(gefördert vom Fonds zur Förderung der wissenschaftlichen Forschung - Projekt Nr. P12911-INF)
Just as classical ring theory is used to study linear functions, nearrings (i.e., rings lacking one distributive law) form the appropriate algebraic structure to model nonlinear functions on groups, which arise quite naturally in many areas (). Computations with such functions are usually done by local approximations with linear functions. This classical and successful approach is not possible, however, if there is no meaningful notion of derivative, in particular in the case of functions with finite or at least discrete domains. Thus nearring theory should help. Unfortunately, the algorithmic aspect of nearring theory is not well developed.
As the main practical outpout of the FWF-project P11486-TEC, the computer package SONATA (System of Nearrings and their Applications) has been released and is now used by various researches worldwide. It contains a large library of nearrings and algorithms to compute with them. Its presentation to the scientific community probably was the highlight at the Conference on Nearrings and Nearfields at Stellenbosch, South Africa. With the help of this package, some theoretical questions could be answered. Many researchers encouraged us to develop this package further, and also promised to support our efforts.
Motivated by this success, we plan to take a closer view at the algorithmic aspects of nearring theory as well as to using computers for theoretical investigations in this area, within a new project. According to our experience from developing SONATA and to the suggestions from other researchers, the main directions of further development should be: