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.