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Thornforking as local forking
, 2007
"... We introduce the notion of a preindependence relation between subsets of the big model of a complete firstorder theory, an abstraction of the properties which numerous concrete notions such as forking, dividing, thornforking, thorndividing, splitting or finite satisfiability share in all complete ..."
Abstract

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We introduce the notion of a preindependence relation between subsets of the big model of a complete firstorder theory, an abstraction of the properties which numerous concrete notions such as forking, dividing, thornforking, thorndividing, 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 thornforking 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 BenYaacov’s ‘kinconsistency witnesses’) rather than the forking formulas. It follows that every theory with an Msymmetric lattice of algebraically closed sets (in T eq) is rosy, with a simple lattice theoretical interpretation of thornforking.
Simplicity simplified
 Rev. Colombiana Mat., 41(Numero Especial):263–277
, 2007
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