×

zbMATH — the first resource for mathematics

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
PDF BibTeX XML Cite
Full Text: DOI
References:
[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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.