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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 \(\mathbb{F}_q\).

03C45 Classification theory, stability, and related concepts in model theory
03C35 Categoricity and completeness of theories
03C60 Model-theoretic algebra
Full Text: DOI
[1] Constructing {\(\omega\)}-stable structures: Rank 2 fields 65 pp 371– (2000)
[2] DOI: 10.4064/fm170-1-1 · Zbl 0994.03030
[3] Uncountably categorical theories 117 (1993)
[4] L’égalité au cube 66 pp 1647– (2001)
[5] Geometric stability theory (1996)
[6] Le carré de l’égalité 64 pp 1339– (1999)
[7] DOI: 10.1016/0168-0072(93)90171-9 · Zbl 0804.03020
[8] DOI: 10.1007/BF02808211 · Zbl 0773.12005
[9] Model completeness of the new strongly minimal sets 64 pp 946– (1999)
[10] Finite structures with few types 152 (2003)
[11] The last word on elimination of quantifiers in modules 55 pp 670– (1990)
[12] DOI: 10.1016/0168-0072(85)90023-5 · Zbl 0566.03022
[13] Skew field constructions 27 (1977) · Zbl 0355.16009
[14] DOI: 10.1016/j.apal.2003.10.003 · Zbl 1042.03026
[15] Fundamentals of stability theory (1988) · Zbl 0685.03024
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