# The interplay between Logic and Algebra

This research topic is mainly in the focus of Professor Strüngmann. He is interested in the interplay between Logic and Algebra and applications in Theoretical Computer Sciences. In particular, the research focuses on structure theorems and realisation theorems for Abelian Groups and combinatorial as well as model theoretical methods.

One of the main topics is the category of almost completely decomposable groups (i.e. finite extensions of completely decomposable groups of finite rank) and generalisations of these. Among those are the Butler groups of arbitrary rank. In the finite rank case Butler groups are strongly related to representations of finite partially ordered sets via the so-called Butler functor. In the infinite rank case the picture is different and various suitable definitions of Butler groups exist and differ in certain models of set theory. The studies are related to the structure of Butler groups and their endomorphism rings in models of set theory like Gödel's universe or Martin's Axiom or complicated forcing extensions.

Similarly Strüngmann is interested in the structure of the group of extensions Ext(G,H) for R-modules G and H. Since the solution of the famous Whitehead problem by Saharon Shelah it is well-known that this structure, in particular the vanishing of Ext(G,Z), depends on the underlying set theory. In some models a characterisation of Ext(G,Z) is known - in others its understanding is beyond reach. Again, infinite combinatorics and set theory can be used to prove structure theorems like singular compactness theorems, the existence of universal modules etc. This has also applications in tilting and cotilting theory.

Finally, transitivity properties of modules are subject of the investigations. Transitivity, weak transitivity and full transitivity provide examples of modules with a rich structure and the property that every two elements can be mapped onto each other under certain natural and necessary assumptions. The notion of transitivity and full transitivity goes back to I. Kaplansky while weak transitivity is a new concept and is of a categorical nature.

Since most of the constructions and techniques that are used involve (infinite) combinatorics, geometric objects and set-theoretic as well as model-theoretic arguments Strüngmann is also very much interested in these areas, e.g. cardinal arithmetic and axiomatic set theory.