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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 $ {(f+g)}\circ h = f\circ h + g\circ h$ holds by definition, while $ f\circ(g+h)= f\circ g + f\circ h$ holds for all $ g,h\in \mathbb{R}^{\mathbb{R}}$ if and only if $ f$ is linear, i.e., if $ f(x+y)=f(x)+f(y)$ for all $ x,y\in \mathbb{R}$. Structures of the type like  $ \mathbb{R}^{\mathbb{R}}$ are called near-rings:

Definition 1   A set $ N$, together with two operations $ +$ and $ \circ$, is a near-ring provided that $ (N,+)$ is a group (not necessarily abelian), $ (N,\circ)$ is a semigroup, and $ (n+n')\circ n'' = n\circ
n''+n'\circ n''$ holds for all $ n,n',n''\in N$.

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  $ \mathbb{R}^{\mathbb{R}}$ already above. This and many other occasions show that near-rings form the ``nonlinear counterpart to ring theory''. More generally, take a group $ (G,+)$; then $ (G^{G},+,\circ)$ is a near-ring, and every near-ring can be embedded in some $ (G^{G},+,\circ)$. More examples of near-rings are given by $ (R[x],+,\circ)$, where $ R$ 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 $ (\mathbf{M}_S(G),+,\circ) = \{f\colon G\to G\ \vert\ f\circ s=s\circ f$ for all $ s\in S \}$, where $ (G,+)$ is a group and $ S\leq \textnormal{End}(G)$.

Actually, the first near-rings considered were near-fields, near-rings  $ (N,+,\circ)$ in which $ (N^{*},\circ)$ 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 $ (\mathbf{M}_S(G),+,\circ)$, where $ S$ 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 $ N$ is called planar if all equations $ x\circ a=x\circ b+c$ ( $ a,b,c\in N$, not $ x\circ a=x\circ b$ for all $ x $), have exactly one solution $ x\in N$.

This condition is motivated by geometry. It implies that two ``non-parallel lines $ y=x\circ a + c_{1}$ and $ y=x\circ b+c_{2}$ have exactly one point of intersection''. One example is $ (\mathbb{C},+,*)$, where $ a*b:=a\cdot\vert b\vert$. 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  $ \mathbb{F}[q]$ and choose a generator $ g$ for its multiplicative group. Choose a factorization $ q-1=st$. Define a new multiplication $ *_{t}$ in  $ \mathbb{F}[q]$ as $ g^{a}*_{t}g^{b} := g^{a+b-b_{t}}$, where $ b_{t}$ denotes the remainder of $ b$ on division by $ t$; if one factor is 0, the product should be 0. Then $ (\mathbb{F}[q],+,*_{t})$ is a planar near-ring.

Example 2   We choose $ \mathbb{F}[7]$, and $ t=2$ as divisor of $ 7-1$, so $ g=3$ serves well as a generator. From this, we get the multiplication table

$\displaystyle \begin{array}{c\vert ccccccc}
*_{2} & 0 & 1 & 2 & 3 & 4 & 5 & 6 \...
5 & 0 & 5 & 3 & 5 & 6 & 6 & 3 \\
6 & 0 & 6 & 5 & 6 & 3 & 3 & 5
\end{array} $

Example 3   We want to test combinations of  $ 7$ 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  $ 7$ and form the sets $ B_{i}=a*_{2}N^{*}+b$, $ a\not\equiv 0$, and abbreviate $ a*_{2}b$ by $ ab$:

$\displaystyle \begin{array}{rlrl}
1N^{*} + 0 & = \{1,2,4\}=:B_{1}, &
3N^{*} + ...
...{*} + 6 & = \{0,1,3\}=:B_{7}, &
3N^{*} + 6 & = \{2,4,5\}=:B_{14}.
\end{array} $

We see: these blocks form a so-called balanced incomplete block design (BIB-design) with $ v=7$ points (namely $ 0,1,2,3,4,5,6$) and $ b=14$ blocks; each point lies in exactly $ r=6$ blocks; each block contains precisely $ k=3$ elements, and every pair of different points appears in $ \lambda=2$ blocks. ($ v$,$ b$,$ r$,$ k$,$ \lambda$) are called the parameters of the design. In order to solve our fertilizer problem, we divide the whole experimental area into $ 14$ experimental fields, which we number by $ 1,2, \dots, 14$. We then apply precisely the fertilizers $ F_{i}$ to a field $ k$ if $ i\in B_{k}$, $ i=0,1,2,
\dots 6$.
Field 1 2 3 4 5 6 7 8 9 10 11 12 13 14
$ F_{0}$       x   x x   x x   x    
$ F_{1}$ x       x   x     x x   x  
$ F_{2}$ x x       x         x x   x
$ F_{3}$   x x       x x       x x  
$ F_{4}$ x   x x         x       x x
$ F_{5}$   x   x x     x   x       x
$ F_{6}$     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 $ 3$ fertilizers, and every fertilizer is applied to $ 6$ 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  $ \mathbf{A}$ of the design: compute $ (\mathbf{A}\mathbf{A}^t)^{-1}\mathbf{A}\mathbf{y}^t=:(\beta_{0}, \dots ,\beta_{v-1})^t$. Then Linear Algebra tells us that $ \beta_{0}, \dots,
\beta_{v-1}$ are the best estimates for the effects of the ingredients $ I_{0}, \dots,
I_{v-1}$. In our case, we get the best estimates for the effects of $ F_{0},\dots, F_{6}$ as $ (3.6, 2.7, 0.3, 1.0, 0.1, 4.8, 0.1) $. If we take $ F_{0},F_{1},F_{3},F_{5}$, we can expect a yield of $ 3.6+2.7+1.0+4.8=12.1$, which is considerably better than the yields $ y_{i}$ which we got in our experiment. A more detailed statistical analysis gives confidence intervals for the effects of $ F_{i}$, 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  $ \mathbf{A}$ are $ 0{-}1$-sequences, so they can be taken as a binary code  $ C_{\mathbf{A}}$. This is a nonlinear, constant weight $ r$ and constant distance   $ d=2(r-\lambda) $ code of length  $ b$ with $ v$ codewords. These codes have the largest number of codewords among all codes with equal weight $ r$, length $ b$, and fixed minimal distance  $ d$. So they also solve discrete sphere packing problems!

next up previous
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