Bi-Colored Expansions of Geometric Theories

Abstract

This paper concerns the study of Bi-colored expansions of geometric theories in the light of the Fraïssé-Hrushovski construction method. Substructures of models of a geometric theory $T$ are expanded by a color predicate $p$, and the dimension function associated with the pre-geometry of the $T$-algebraic closure operator together with a real number $0<\alpha ⩽1$ is used to define a pre-dimension function ${\delta }_{\alpha }$. The pair $({\mathcal{K}}_{\alpha }^{+},{⩽}_{\alpha })$ consisting of all such expansions with a hereditary positive pre-dimension along with the notion of substructure ${⩽}_{\alpha }$ associated to ${\delta }_{\alpha }$ is then used as a natural setting for the study of generic bi-colored expansions in the style of Fraïssé-Hrushovski construction. Imposing certain natural conditions on $T$, enables us to introduce a complete axiomatization ${\mathbb{T}}_{\alpha }$ for the class of rich structures in this class. We will show that if $T$ is a dependent theory (NIP) then so is ${\mathbb{T}}_{\alpha }$. We further prove that whenever $\alpha$ is rational the strong dependence transfers to ${\mathbb{T}}_{\alpha }$. We conclude by showing that if $T$ defines a linear order and $\alpha$ is irrational then ${\mathbb{T}}_{\alpha }$ is not strongly dependent.