设$\mathscr{T}_{n}$是$X_{n}=\{1, 2, \cdots, n\}$上的全变换半群, 对任意$\leqslant r \leqslant n$, 令$\mathscr{F}_{(n, r)}=\left\{\alpha \in \mathscr{T}_{n}: i \alpha=i, \quad \forall i \in\{1, 2, \cdots, r\}\right\}, $则$\mathscr{F}_{(n, r)}$是$\mathscr{T}_{n}$的子半群。本文将研究半群$\mathscr{F}_{(n, r)}$的核$\mathscr{C} \mathscr{F}_{(n, r)}=\left\langle E\left(\mathscr{F}_{(n, r)}\right)\right\rangle$, 其中$E\left(\mathscr{F}_{(n, r)}\right)=\left\{\alpha \in \mathscr{F}_{(n, r)}: \alpha^{2}=\alpha\right\}$, 通过对$\mathscr{F}_{(n, r)}$幂等元的分析, 得到半群$\mathscr{C} \mathscr{F}_{(n, r)}$的秩和幂等元秩都为$\frac{(n-r)(n-r-1)}{2}+r(n-r)+1$。
