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保持序和等价关系的自然偏序变换半群 被引量:9

Naturally Ordered Transformation Semigroups Preserving Order and an Equivalence Relation
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摘要 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.
出处 《数学学报(中文版)》 SCIE CSCD 北大核心 2012年第2期235-250,共16页 Acta Mathematica Sinica:Chinese Series
基金 国家自然科学基金资助项目(10971086)
关键词 自然偏序 相容性 极大(小)元 覆盖元 natural order compatibility the maximal (minimal) elements the covering elements
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