On linearly ordered structures of finite rank

Journal of Mathematical Logic 9 (2):201-239 (2009)
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Abstract

O-minimal structures have long been thought to occupy the base of a hierarchy of ordered structures, in analogy with the role that strongly minimal structures play with respect to stable theories. This is the first in an anticipated series of papers whose aim is the development of model theory for ordered structures of rank greater than one. A class of ordered structures to which a notion of finite rank can be assigned, the decomposable structures, is introduced here. These include all ordered structures definable in o-minimal structures. The principal result in this paper, Theorem 5.1, asserts roughly that a decomposable structure [Formula: see text] can be partitioned into finitely many definable subsets such that on each set the restriction of < is a "twisted lexicographic" order. As a consequence, for all n and linear orders ≺ definable on a subset X ⊆ Mn in an o-minimal structure [Formula: see text], there is a definable partition of X such that the restriction of ≺ to each set in the partition is "lexicographic".

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Citations of this work

The Marker–Steinhorn Theorem via Definable Linear Orders.Erik Walsberg - 2019 - Notre Dame Journal of Formal Logic 60 (4):701-706.
Definable one-dimensional topologies in O-minimal structures.Ya’Acov Peterzil & Ayala Rosel - 2020 - Archive for Mathematical Logic 59 (1-2):103-125.

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Characterizing Rosy Theories.Clifton Ealy & Alf Onshuus - 2007 - Journal of Symbolic Logic 72 (3):919 - 940.
Properties and Consequences of Thorn-Independence.Alf Onshuus - 2006 - Journal of Symbolic Logic 71 (1):1 - 21.
Definable types in o-minimal theories.David Marker & Charles I. Steinhorn - 1994 - Journal of Symbolic Logic 59 (1):185-198.
Definable Types in $mathscr{O}$-Minimal Theories.David Marker & Charles I. Steinhorn - 1994 - Journal of Symbolic Logic 59 (1):185-198.
Forking and independence in o-minimal theories.Alfred Dolich - 2004 - Journal of Symbolic Logic 69 (1):215-240.

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