What we intend to prove - Questions in classical near-ring theory

Since the software produced in this project gives us the chance to compute with interesting examples of near-rings, we hope to get more insight into some of the following problems:

• Polynomial functions
  The set of all polynomial functions on an -group G forms a near-ring with respect to pointwise addition and functional composition (cf. [LN73]). Quite a lot is known about polynomials on commutative rings with identity, but it gets a lot darker if we consider polynomial functions on groups: The number of polynomial functions on a group G is known for abelian groups; it has only recently been computed for the (very special class of) extraspecial p-groups ([Saa95]).

  Polynomial functions have no tag on them: it is therefore interesting to find properties that characterize polynomial functions. We have a ``negative'' result in this direction: Let be the near-ring of all polynomial functions on the ring of the integers. Then there exists an uncountable sub-near-ring of which is contained in precisely the same varieties (of near-rings) as . Hence equational properties cannot single out polynomial functions. On the other hand, there are groups on which a function f is a polynomial function if and only if f is congruence-preserving. Such a group is called affine complete. Stated in the language of [HN77], a group is affine complete iff . For a finite group G we may therefore define the degree of polynomial completeness as the smallest natural number n for which we have . This leads us to interpolation theory:

• Interpolation problems
  In [Aic95a], we have defined the local interpolation near-rings similar to local polynomial functions, which are introduced in [HN77]. The chain of local interpolation near-rings provides a frame-work for studying the density theorems in near-ring theory. Therefore, we continue our investigation of this chain and of the equalities occurring in it.

This introduction allows us to state some of the problems we want to attack:

• Which groups are affine complete ? Since this question is answered only for abelian, hamiltonian and completely non-abelian groups (cf. [Kaa78]), also partial answers for other classes of groups are nice.
• What can be said about the degree of polynomial completeness for special classes of groups ? A complete answer is again known only for abelian, hamiltonian and completely non-abelian groups.

Density properties are - via the Chinese Remainder Theorem - connected to properties of the lattice of left ideals. A problem in this area is:

• Does every near-ring have a distributive lattice of left ideals ? This is an open question; existing ``proofs'' are known to have gaps, therefore it might be reasonable to look for a counterexample.

Another problem crying for an example is the following:

• Do there exist 2-tame near-rings (cf. [Pil83]) that are not 3-tame ? Strong properties can be proved for 2-tame near-rings; seemingly stronger properties for 3-tame near-rings, but no example of a 2-tame near-ring that is not 3-tame is known. This problem has also been listed in [Sco95a] as an interesting open problem.

We also intend to treat completely different questions that belong to the geometric branch of near-ring theory. These problems are more related to the work of Roland Eggetsberger ([Egg92]).

• For which parameters do the methods explained in [Cla92] allow to construct a BIB-design ? The practical output of this question will be a function in the near-ring package that yields the required design.