半群SPOn中理想的非群元秩和相关秩

设自然数n≥3,SPOn是有限链[n]上的严格部分保序奇异变换半群.对任意的r(0≤r≤n-2),记N(n,r)={α∈SPOn:|Imα|≤r}为半群SPOn的双边理想.通过对其非群元和格林关系的分析,分别获得了半群N(n,r)的极小非群元生成集、非群元秩和非幂等元秩.进一步确定了当0≤l≤r时,半群N(n,r)关于其理想N(n,l)的相关秩.

Non-group rank and relative rank of each ideal of the semigroup

Let SPOn be the semigroup of all strictly partial order-preserving singular transformations on a finite-chain [n] if n≥3,and let N(n,r)={α∈SPOn:|Imα|≤r} be the two-sided ideal of the semigroup SPOn if for an arbitrary integer r such that 0≤r≤n-2.By analyzing the non-group elements and Green’s relations,the minimal non-group elements generating set,non-group rank and non-idempotent rank of the semigroup N(n,r) are obtained,respectively.Furthermore,the relative rank of the semigroup N(n,r) with respect to itself each ideal N(n,l) is determined if 0≤l≤r.