半群${\mathcal{T}}_{{ℱ}_m}\left(n\right)$的性质

Property of Semigroup ${\mathcal{T}}_{{ℱ}_m}\left(n\right)$

摘要: 设${\mathcal{S}}_n$ 和${\mathcal{T}}_n$ 分别是$X_n=\left\{1,2,\cdots ,n\right\}$ 上的对称群和全变换半群。对于$1\le m\le n-1$ ，记$X_m=\left\{1,2,\cdots ,m\right\}$ 且$X_{n-m}=X_n\backslash X_m$ 。令${\mathcal{T}}_{{ℱ}_m}\left(n\right)=\left\{\alpha \in {ℱ}_{\left(n,m\right)}:存在1\le j\le m<i\le n使得i\alpha =j\right\}$ ，则显然${\mathcal{T}}_{F_m}\left(n\right)$ 是${\mathcal{T}}_n$ 的子半群。本文刻画了该半群的格林关系和理想。

Abstract: Let ${\mathcal{S}}_n$ and ${\mathcal{T}}_n$ be the symmetric group and the full transformation semigroup on $X_n=\left\{1,2,\cdots ,n\right\}$ , respectively. For $1\le m\le n-1$ , we denote by $X_m$ the set $\left\{1,2,\cdots ,m\right\}$ and by $X_{n-m}$ the set $X_n\backslash X_m$ . Put ${\mathcal{T}}_{{ℱ}_m}\left(n\right)=\left\{\alpha \in {ℱ}_{\left(n,m\right)}:存在1\le j\le m<i\le n使得i\alpha =j\right\}$ , then ${\mathcal{T}}_{{ℱ}_m}\left(n\right)$ is a subsemigroup of $T_n$ . In this paper, we describe the Green relation and ideals of the semigroup ${\mathcal{T}}_{F_m}\left(n\right)$ .