What we intend to program - Motivations for a computational near-ring theory

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:

• We take advantage of the power (efficiency) of computational group theory. Since near-rings are now simply groups, all the implemented group-operations, such as e.g. Intersection can be readily used for near-rings.

There is, of course, the following draw-back:

• We cannot compute a near-ring whose additive generators we do not know.

This disadvantage, however, is less painful than one might think: Of course, it is possible to construct a near-ring whose additive generators are not known as an intersection of near-rings that can already be computed. My algorithms computing centralizer near-rings and the near-rings of compatible functions are based on this fact: However, I compute them not as the intersection of some near-rings, but as the intersection of suitable subgroups of .

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:

• A GAP-library for computing with near-rings, including the most important applications of near-rings. This includes to some extent working with rings (cf. [Eck95]).
• First steps into computational near-ring theory.