[過去ﾛｸﾞ] 純粋・応用数学（含むガロア理論）8 (942ﾚｽ)
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580(1): 現代数学の系譜 雑談 ◆yH25M02vWFhP 2021/05/29(土)08:12 ID:fi/E4J7v(2/25) AA×
>>579

https://en.wikipedia.org/wiki/Well-founded_relation

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580: 現代数学の系譜 雑談 ◆yH25M02vWFhP [] 2021/05/29(土) 08:12:50.16 ID:fi/E4J7v >>579 つづき ＜英語版＞ https://en.wikipedia.org/wiki/Well-founded_relation Well-founded relation (抜粋) In mathematics, a binary relation R is called well-founded (or wellfounded) on a class X if every non-empty subset S ⊆ X has a minimal element with respect to R, that is, an element m not related by sRm (for instance, "s is not smaller than m") for any s ∈ S. Equivalently, assuming the axiom of dependent choice, a relation is well-founded if it contains no countable infinite descending chains: that is, there is no infinite sequence x0, x1, x2, ... of elements of X such that xn+1 R xn for every natural number n.[1][2] References [1] "Infinite Sequence Property of Strictly Well-Founded Relation". ProofWiki. Retrieved 10 May 2021. In order theory, a partial order is called well-founded if the corresponding strict order is a well-founded relation. If the order is a total order then it is called a well-order. In set theory, a set x is called a well-founded set if the set membership relation is well-founded on the transitive closure of x. The axiom of regularity, which is one of the axioms of Zermelo?Fraenkel set theory, asserts that all sets are well-founded. A relation R is converse well-founded, upwards well-founded or Noetherian on X, if the converse relation R?1 is well-founded on X. In this case R is also said to satisfy the ascending chain condition. In the context of rewriting systems, a Noetherian relation is also called terminating. つづく http://rio2016.5ch.net/test/read.cgi/math/1620904362/580
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