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: