O-minimalism

TL;DR

This paper introduces the concept of o-minimalistic structures, explores their properties through the DCTC theory, and investigates invariants like the Grothendieck ring and discretely valued Euler characteristics.

Contribution

It develops a comprehensive framework for o-minimalistic structures, including criteria for tameness and new invariants such as the Grothendieck ring and Euler characteristics.

Findings

DCTC theory describes key properties of o-minimalistic structures.

Criteria for tameness based on cell decomposition failure.

Introduction of discretely valued Euler characteristics.

Abstract

An ordered structure is called o-minimalistic if it has all the first-order features of an o-minimal structure. We propose a theory, DCTC (Definable Completeness/Type Completeness), that describes many properties of o-minimalistic structures (dimension theory, monotonicity, Hardy structures, quasi-cell decomposition). Failure of cell decomposition leads to the related notion of a tame structure, and we give a criterium for an o-minimalistic structure to be tame. To any o-minimalistic structure, we can associate its Grothendieck ring, which in the non-o-minimal case is a non-trivial invariant. To study this invariant, we identify a third o-minimalistic property, the Discrete Pigeonhole Principle, which in turn allows us to define discretely valued Euler characteristics.