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Fusion over sublanguages. (English) Zbl 1100.03024
Summary: Generalising Hrushovski’s fusion technique we construct the free fusion of two strongly minimal theories $$T_1,T_2$$ intersecting in a totally categorical sub-theory $$T_0$$. We show that if, e.g., $$T_0$$ is the theory of infinite vector spaces over a finite field then the fusion theory $$T_\omega$$ exists, is complete and $$\omega$$-stable of rank $$\omega$$. We give a detailed geometrical analysis of $$T_\omega$$, proving that if both $$T_1,T_2$$ are 1-based then $$T_\omega$$ can be collapsed into a strongly minimal theory if some additional technical conditions hold – all trivially satisfied if $$T_0$$ is the theory of infinite vector spaces over a finite field $$\mathbb{F}_q$$.

##### MSC:
 03C45 Classification theory, stability, and related concepts in model theory 03C35 Categoricity and completeness of theories 03C60 Model-theoretic algebra
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