Stationarily ordered types and the number of countable models

Abstract

We introduce notions of stationarily ordered types and theories; the latter generalizes weak o-minimality and the first 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 developed techniques are applied to prove that in the case of a binary, stationarily ordered theory with fewer than $2^{{\aleph }_0}$ 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.

arXiv commentRevised version accepted for publication in Annals of Pure and Applied Logic

References in corpus (3)

• Witnessing dp-rank
• Around Rubin's "Theories of linear order"
• Asymmetric regular types

Cited by in corpus (4)

• Around Rubin's "Theories of linear order"
• Vaught's conjecture for theories of discretely ordered structures
• Does weak quasi-o-minimality behave better than weak o-minimality?
• Weakly o-minimal types