半群CSP_(n,k)的秩和k方幂等元秩

On the Rank and the k-Idempotent Rank of the Semigroup,CSP_(n,k)

设自然数n≥3,P_(n)和S_(n)是有限链X_(n)上的部分变换半群和对称群.对任意的正整数k满足3≤k≤n,令C_(k)=g_(k)是X_(n)上的k-局部循环群且CSP_(n,k)=C_(k)∪(P_(n)\S_(n)),易证CSP_(n,k),是部分变换半群P_(n)的子半群.通过分析半群CSP_(n,k),的格林关系和幂等元,获得了半群CSP_(n,k),的极小生成集和k方幂等元极小生成集,进一步确定了半群CSP_(n,k),的秩和k方幂等元秩.

Let P_(n) and S_(n) be partial transformation semigroup and symmetry group on a finite chain X_(n),respectively,if natural number n≥3.Let C_(k)=(g_(k))be a k-locally cyclic group on X_(n),and letCSP_(n,k)=C_(k)∪(P_(n)\S_(n)),if for any positive integer k such that 3≤k≤n.It is easy to prove that,CSP_(n,k) is a subsemigroup of the partial transformation semigroup P_(n).Through an analysis of the Green’s relation and the idempotent of the semigroup,CSP_(n,k),the minimal generating set and the minimal generating set of k-idempotent are obtained.Further,the rank and the k-idempotent rank of the semigroup,CSP_(n,k) is further confirmed.