半群C_(n,r)(k)的秩和幂等元秩

Rank and idempotent rank of the semigroups C_(n,r)(k)

设[n]={1,,2,…,n},Cn是[n]上的保序且降序变换半群,k∈[n],令Cn(k)={α∈Cn:kα=k},则Cn(k)是Cn的子半群。对任意的1≤r≤n-1,考虑Cn,r(k)={α∈Cn(k):|im(α)|≤r}的秩和幂等元秩,证明了半群Cn,r(k)是由秩为r的幂等元生成的,并得到了Cn,r(k)的秩和幂等元秩均为Cr-2n-2。

Abstract： Let [n]={1,,2,…,n},Cn be the decreasing and order-preserving transformation semigroup on [n],let k∈[n], Cn（k） ={α∈Cn：kα=k} thenCn （k） is a subsemigroup of Cn . For an arbitrary integer r such that 1≤r≤n-1 , the rank and idempotent rank of the semigroup Cn,r （ k ） ={α∈Cn（k）：｜im（α）｜≤r} were studied. We showed that Cn,r （k） is generated by idempotents of rank r. Furthermore, we obtained that the rank and idempotent rank of Cn,r （k） are both equal to Cr-2n-2.