Authors: Moconja, Slavko
Tanović, Predrag 
Affiliations: Mathematical Institute of the Serbian Academy of Sciences and Arts 
Title: Stationarily ordered types and the number of countable models
Journal: Annals of Pure and Applied Logic
Volume: 171
Issue: 3
Issue Date: 1-Mar-2020
Rank: M21
ISSN: 0168-0072
DOI: 10.1016/j.apal.2019.102765
Abstract: 
We introduce the notions of stationarily ordered types and theories; the latter generalizes weak o-minimality and the former is a relaxed version of weak o-minimality localized at the locus of a single type. We show that forking, as a binary relation on elements realizing stationarily ordered types, is an equivalence relation and that each stationarily ordered type in a model determines some order-type as an invariant of the model. We study weak and forking non-orthogonality of stationarily ordered types, show that they are equivalence relations and prove that invariants of non-orthogonal types are closely related. The techniques developed are applied to prove that in the case of a binary, stationarily ordered theory with fewer than 2ℵ0 countable models, the isomorphism type of a countable model is determined by a certain sequence of invariants of the model. In particular, we confirm Vaught's conjecture for binary, stationarily ordered theories.
Keywords: Coloured order | dp-Minimality | Shuffling relation | Stationarily ordered type | Vaught's conjecture | Weakly quasi-o-minimal theory
Publisher: Elsevier
Project: Representations of logical structures and formal languages and their application in computing 
Algebraic, logical and combinatorial methods with applications in theoretical computer science 
Narodowe Centrum Nauki, Grant no. 2016/22/E/ST1/00450

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