O-minimalism
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.
Cite
@article{arxiv.1106.1196,
title = {O-minimalism},
author = {Hans Schoutens},
journal= {arXiv preprint arXiv:1106.1196},
year = {2014}
}