半群S_n(k)的秩
Ranks of the Semigroup S_n(k)

设Sing_n是[n]上的奇异变换半群.对任意1≤k≤n-1, 研究半群 ${S_n}\left( k \right) = \left\{ {\alpha \in {\text{Sin}}{{\text{g}}_n}:\forall x \in \left[ n \right],x \leqslant k \Rightarrow x\alpha \leqslant k} \right\}$ 证明了S_n(k)是由秩为n-1的幂等元生成的, 并得到半群S_n(k)(k≠2) 的秩和幂等元秩都为$\frac{{n(n - 1)}}{2}$.同时, 得到了半群S_n(2) 的秩和幂等元秩都为$\frac{{n(n - 1)}}{2}$.
Let Sing_n be a singular transformation semigroup on [n]. For an arbitrary integer 1≤k≤n-1, the rank and idempotent rank of the semigroup S_n(k)={α∈Sing_n:？x∈[n], x≤k$ \Rightarrow $xα≤k} are studied. We show that the semigroup S_n(k) is generated by the idempotents of rank n-1, and obtain that the rank and idempotent rank of the semigroup S_n(k)(k≠2) are both equal to $ \frac{n(n-1)}{2} $, and the rank and idempotent rank of the semigroup S_n(2) are both equal to $ \frac{n(n-1)}{2} $