A poset hierarchy

• Autorzy: Mirna Džamonja , Katherine Thompson
• Języki publikacji: 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

EN

Set theory; ordered sets; κ-dense; ηα-orderings; 03E04; 06A05; 06A06

• Wydawca: De Gruyter Open
• Czasopismo: Open Mathematics
• Rocznik: 2006
• Tom: 4
• Numer: 2
• Strony: 225-241

Daty

wydano

2006-06-01

online

2006-06-01

Twórcy

autor

Mirna Džamonja

• University of East Anglia

autor

Katherine Thompson

• University of Vienna

Bibliografia

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• [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.
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Identyfikatory

DOI

10.1007/s11533-006-0001-1