# A poset hierarchy

Mirna Džamonja; Katherine Thompson

Open Mathematics (2006)

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

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topMirna 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

top- [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. Zbl0934.06005
- [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
- [3] R. Bonnet and M. Pouzet: “Linear Extensions of Ordered Sets”, In: Ordered Sets, D. Reidel Publishing Company, 1982, pp. 125–170. Zbl0499.06002
- [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. Zbl0188.04203
- [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. Zbl0713.03013
- [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] 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. Zbl0965.03059
- [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 Zbl0087.27001
- [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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