设 X 为一个集合,TX 为 X 上的全变换半群.设 E 是 X 上的一个等价关系,定义
TE(X) = {f ∈TX: ∀(x,y)∈ E,(f(x),f(y))∈ E},
则 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) ={f ∈TE(X):∀x,y ∈ X,x ≤ y 蕴涵 f(x) ≤ f(y)}
是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) = {f ∈TX: ∀(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) ={f ∈TE(X):∀x,y ∈ X,x ≤ y 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
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Key words
natural order /
compatibility /
the maximal (minimal) elements /
the covering elements
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参考文献
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脚注
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