Metric Linear Orders and O-Minimality
Aaron Anderson, Diego Bejarano
TL;DR
The work provides a unified, model-theoretic treatment of metric linear and cyclic orders in continuous logic, introducing complete theories ${\textsf{MLO}}$, ${\textsf{MDLO}}$, ${\textsf{UDLO}}$, and their ultrametric companions ${\textsf{DU}}$, ${\textsf{ULO}}$, ${\textsf{UDCO}}$, enabling quantifier elimination and approximate categoricity results. It develops a robust framework of regulated functions to capture o-minimality in metric contexts, proving definable sets and predicates in dimension one are governed by strong/weak regulation and establishing distality for weakly o-minimal metric theories. The paper then extends these ideas to cyclic orders and to ordered metric valued fields, showing cyclic o-minimality for projective-lines of metric valued fields and weak o-minimality for valuation-ring frameworks, and discusses potential Ax–Kochen–Ershov-type principles via value-group linear orders. Together, these results illuminate the interaction between metric structure and order in continuous logic, with concrete applications to real closed metric valued fields and their projective/valuation-theoretic avatars. The construction of model companions, QE, and approximate categoricity provides a solid foundation for further study of ordered metric structures and their analytic applications in continuous logic.
Abstract
In continuous logic, there are plenty of examples of interesting stable metric structures. However, on the other side of the SOP line, there are only a few metric structures where order is relevant, and orders often appear in different ways. We now present a unified approach to linear and cyclic orders in continuous logic. We axiomatize theories of metric linear and cyclic orders, and in the ultrametric case, find generic completions, analogous to the complete theory DLO. We then characterize which expansions of metric linear orders should be considered o-minimal in terms of regulated functions. We prove versions several key properties of o-minimal structures, such as definable completeness and distality, in the context of o-minimal metric structures. Finally we find examples of o-minimal (and weakly o-minimal) metric structures by turning to real closed metric valued fields.