Abstract
If ℘ is a poset and every antichain is finite, and if the length of the well-founded poset of antichains is less than ω2 1, then ℘ is the union of countably many chains. We also compute the length of the poset of antichains in the product of two ordinals, αxβ.
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References
P. W. Carruth (1942) Arithmetic of ordinals with application to the theory of ordered Abelian groups, Bull. Amer. Math. Soc. 48, 262–271.
R. P. Dilworth (1950) A decomposition theorem for partially ordered sets, Ann. Math. (2) 51, 161–166.
B. Dushnik and E. W. Miller (1941) Partially ordered sets, Amer. J. Math. 63, 600–610.
G. Hessenberg (1906) Grundbegriffe der Mengenlehre, Abh. der Friesschen Schule. N.S. [1], Heft 4. 220 pp.
A. Levy (1979) Basic Set Theory, Springer, Berlin.
E. C. Milner and K. Prikry (1983) The cofinality of a partially ordered set, Proc. London Math. Soc. (3), 46, 454–470.
M. Perles (1963) A proof of Dilworth's decomposition theorem for partially ordered sets, Israel J. Math. 1, 105–107.
M. Perles (1963) On Dilworth's theorem in the infinite case, Israel J. Math. 1, 108–109.
M. Pouzet, Parties cofinales des ordres partiels ne contenant pas d'antichaînes infinité, J. London Math. Soc. (to appear).
W. Sierpiński (1958) Cardinal and Ordinal Numbers, Państwowe wydawnictwo naukowe, Warsaw.
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Communicated by F. Galvin
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Abraham, U. A note on Dilworth's theorem in the infinite case. Order 4, 107–125 (1987). https://doi.org/10.1007/BF00337691
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DOI: https://doi.org/10.1007/BF00337691