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2006 | 4 | 2 | 225-241
Tytuł artykułu

A poset hierarchy

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EN
Abstrakty
EN
This article extends a paper of Abraham and Bonnet which generalised the famous Hausdorff characterisation of the class of scattered linear orders. They gave an inductively defined hierarchy that characterised the class of scattered posets which do not have infinite incomparability antichains (i.e. have the FAC). We define a larger inductive hierarchy κℌ* which characterises the closure of the class of all κ-well-founded linear orders under inversions, lexicographic sums and FAC weakenings. This includes a broader class of “scattered” posets that we call κ-scattered. These posets cannot embed any order such that for every two subsets of size < κ, one being strictly less than the other, there is an element in between. If a linear order has this property and has size κ it is unique and called ℚ(κ). Partial orders such that for every a < b the set {x: a < x < b} has size ≥ κ are called weakly κ-dense, and posets that do not have a weakly κ-dense subset are called strongly κ-scattered. We prove that κℌ* includes all strongly κ-scattered FAC posets and is included in the class of all FAC κ-scattered posets. For κ = ℵ0 the notions of scattered and strongly scattered coincide and our hierarchy is exactly aug(ℌ) from the Abraham-Bonnet theorem.
Słowa kluczowe
Wydawca
Czasopismo
Rocznik
Tom
4
Numer
2
Strony
225-241
Opis fizyczny
Daty
wydano
2006-06-01
online
2006-06-01
Twórcy
Bibliografia
  • [1] U. Abraham and R. Bonnet: “Hausdorff’s Theorem for Posets That Satisfy the Finite Antichain Property”, Fundamenta Mathematica, Vol. 159(1), (1999), pp. 51–69.
  • [2] F. Hausdorff: “Grundzüge einer Theorie der geordneten Mengenlehre” (in German), Mathematische Annalen, Vol. 65, (1908), pp. 435–505. http://dx.doi.org/10.1007/BF01451165
  • [3] R. Bonnet and M. Pouzet: “Linear Extensions of Ordered Sets”, In: Ordered Sets, D. Reidel Publishing Company, 1982, pp. 125–170.
  • [4] R. Bonnet and M. Pouzet: “Extension et stratification d’ensembles dispersés” (in French), C.R.A.S., Paris, Série A, Vol. 168, (1969), pp. 1512–1515.
  • [5] S. Shelah: Nonstructure Theory, to appear.
  • [6] S. Shelah: Classification Theory, Revised ed., Studies in Logic and Foundations of Mathematics, Vol. 92, North-Holland, 1990.
  • [7] G. Asser, J. Flachsmeyer and W. Rinow: Theory of Sets and Topology; In honour of Felix Hausdorff, Deutscher Verlag der Wissenschaften, 1972.
  • [8] H.J. Kiesler and C.C. Chang: Model Theory, 3rd ed., Studies in Logic and Foundations of Mathematics, Vol. 73, Elsevier Science B.V., 1990.
  • [9] R. Fraïssé: Theory of Relations, Revised ed., Studies in Logic and Foundations of Mathematics, Vol. 145, Elsevier Science, B.V., 2000.
  • [10] J. Rosenstein: Linear Orderings, Pure and Applied Mathematics, Academic Press, 1982.
  • [11] E. Mendelson: “On a class of universal ordered sets”, Proc. Amer. Math. Soc., Vol. 9, (1958), pp. 712–713. http://dx.doi.org/10.2307/2033073
  • [12] M. Kojman and S. Shelah: “Non-existence of Universal Orders in Many Cardinals”, J. Symbolic Logic, Vol. 57(3), (1992), pp. 875–891.
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.doi-10_1007_s11533-006-0001-1
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