Independence, via limits

Gwyneth Harrison-Shermoen (Leeds)

Frank Adams 1,

The concept of an independence relation (a ternary relation among sets, satisfying certain properties) generalises that of linear independence in vector spaces and algebraic independence in fields, and gives us a way to determine what behaviour is "generic" in a given theory. Kim and Pillay showed that if a theory has an abstract independence relation satisfying an extra property (the “independence theorem over a model”), then the theory is simple and the independence relation is non-forking independence. There are, however, non-simple theories with relations that satisfy quite a few of the desired properties for a notion of independence. Given a large model M of some theory T, and following work of N. Granger (on two-sorted theories of infinite-dimensional vector spaces over an algebraically closed field and with a bilinear form), we describe a method of lifting independence relations from the (tame) theories of substructures of M to a reasonably well-behaved notion of independence in M.

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