Imaginary modules

Abstract

It was a fundamental insight of Shelah that the equivalence classes of a definable equivalence relation on a structure M often behave like (and indeed must be treated like) the elements of the structure itself, and that these so-called imaginary elements are both necessary and sufficient for developing many aspects of stability theory. The expanded structure Meq was introduced to make this insight explicit and manageable. In Meq we have “names” for all things (subsets, relations, functions, etc.) “definable” inside M. Recent results of Bruno Poizat allow a particularly simple and more or less algebraic modification of Meq in the case that M is a module. It is the purpose of this paper to describe this nearly algebraic structure in such a way as to make the usual algebraic tools of the model theory of modules readily available in this more general context. It should be pointed out that some of our discussion has been part of the “folklore” of the subject for some time; it is certainly time to make this “folklore” precise, correct, and readily available.Modules have proved to be good examples of stable structures. Not, we mean, in the sense that they are well-behaved (which, in the main, they are), but in the sense that they are straightforward enough to provide comprehensible illustrations of concepts while, at the same time, they have turned out to be far less atypical than one might have supposed. Indeed, a major feature of recent stability theory has been the ubiquitous appearance of modules or more general “abelian structures” in abstract stable structures.