Super/rosy [formula omitted]-theories and classes of finite structures

Abstract

We recover the essentials of þ-forking, rosiness and super-rosiness for certain amalgamation classes K, and thence of finite-variable theories of finite structures. This provides a foundation for a model-theoretic analysis of a natural extension of the “Lk-Canonization Problem” – the possibility of efficiently recovering finite models of T given a finite presentation of an Lk-theory T. Some of this work is accomplished through different sorts of “transfer” theorem (of varying degrees of subtlety) to the first-order theory Tlim of the direct limit. Our results include, to start with, a recovery of the basic technology of þ-independence (analogous to Onshuus (2006) [15]) using a rather straightforward transfer. We also recover an analog of the “þ-Independence theorems” of Ealy and Onshuus (2007) [7] for amalgamation classes and their limits by showing how to transfer/lift an abstract independence relation ▪ on the amalgamation class to the limit theory Tlim. We also work out an appropriate notion of Local Character for independence relations over classes finite structures, and we use this to verify that rosiness and super-rosiness-with-finite-Uþ-ranks coincide in these amalgamation classes and their limit theories.