Near-rings are ?rings with just one distributive law and with possibly non-commutative
addition?. Standard examples are collections M(G) of all mappings from a group (G,+)
into itself, with point-wise addition and composition of mappings. If (G,+) is abelian
and if one only takes endomorphisms (?linear maps?), one gets rings; so near-rings can
be viewed as the ?non-linear generalizations of rings?.
In this talk, I want to present an especially
useful class of near-rings N, the planar ones. The are characterized by the property
that all equations xa = xb + c have a unique solution x, unless xa = xb holds for all
a,b ? N. If N ? = N \ {0}and one takes the collection B of all subsets of N of the
form aN ? + b (with a not = 0) and their translates, one gets balanced incomplete block
designs. It will be described why they are extremely useful for the design of statistical
experiments, especially in biology and medicine.
Forschungs-