Abstract
We pursue the fine analysis of the quasi-orderings ≼∃ ∀ and ≼∀ ∃ on the power set of a quasi-ordering (Q,≼). We set X≼∃ ∀ Y if every x∈X is majorized in ≼ by some y∈Y, and X≼∀ ∃ Y if every y∈Y is minorized in ≼ by some x∈X. We show that both these quasi-orderings are α-wqo if and only if the original quasi-ordering is (α ⋅ ω)-wqo. For ≼∀ ∃ this holds also restricted to finite subsets, thus providing an example of a finitary operation on quasi-orderings which does not preserve wqo but preserves bqo.
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Marcone, A. Fine Analysis of the Quasi-Orderings on the Power Set. Order 18, 339–347 (2001). https://doi.org/10.1023/A:1013952225669
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DOI: https://doi.org/10.1023/A:1013952225669