半群I_n^k的秩和平方幂等元秩

On the Rank and Quasi-Idempotent Rank of Semigroup I_n^k

摘要: 设自然数n≥3，I_n和S_n是有限集X_n={1,2,…,n}上的对称逆半群和置换群。对任意的正整数k满足1≤k≤n，令S_k={α∈S_n:∀x∈{k+1,…,n},xα=x}。易见，S_k是S_n的子群，则称S_k是X_n上的k-局部置换群，再令I_n^k=S_k∪(I_n\S_n)。易证，I_n^k是对称逆半群I_n的子半群。通过分析半群 的格林关系和平方幂等元，获得了半群 的极小生成集和平方幂等元极小生成集。进一步，确定了半群I_n^k的秩和平方幂等元秩。

Abstract: Let I_n and S_n be symmetric inverse semigroup and permutation group on the finite set X_n={1,2,…,n} if nature number n≥3, respectively. For any positive integer k that satisfies 1≤k≤n, let S_k={α∈S_n:∀x∈{k+1,…,n},xα=x}. It is easy to prove that S_k is a subgroup of S_n, then S_k is called the k-local permutation group on X_n, and then let I_n^k=S_k∪(I_n\S_n), it is easy to prove that is a subsemigroup of symmetric inverse semigroup I_n^k. By analyzing the Green’s relations and the quasi idempotent of the semigroup I_n^k, the minimal generating set and the minimal generating set of quasi idempotent be obtained, respectively. Further, the rank and quasi idempotent rank are definite, respectively.