Fusion over sublanguages

Abstract

Generalising Hrushovski’s fusion technique we construct the free fusion of two strongly minimal theories T₁,T₂ intersecting in a totally categorical sub-theory T₀. We show that if, e.g., T₀ is the theory of infinite vector spaces over a finite field then the fusion theory T_ω exists, is complete and ω-stable of rank ω. We give a detailed geometrical analysis of T_ω, proving that if both T₁, T₂ are 1-based then, T_ω can be collapsed into a strongly minimal theory, if some additional technical conditions hold—all trivially satisfied if T₀ is the theory of infinite vector spaces over a finite field 𝔽_q.