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We say that C is a chain if for every c, d ∈ C, we have c ≤ d or d ≤ c. The chain C is an ascending chain if the elements of C are indexed by N such that ci ≤ ci+1 and there does not exist d ∈ C such that ci ≨ d ≨ ci+1, for all i ∈ N. (“Ascending chain ” does not seem to be formally defined in the texts I’ve read. However, this definition makes precise the often made statement, “there are no infinite ascending chains ” when ACC holds.) Descending chains are defined similarly. Theorem 2. Let Σ ̸ = ∅ be a partially ordered set, ordered by ≤. Then the following are equivalent (1) For every ∅ ̸ = S ⊂ Σ, there exists a maximal M ∈ S. (2) For every chain C ⊂ Σ, C ̸ = ∅, there exists an upper bound M ∈ C. That is, there is M ∈ C such that for all c ∈ C, we have c ≤ M. (Note that [R, p. 25] defines an upper bound to have c < M for all c ∈ C, but it is incorrect not to allow M ∈ C, and in fact we always have M ∈ C when ACC holds.) (2’) If ∅ ̸ = C ⊂ Σ is an ascending chain with elements ci for all i ∈ N, the

Year: 2011

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