Although near-rings are a well-developed branch of algebra, little effort has been spent on treating near-rings by the help of a computer. This is most regrettable in the view of the research in both the theory and the applications of near-rings. Since good algorithms for computing with groups, in particular permutation groups, are now available, i.e. invented and implemented, my first approach to computing with near-rings was to reduce the near-ring problems to group-theoretic problems and to solve those using GAP.
My strategy was the following: Actually, there is no loss of generality in considering only sub-near-rings of the near-rings M(G), where M(G) is the near-ring of all selfmaps on a group G. Since any function on G can be viewed as an element of the direct product
the additive group of any near-ring can be identified with a subgroup of this direct product. This viewpoint gives us immediately a possibility to compute a representation of the nearrings F for which the generators of the additive group (F, +) are known. This viewpoint has the following advantage:
Although this approach yields very efficient algorithms, I want to compare it with another approach: Let F be a sub-near-ring of M(G). Viewing F as an F-group, we may factor out the noetherian quotient . The factor F/N is isomorphic to some subgroup of G; N can recursively be treated in the same way. The outcoming way of representing F is a bit in the spirit of [Sim70] and hopefully, we shall find concepts for representing near-rings that are similar to the concepts of base and strong generating sets, which are the key-techniques in computing with permutation-groups (cf. [Neu82]). We intend to take theoretic studies into this direction.
A typical theoretic question arising in implementing near-rings is the problem whether a near-ring is distributively generated (cf. [Pil83]). Finding an efficient algorithm deciding this property could also be a contribution to the open theoretical questions around this point. It is e.g. unknown whether any distributively generated near-ring can be embedded into an endomorphism near-ring E(G).
Summarizing, the ``algorithmic'' part of this project will have the following output: