一类变换半群的正则元和Green关系

设T_X为X上的全变换半群,E为X上的等价关系,令T_E(X)={f∈T_X:■(x,y)∈E,(f(x),f(y))∈E},则T_E(X)是T_X的子半群,如果X是一个全序集,E是X上的一个凸等价关系,设OP_E(X)为T_E(X)中所有保向映射作成的半群。对于有限全序集X上一类特殊的凸等价关系E,本文刻画了半群OP_E(X)的正则元的特征,并且描述了这个半群上的Green关系。

Regularity and Green's Relations on a Special Transformation Semigroup

Let ${\cal T}_{X}$ be the full transformation semigroup on a set $X$, and $E$ an equivalence on $X$. Let $T_{E}(X)=\{f\in{\cal T}_{X}:\,\forall \, (x,y)\in E,(f(x),f(y))\in E\}.$ Then $T_{E}(X)$ forms a subsemigroup of ${\cal T}_{X}$. If $X$ is a totally ordered set and $E$ is a convex equivalence on $X$, then let $OP_{E}(X)$ be a semigroup consisting of all the orientation-preserving maps in $T_{E}(X)$. In this paper, for the special convex equivalence $E$ on a finite totally ordered set $X$, we describe the regular elements and characterize Green's relations on the semigroup $OP_E(X)$.