Abstract
We study the possible order types of chains of ideals in an ordered set. Our main result is this. Given an indecomposable countable order type α, there is a finite listA α1 , ...,A α n of ordered sets such that for every ordered setP the setJ(P) of ideals ofP, ordered by inclusion, contains a chain of type α if and only ifP contains a subset isomorphic to one of theA #x03B1;1 , ...,A α n . The finiteness of the list relies on the notion of better quasi-ordering introduced by Nash-Williams and the properties of scattered chains obtained by Laver.
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References
G. Birkhoff (1967)Lattice Theory, 3rd edn., American Mathematics Society, Providence, RI.
R. Bonnet (1975) On the cardinality of the set of initial intervals (Coll. Math. Soc. J. Boyali) inInfinite and Finite Sets (ed. Kesztheley), American Maths Society Pub., pp. 189–198.
R. Bonnet and M. Pouzet (1982) Linear extensions of ordered sets, inOrdered Sets (ed. I. Rival) D. Reidel, Dordrecht, pp. 125–170.
D. Duffus, M. Pouzet and I. Rival (1981) Complete ordered sets with no infinite antichains,Discrete Math. 35, 39–52.
P. Erdös and A. Hajnal (1962) On a classification of denumerable order types and an application to the partition calculus,Fund. Math. 51, 117–129.
R. Fraisse (1984)The Theory of Relations, North-Holland, Amsterdam (to appear).
A. Gleyzal (1940) Order types and structure of orders,Trans. Amer. Math. Soc. 48, 451–466.
G. Hausdorff (1908) Grundzüge einer theorie der Geordnete Mengen,Math. Ann. 65, 435–505.
J. B. Kruskal (1972) The theory of well-quasi-ordering, a frequently discovered concept,J. Comb. Th. (A) 13, 297–305.
R. Laver (1971) On Fraisse's order type conjecture,Ann. Math. 93, 89–111.
E. C. Milner and M. Pouzet: The Erdös-Dushnik-Miller theorem for topological graphs and orders,Order (to appear).
E. C. Milner and K. Prikry (1983) The cofinality of a partially ordered set,Proc. London Math. Soc. 3, 454–470.
C. St. J. A. Nash-Williams (1965) On well-quasi-ordering infinite trees,Proc. Camb. Philos. Soc. 60, 697–720.
M. Pouzet (1972) Sur les premeilleurordres,Ann. Inst. Fourier 22 (2) 1–20.
M. Pouzet (1978) Condition de chaîne en théorie des relations,Israel J. Math. 30, 65–84.
J. G. Rosenstein (1982)Linear Ordering, Academic Press, New York.
N. Zaguia (1983) Chaines d'ideaux et de sections initiales d'un ensemble ordonné, These de 3e Cycle; N. 1286, Universite Claude Bernard, Lyon.
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Communicated by I. Rival
The results presented. here constitute the second chapter of the third cycle thesis presented by the second author before the Claude Bernard University, Lyon (July 1983).
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Pouzet, M., Zaguia, N. Ordered sets with no chains of ideals of a given type. Order 1, 159–172 (1984). https://doi.org/10.1007/BF00565651
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DOI: https://doi.org/10.1007/BF00565651