Bi-Colored Expansions of Geometric Theories

TL;DR

This paper develops a framework for bi-colored expansions of geometric theories using Fra"{i}ssé-Hrushovski constructions, analyzing dependence properties and providing axiomatizations for these expanded structures.

Contribution

It introduces a new class of bi-colored expansions with a pre-dimension function, axiomatizes their rich structures, and studies their dependence properties based on the original theory.

Findings

Dependent theories lead to dependent expansions.

Rational alpha preserves strong dependence.

Irrational alpha can result in non-strongly dependent structures.

Abstract

This paper concerns the study of Bi-colored expansions of geometric theories in the light of the Fra\"{i}ss\'{e}-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 $0<\alpha\leqslant 1$ is used to define a pre-dimension function $\delta_{\alpha}$. The pair $(\mathcal{K}_{\alpha}^{+},\leqslant_{\alpha})$ consisting of all such expansions with a hereditary positive pre-dimension along with the notion of substructure $\leqslant_{\alpha}$ associated to $\delta_{\alpha}$ is then used as a natural setting for the study of generic bi-colored expansions in the style of Fra\"{i}ss\'{e}-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.