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 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:
This introduction allows us to state some of the problems we want to attack:
Density properties are - via the Chinese Remainder Theorem - connected to properties of the lattice of left ideals. A problem in this area is: