半群OIk（n,r）的秩和相关秩

On Rank and Relative Rank of the Semigroup OI ■

设自然数n≥3,OI^kn是有限链[n]上的双边k型-保序严格部分一一变换半群.对任意的1≤k≤n-1,0≤r≤n-1,记OI^k(n,r)={α∈OI^kn:|im(α)|≤r}为半群OI^kn的双边理想.通过对秩为r的元素和格林关系的分析,分别获得了半群OI^k(n,r)的极小生成集和秩.进一步确定了当0≤l≤r时,半群OI^k(n,r)关于其理想OI^k(n,l)的相关秩.

Let OI^ k n be the semigroup of all bilateral k type-order-preserving one-to-one partial transformations on a finite-chain [n]( n>3), and let OI^ k (n, r)={α∈ OI^ k n:| im (α)|≤r} be the two-sided ideal of the semigroup OI^ k (n, r) for an arbitrary integer k,r such that 1≤k≤n-1 , 0≤r≤n-1 . By analyzing the elements of rank r and Green s relations, the minimal generating sets and the rank of the semigroup OI^ k (n, r) are obtained, respectively. Furthermore, the relative rank of the semigroup OI^ k (n, r) with respect to itself each ideal OI^ k (n, l) is determined for 0≤l≤r .