  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.
