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Computing with near-rings - Algorithms and Implementation

(gefördert vom Fonds zur Förderung der wissenschaftlichen Forschung - Projekt Nr. P12911-INF)

Erhard Aichinger
Franz Binder
Jürgen Ecker
Peter Mayr
Christof Nöbauer


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 ([29]). 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:

  1. theoretical investigation of special nearrings, e.g. on finite p-groups, based on the computed results now available;
  2. enlarging our library of nearrings;
  3. computing very big nearrings of specific type, e.g. centralizer nearrings, matrix nearrings, transformation nearrings of sizes above , or even infinite nearrings of specific type such polynomial nearrings or finitely generated nearrings;
  4. special algorithms for the important case of nearrings with identity;
  5. enhanced graphic representation of ideal and subnearring lattices;
  6. facilities for extensive hypothesis testing, such as generating nearrings with specific properties at random;
  7. application of the system to solve specific problems in nearring theory;
  8. algorithmic methods for determining the size of a nearring generated by some actions on a group
  9. maintenance of SONATA, including user support.

Project Description

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Next: State of the Art

Juergen Ecker
Wed May 13 09:13:56 CEST 1998