A poset hierarchy

Mirna Džamonja; Katherine Thompson

Open Mathematics (2006)

  • Volume: 4, Issue: 2, page 225-241
  • ISSN: 2391-5455

Abstract

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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.

How to cite

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Mirna Džamonja, and Katherine Thompson. "A poset hierarchy." Open Mathematics 4.2 (2006): 225-241. <http://eudml.org/doc/268922>.

@article{MirnaDžamonja2006,
abstract = {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.},
author = {Mirna Džamonja, Katherine Thompson},
journal = {Open Mathematics},
keywords = {03E04; 06A05; 06A06},
language = {eng},
number = {2},
pages = {225-241},
title = {A poset hierarchy},
url = {http://eudml.org/doc/268922},
volume = {4},
year = {2006},
}

TY - JOUR
AU - Mirna Džamonja
AU - Katherine Thompson
TI - A poset hierarchy
JO - Open Mathematics
PY - 2006
VL - 4
IS - 2
SP - 225
EP - 241
AB - 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.
LA - eng
KW - 03E04; 06A05; 06A06
UR - http://eudml.org/doc/268922
ER -

References

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  2. [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 Zbl39.0099.01
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  7. [7] G. Asser, J. Flachsmeyer and W. Rinow: Theory of Sets and Topology; In honour of Felix Hausdorff, Deutscher Verlag der Wissenschaften, 1972. Zbl0256.00006
  8. [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. [9] R. Fraïssé: Theory of Relations, Revised ed., Studies in Logic and Foundations of Mathematics, Vol. 145, Elsevier Science, B.V., 2000. Zbl0965.03059
  10. [10] J. Rosenstein: Linear Orderings, Pure and Applied Mathematics, Academic Press, 1982. 
  11. [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 Zbl0087.27001
  12. [12] M. Kojman and S. Shelah: “Non-existence of Universal Orders in Many Cardinals”, J. Symbolic Logic, Vol. 57(3), (1992), pp. 875–891. Zbl0790.03036

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