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AROUND RUBIN’S “THEORIES OF LINEAR ORDER”

Part of: Ordered sets Model theory

Published online by Cambridge University Press:  27 October 2020

PREDRAG TANOVIĆ ,
SLAVKO MOCONJA  and
DEJAN ILIĆ
PREDRAG TANOVIĆ
Affiliation:
MATHEMATICAL INSTITUTE SANU KNEZ MIHAILOVA 36, BELGRADE11000, SERBIAE-mail: tane@mi.sanu.ac.rs
SLAVKO MOCONJA
Affiliation:
UNIVERSITY OF BELGRADE, FACULTY OF MATHEMATICS STUDENTSKI TRG 16, BELGRADE11000, SERBIA INSTYTUT MATEMATYCZNY, UNIWERSYTET WROCłAWSKI PL. GRUNWALDZKI 2/4, WROCŁAW 50-384, POLANDE-mail: slavko@matf.bg.ac.rs
DEJAN ILIĆ
Affiliation:
UNIVERSITY OF BELGRADE, FACULTY OF TRANSPORT AND TRAFFIC ENGINEERING VOJVODE STEPE 305, BELGRADE11000, SERBIAE-mail: d.ilic@sf.bg.ac.rs
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Abstract

Let $\mathcal M=(M,<,\ldots)$ be a linearly ordered first-order structure and T its complete theory. We investigate conditions for T that could guarantee that $\mathcal M$ is not much more complex than some colored orders (linear orders with added unary predicates). Motivated by Rubin’s work [5], we label three conditions expressing properties of types of T and/or automorphisms of models of T. We prove several results which indicate the “geometric” simplicity of definable sets in models of theories satisfying these conditions. For example, we prove that the strongest condition characterizes, up to definitional equivalence (inter-definability), theories of colored orders expanded by equivalence relations with convex classes.

Keywords

linearly ordered structuresbinary theorieslinear binaritylinear finitenesscolored ordersconvex equivalence relations

MSC classification

Primary: 03C64: Model theory of ordered structures; o-minimality
Secondary: 06A05: Total order
Type
Articles
Information
The Journal of Symbolic Logic , Volume 85 , Issue 4 , December 2020 , pp. 1403 - 1426
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Copyright
© The Association for Symbolic Logic 2020

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References

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