保持序和等价关系的自然偏序变换半群

Naturally Ordered Transformation Semigroups Preserving Order and an Equivalence Relation

设X为一个集合,■_X为X上的全变换半群.设E是X上的一个等价关系,定义T_E(X)={f∈■_X:■(x,y)∈E,(f(x),f(y))∈E},则T_E(X)是由等价关系E所确定的■_X的子半群.本文中,所考虑的集合X是一个有限全序集,同时E是非平凡的且所有的E-类都是凸集.显然■_E(X)={f∈T_E(X):■_x,y∈X,x≤y蕴涵f(x)≤f(y)}是T_E(X)的一个子半群.我们赋予■_E(X)自然偏序并讨论何时■_E(X)中的两个元素是关于这个偏序是相关的,然后确定■_E(X)中那些关于≤是相容的元素.此外,还描述了极大(极小)元和覆盖元.

Let X be a set and Jx the full transformation semigroup on X. Let E be an equivalence on X and define TE（x）={f∈Jx：A（x,y）∈E,f（x）,f（y）∈E} Then TE（X） is a subsemigroup of JX determined by the equivalence E. In this paper, the set X under consideration is a totally ordered finite set, while the equivalence E is non-trivial and all E-classes are convex. It is clear that OE（X）={f∈TE（x）：Ax,y∈X,x≤y implics f（x）≤f（y）） is a subsemigroup of TE（X）. We endow OE（X） with the so-called natural order ≤ and discuss when two elements in rYE（X） are related under this order, then determine those elements of OE（X） which are compatible with ≤. Also, the maximal （minimal） elements and the covering elements are described.