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Preprint Number 336
336. Hans Schoutens
Submission date: 6 June 2011.
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.
Mathematics Subject Classification: 03C64.
Keywords and phrases: o-minimality, tame structures, Grothendieck ring
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