关于一类纯正半群的秩

On the Rank of the Certain Orthodox Semigroups

设Xn={1,2,…,n}(n≥4)是一个自然序集,Wn是Xn上的保序压缩奇异变换半群,RWn是Wn的所有正则元构成的正则子半群.利用Green等价关系和蛋合图,证明了RWn的理想Ir={αRWn:imα≤r}(1≤r≤n-1)的秩为n-r+1.

Let Xn={1,2,…,n}（n≥4） be natural order set, Wn be semigroup of the order-preserving and compressing singular transformations on Xn ,RWn be regular subsemigroup consisting of all regular elements in Wn. By Green equivalence and eggbox graph, it is shown that the rank of ideals Ir={α∈RWn：｜imα｜≤r}（1≤r≤n-1） be n-r＋1.