中文摘要：

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.

英文摘要：

Let SPOnbe 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.