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Supersimple omega-categorical theories and pregeometries
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.ORCID iD: 0000-0002-9838-3403
2019 (English)In: Annals of Pure and Applied Logic, ISSN 0168-0072, E-ISSN 1873-2461, Vol. 170, no 12, article id 102718Article in journal (Refereed) Published
Abstract [en]

We prove that if T is an omega-categorical supersimple theory with nontrivial dependence (given by forking), then there is a nontrivial regular 1-type over a finite set of reals which is realized by real elements; hence forking induces a nontrivial pregeometry on the solution set of this type and the pregeometry is definable (using only finitely many parameters). The assumption about omega-categoricity is necessary. This result is used to prove the following: If V is a finite relational vocabulary with maximal arity 3 and T is a supersimple V-theory with elimination of quantifiers, then T has trivial dependence and finite SU-rank. This immediately gives the following strengthening of [18, Theorem 4.1]: if M is a ternary simple homogeneous structure with only finitely many constraints, then Th(M) has trivial dependence and finite SU-rank. (C) 2019 Published by Elsevier B.V.

Place, publisher, year, edition, pages
Elsevier, 2019. Vol. 170, no 12, article id 102718
Keywords [en]
Model theory, Simple theory, Pregeometry, Omega-categorical theory, Elimination of quantifiers, Homogeneous structure
National Category
Algebra and Logic
Identifiers
URN: urn:nbn:se:uu:diva-399083DOI: 10.1016/j.apal.2019.102718ISI: 000491614200004OAI: oai:DiVA.org:uu-399083DiVA, id: diva2:1379285
Available from: 2019-12-16 Created: 2019-12-16 Last updated: 2019-12-16Bibliographically approved

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Koponen, Vera

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