Some Model Theoretic Properties of Non-AC Generic Structures

In the context of Hrushovski constructions we take a language $ \mathcal{L} $ with a ternary relation $ R $ and consider the theory of the generic models $ M^{*}_{\alpha}, $ of the class of finite $ \mathcal{L}$-structures equipped with predimension functions $ \delta_{\alpha}, $ for $ \alpha\in(0,1]\cap\mathbb{Q} $. The theory of generic structures of non-AC smooth classes have been investigated from different points of view, including decidability and their power in interpreting known structures and theories. For a rational $ \alpha\in(0,1], $ first we prove that the theory of $ M^{*}_{\alpha} $ admits a quantifier elimination down to a meaningful class of formulas, called \textit{closure formulas}; and on the other hand we prove that $ Th(M^{*}_{\alpha}) $ does not have the finite model property.