半群PHn的每个星理想的秩和幂等元秩

设自然数n≥3, PHn是自然序集Xn=｛1,2,3,…,n｝上的保降序且保序有限部分奇异变换半群, 对0≤r≤n-1时, 记P(n,r)=｛α∈PHn:|imα|≤r｝ 为半群PHn的双边星理想。通过对其幂等元的分析, 分别刻划了半群P(n,r)的极小幂等生成集, 秩和幂等元秩。进一步证明了当0≤l≤r时, 半群P(n,r)关于它的每个星理想P(n,l)的相关秩。

On the rank and idempotent rank of each star ideal of the semigroup PHn

 Let PHn be the semigroups of order-decreasing and order preserving finite partial singular transformations on the natural order set Xn=｛1,2,3,…,n｝ if n≥3, and let P(n,r)=｛α∈PHn:|imα|≤r｝ be the two-sided star ideal of the semigroup PHn if 0≤r≤n-1. By analyzing the idempotent elements, the minimal idempotent generating set, rank and idempotent rank of the semigroup P(n,r) are characterized, respectively. Furthermore, it is shown that for 0≤l≤r, the relative rank of the semigroup P(n,r) with respect to itself each star ideal P(n,l).