变换半群POE_n的理想POE(n,r)的秩与幂等元秩

The Rank and Idempotent Rank of Ideal POE(n,r)of Transformation Semigroup POEn

设POEn是有限链Xn上的升序且保序部分变换半群.对于任意的3≤r≤n-1,本文证明了变换半群POEn的理想POE(n,r)={α∈POE_n:|Im(α)|≤r}是由秩为r的幂等元生成的,其秩和幂等元秩均为n∑nk=(nk)(k-1r-1).

Let POEnbe the semigroup of all extensive and order-preserving partial transformation on a finite chain Xn.In this paper,we show that the ideal POE（n,r）={α∈POEn：｜Im（α）｜≤r}（3≤r≤n-1）of transformation semigroup POEnis generated by idempotents of rank r and we prove that its rank and idempotent rank are both equal to n∑nk=（nk）（k-1r-1）.