In mathematics, a pantachy or pantachie (from the Greek word πανταχη meaning everywhere) is a maximal totally ordered subset of a partially ordered set, especially a set of equivalence classes of sequences of real numbers. The term was introduced by du Bois-Reymond (1879, 1882) to mean a dense subset of an ordered set, and he also introduced "infinitary pantachies" to mean the ordered set of equivalence classes of real functions ordered by domination, but as Felix Hausdorff pointed out this is not a totally ordered set.[1] Hausdorff (1907) redefined a pantachy to be a maximal totally ordered subset of this set.
Notes
- ^ Hausdorff declared that "the infinitary pantachie in the sense of Du Bois-Reymond does not exist" (1907), p. 107.
References
- du Bois-Reymond, Paul (1879), "Erläuterungen zu den Anfangsgründen der Variationsrechnung", Mathematische Annalen, 15 (2): 283–314, doi:10.1007/BF01444144, S2CID 124616845
- du Bois-Reymond, P. (1882), Die allgemeine Funktionentheorie, Tübingen
- Hausdorff (1907), "Untersuchungen über Ordnungstypen IV, V", Ber. über die Verhandlungen der Königl. Sächs. Ges. Der Wiss. Zu Leipzig. Math.-phys. Klasse, 59: 84–159 English translation in Hausdorff (2005)
- Hausdorff, F. (1914), Grundzüge der Mengenlehre, Leipzig: Veit & Co
- Hausdorff, Felix (2005), Plotkin, J. M. (ed.), Hausdorff on ordered sets, History of Mathematics, vol. 25, Providence, RI: American Mathematical Society, ISBN 0-8218-3788-5, MR 2187098
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