Department of Mathematics
Server

I will start with some brief remarks on the structure of tt modules, elementary duality, and what this implies about the structure of modules with ACC on definable subgroups.

There is a natural correspondence between finitely presented modules and pp formulas, called "free realizations" of pp formulas. This leads naturally into W. Zimmermann's characterizations [1] of various kinds of definable or infinitely definable subgroups of modules via certain kinds of functors defined from "pointed modules".

These functors can be used to define various relativized chain conditions. Ph. Rothmaler and I are trying to extract the model-theoretic content of Zimmermann's results. U;ltimately the goal is to develop a theory for modules with ACC on definable subgroups as rich as that for modules with DCC on definable subgroups (tt modules). Zimmermann's methods may give methods for dealing with infinite sets of pp formulas, or sets of formlulas with infinitely many free variables, that are not apparent from a purely model-theortic viewpoint.

[1] Wolfgang Zimmermann, Modules with Chain Conditions for Finite Matrix Subgroups, J. Alg. (190), 68–87 (1997).