半群P_(n)^(k)的秩和平方幂等元秩

On the rank and the quasi-idempotent rank of the semigroup P_(n)^(k)

设自然数n≥3,P_(n)和S_(n)分别是有限集X_(n)={1,2,…,n}上的部分变换半群和置换群.对任意的正整数k满足1≤k≤n,令S_(k)={α∈S_(n):x∈{k+1,…,n},xα=x}.易见S_(k)是S_(n)的子群,称S_(k)是X_(n)上的k-局部置换群.再令P_(n)^(k)=S_(k)∪(P_(n)\S_(n)),易证P_(n)^(k)是部分变换半群P_(n)的子半群,通过分析半群P_(n)^(k)的格林关系和平方幂等元,获得了半群P_(n)^(k)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P_(n)^(k)的秩和平方幂等元秩.

Let Pn and Sn be partial transformation semigroup and permutation group,respectively,on finite set X_(n)={1,2,…,n},if nature number n≥3.Let S_(k)={α∈Sn:x∈{k+1,…,n},xα=x},it’s easy to see that S_(k)is the subgroup of S_(n)and is called the k-local permutation group on X_(n).Let P_(n)^(k)=S_(k)∪(P_(n)\S_(n)),if for arbitrary integer k such that 1≤k≤n,it’s easy to prove that P_(n)^(k) is the sub semigroup of partial transformation Semigroup P_(n).By analyzing the Green’s relations and the quasi-idempotent of the Semigroup P_(n)^(k),the minimal generating set and the minimal generating set of Quasi-idempotent is obtainer,respectively.Furtherly,the rank and the quasi-idempotent rank of the semigroup P_(n)^(k) is confirmed,respectively.