純粋・応用数学（含むガロア理論）8

純粋・応用数学（含むガロア理論）8at MATH

純粋・応用数学（含むガロア理論）8 - 暇つぶし2ch642:tructure. Then R is a strictly well-founded relation if and only if there is no infinite sequence ?an? of elements of S such that: ∀n∈N:an+1 R an Proof Reverse Implication Suppose R is not a strictly well-founded relation. So by definition there exists a non-empty subset T of S which has no strictly minimal element. Let a∈T. Since a is not strictly minimal in T, we can find b∈T:bRa. This holds for all a∈T. Hence the restriction R↑T×T of R to T×T is a right-total endorelation on T. So, by the Axiom of Dependent Choice, it follows that there is an infinite sequence ?an? in T such that: ∀n∈N:an+1 R an It follows by the Rule of Transposition that if there is no infinite sequence ?an? of elements of S such that: ∀n∈N:an+1 R an then R is a strictly well-founded relation. □ つづく

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