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Well-order

在阅读维基百科Recursive definitionForm of recursive definitions章节时,发现了这个概念,遂对它进行了整理。

维基百科Well-order

笔记

Well-order VS total order

原文对Well-order 的定义如下:

In mathematics, a well-order (or well-ordering or well-order relation) on a set S is a total order on S with the property that every non-empty subset of S has a least element in this ordering.

显然,它是在 total order 的基础上添加了一个限制,那这限制有何意义呢?是保证S的每个子集依然满足 total order

其实理解well order的一个很好的方法是结合实例来进行理解,set theory所描述的tree就是一个典型的well order,在维基百科Tree (set theory)

In set theory, a tree is a partially ordered set (T, <) such that for each tT, the set {sT : s < t} is well-ordered by the relation <. Frequently trees are assumed to have only one root (i.e. minimal element), as the typical questions investigated in this field are easily reduced to questions about single-rooted trees.

维基百科Well-founded relation

并没有理解维基百科Well-founded relation