Next: 2.1.3 Introduction to Frobenius Up: 2.1 State of the Previous: 2.1.1 An overview of

## 2.1.2 Introduction to near-rings

Planar near-rings form a special class of algebraic systems. Here is a short account on the basics of near-rings in general and planar near-rings in particular.

The set of all real functions, together with pointwise addition and composition, does not form a ring. Observe that holds by definition, while holds for all if and only if is linear, i.e., if for all . Structures of the type like  are called near-rings:

Definition 1   A set , together with two operations  and , is a near-ring provided that is a group (not necessarily abelian), is a semigroup, and holds for all .

Of course, every ring is a near-ring; hence near-rings are generalized rings. Two ring axioms are missing: the commutativity of addition and (much more important) the other distributive law.

Example 1   We have met the near-ring  already above. This and many other occasions show that near-rings form the nonlinear counterpart to ring theory''. More generally, take a group ; then is a near-ring, and every near-ring can be embedded in some . More examples of near-rings are given by , where  is a commutative ring with identity, by the collection of continuous real functions, by the differentiable ones, or by the polynomial functions. Other important examples (see below) are centralizer near-rings for all , where is a group and .

Actually, the first near-rings considered were near-fields, near-rings  in which forms a group. In 1905, L. E. Dickson constructed the first proper near-fields by distorting'' the multiplication in a field. These types of near-fields are now called Dickson near-fields. Two years later, Veblen and Wedderburn used near-fields to coordinatize geometric planes. In a monumental paper, H. Zassenhaus showed in 1936 that all finite near-fields are Dickson ones, with the exception of 7 (well-known) cases. 51 years later, Zassenhaus showed that there do exist non-Dickson infinite near-fields of every prime characteristic.

For near-rings, we now have a sophisticated structure theory. Quite often, the results look similar to the ring case, but the proofs are completely different. For instance, matrix rings (the stuff rings are made of'') have to be replaced by centralizer near-rings , where is a fixed-point free automorphism group.

These types of automorphisms turn up in another context as well; this one has nice applications to real-world problems.

Definition 2   A near-ring  is called planar if all equations ( , not for all ), have exactly one solution .

This condition is motivated by geometry. It implies that two non-parallel lines and have exactly one point of intersection''. One example is , where . There are various methods to construct planar near-rings (see [Cla92,Pil83,Pil96]). Most of them use fixed-point-free automorphism groups; we just present the easiest way.

Definition 3   Construction method for finite planar near-rings. Take a finite field  and choose a generator  for its multiplicative group. Choose a factorization . Define a new multiplication  in  as , where denotes the remainder of  on division by ; if one factor is 0, the product should be 0. Then is a planar near-ring.

Example 2   We choose , and as divisor of , so serves well as a generator. From this, we get the multiplication table

Example 3   We want to test combinations of   ingredients for fertilizers. Testing all possible combinations of ingredients requires a huge amount of space and money. So we conduct an incomplete test; but this one should be fair to the ingredients (each ingredient should be applied the same number of times) and fair to the experimental fields (each test-field should get the same number of ingredients). We take the above near-ring of order   and form the sets , , and abbreviate by :

We see: these blocks form a so-called balanced incomplete block design (BIB-design) with points (namely ) and blocks; each point lies in exactly blocks; each block contains precisely elements, and every pair of different points appears in blocks. (,,,,) are called the parameters of the design. In order to solve our fertilizer problem, we divide the whole experimental area into experimental fields, which we number by . We then apply precisely the fertilizers  to a field  if , .
 Field 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Fertilizer x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x Yields: 2.8 6.1 1.2 8.5 8.0 3.9 7.7 5.5 4.0 10.7 3.2 4.9 3.8 5.5
The last row indicates the yields on the experimental fields after performing the experiment. Then we get: every field contains exactly fertilizers, and every fertilizer is applied to fields. Finally, every pair of different fertilizers is applied precisely twice in direct competition. Hence planar near-rings even have applications in agriculture!

The statistical analysis uses the incidence matrix  of the design: compute . Then Linear Algebra tells us that are the best estimates for the effects of the ingredients . In our case, we get the best estimates for the effects of as . If we take , we can expect a yield of , which is considerably better than the yields  which we got in our experiment. A more detailed statistical analysis gives confidence intervals for the effects of , as well as information concerning their synergy effects.

Even more can be done with incidence matrices coming from BIB-Designs. The rows in the matrix  are -sequences, so they can be taken as a binary code  . This is a nonlinear, constant weight  and constant distance   code of length   with codewords. These codes have the largest number of codewords among all codes with equal weight , length , and fixed minimal distance  . So they also solve discrete sphere packing problems!

Next: 2.1.3 Introduction to Frobenius Up: 2.1 State of the Previous: 2.1.1 An overview of
Peter Mayr 2002-08-12