保持序和等价关系的自然偏序变换半群

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.

Naturally Ordered Transformation Semigroups Preserving Order and an Equivalence Relation

Let X be a set and■_X the full transformation semigroup on X.Let E be an equivalence on X and define T_E(X) = {f∈■_X:■(x,y)∈E,(f(x),f(y))∈E}. Then T_E(X) is a subsemigroup of■_X 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 ■_E(X)={f∈T_E(X):■x,y∈X,x≤y implies f(x)≤f(y)} is a subsemigroup of T_E(X).We endow■_E(X) with the so-called natural order≤and discuss when two elements in■_E(X) are related under this order,then determine those elements of■_E(X) which are compatible with≤.Also,the maximal(minimal) elements and the covering elements are described.