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On sets with rank one in simple homogeneous structures
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra and Geometry.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra and Geometry.
2015 (English)In: Fundamenta Mathematicae, ISSN 0016-2736, E-ISSN 1730-6329, Vol. 228, 223-250 p.Article in journal (Refereed) Published
Abstract [en]

We study definable sets D of SU-rank 1 in Meq, where M is a countable homogeneous and simple structure in a language with finite relational vocabulary. Each such D can be seen as a 'canonically embedded structure', which inherits all relations on D which are definable in Meq, and has no other definable relations. Our results imply that if no relation symbol of the language of M has arity higher than 2, then there is a close relationship between triviality of dependence and D being a reduct of a binary random structure. Somewhat more precisely: (a) if for every n≥2, every n-type p(x1,...,xn) which is realized in D is determined by its sub-2-types q(xi,xj)⊆p, then the algebraic closure restricted to D is trivial; (b) if M has trivial dependence, then D is a reduct of a binary random structure.

Place, publisher, year, edition, pages
2015. Vol. 228, 223-250 p.
Keyword [en]
model theory, homogeneous structure, simple theory, pregeometry, rank, reduct, random structure
National Category
Algebra and Logic
Research subject
Mathematics
Identifiers
URN: urn:nbn:se:uu:diva-243006DOI: 10.4064/fm228-3-2ISI: 000352858400002OAI: oai:DiVA.org:uu-243006DiVA: diva2:785724
Available from: 2015-02-03 Created: 2015-02-03 Last updated: 2017-12-05Bibliographically approved
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Ahlman, OveKoponen, Vera

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