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Preprint Number 59

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59. Hans Adler
Thorn-forking as local forking

Submission date: 6 March 2007


We introduce the notion of a preindependence relation between subsets of the big model of a complete first-order theory, an abstraction of the properties which numerous concrete notions such as forking, dividing, thorn-forking, thorn-dividing, splitting or finite satisfiability share in all complete theories. We examine the relation between four additional axioms (extension, local character, full existence and symmetry) that one expects of a good notion of independence.

We show that thorn-forking can be described in terms of local forking if we localise the number k in Kim's notion of 'dividing with respect to k' (using Ben-Yaacov's 'k-inconsistency witnesses') rather than the forking formulas. It follows that every theory with an M-symmetric lattice of (imaginary) algebraically closed sets is rosy, with a simple lattice theoretical interpretation of thorn-forking.

Mathematics Subject Classification: 03C45

Keywords and phrases: Independence relation, forking, thorn-forking

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