　　要：设自然数n≥3,T_(n)和Sn分别是有限集Xn={1,2,…,n}上的全变换半群和置换群.对任意正整数k满足1≤k≤n,记D_(k)=,其中对任意的x∈{1,2,…,k-1}有xgk=x+1,kgf=1且对任意的x∈{k+1,…,n}有xgk=x;对任意的x∈{1,2,…,k}有xδk=k+1-x且对任意的x∈{k+1,…,n}有xδk=x.易见D_(k)是Sn的子群,称D_(k)是Xn上的k-局部二面体群,再记D_(k)T_(n)=D_(k)∪(T_(n)Sn).易证D_(k)T_(n)是全变换半群T_(n)的子半群.通过分析半群D_(k)T_(n)的格林关系和平方幂等元,获得了半群D_(k)T_(n)的极小生成集和平方幂等元极小生成集,进一步确定了半群D_(k)T_(n)的秩和平方幂等元秩.Let T_(n)and Sn are the full transformation semigroup and permutation group on the finite set Xn=[1,2,……-,n),respectively,if nature number n≥3.Denote D_(k)=(gk,o),where there is agk=+1,kgk=1 for any a E(1,2,……,k-1)and gk=a for any a E(k+1,……-,n);ro=k+1-for any E(1,2,……,k)and cok=for anyαE[k+1,……,n),if for any positive integer k such that 1≤k≤n.It is easy to see that D_(k)is a subgroup of Sn,say that D is a k-local dihedral group on Xn,and then write D_(k)T_(n)=D,U(T_(n)Sn).It is easy to prove that D_(k)T_(n)is a subsemigroup of full transformation semigroup T_(n).By analyzing the Green's relation and the quasi-idempotent of the semigroup D_(k)T_(n),the minimal generating set and the minimal generating set of quasi-idempotent be obtained,respectively.Further,the rank and the quasi-idempotent rank of the semigroup D_(k)T_(n)be determined,respectively.

关 键 词：全变换半群 k-局部二面体群 (平方幂等元)极小生成集 (平方幂等元)秩 

分 类 号：O152.7[理学—数学]