保持序和等价关系的自然偏序变换半群

裴惠生, 邓伟娜

数学学报 ›› 2012 ›› Issue (2) : 235-250.

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数学学报 ›› 2012 ›› Issue (2) : 235-250. DOI: 10.12386/A2012sxxb0024
论文

保持序和等价关系的自然偏序变换半群

    裴惠生1, 邓伟娜2
作者信息 +

Naturally Ordered Transformation Semigroups Preserving Order and an Equivalence Relation

    Hui Sheng PEI1, Wei Na DENG2
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文章历史 +

摘要

X 为一个集合,TXX 上的全变换半群.设 E 是 X 上的一个等价关系,定义
TE(X) = {fTX: ∀(x,y)∈ E,(f(x),f(y))∈ E},
TE(X) 是由等价关系 E 所确定的 TX 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
OE(X) ={fTE(X):∀x,yX,xy 蕴涵 f(x) ≤ f(y)}
TE(X) 的一个子半群.我们赋予 OE(X) 自然偏序并讨论何时 OE(X) 中的两个元素是关于这个偏序是相关的,然后确定 OE(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.

Abstract

Let X be a set and TX the full transformation semigroup on X. Let E be an equivalence on X and define
TE(X) = {fTX: ∀(x,y)∈ E,(f(x),f(y))∈ E}. Then TE(X) is a subsemigroup of TX 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) ={fTE(X):∀x,yX,xy implies 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 OE(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.

关键词

自然偏序 / 相容性 / 极大(小)元 / 覆盖元

Key words

natural order / compatibility / the maximal (minimal) elements / the covering elements

引用本文

导出引用
裴惠生, 邓伟娜. 保持序和等价关系的自然偏序变换半群. 数学学报, 2012(2): 235-250 https://doi.org/10.12386/A2012sxxb0024
Hui Sheng PEI, Wei Na DENG. Naturally Ordered Transformation Semigroups Preserving Order and an Equivalence Relation. Acta Mathematica Sinica, Chinese Series, 2012(2): 235-250 https://doi.org/10.12386/A2012sxxb0024

参考文献

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基金

国家自然科学基金资助项目(10971086)

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