CONFERENCE ON
NEARRINGS AND NEARFIELDS
UNIVERSITAET DER BUNDESWEHR HAMBURG
July 30  August 06, 1995
ABSTRACTS
These are versions of the LaTeX abstracts provided for the conference
by the attendees. Some of the laTeX has been left in place because it
is not easily turned into HTML. Some other hacks have been used,
for instance
the use of o for functional composition, instead of the LaTeX \circ.
Use your imagination.
Converted by hand, 21 July 1995 by Tim Boykett.
On Some Density Theorems for Nearrings and
Composition Rings
Erhard Aichinger
We discuss a method from interpolation theory that
allows a quite easy proof of the density results
for nearrings and apply this method to some composition rings.
Finally, we use the Chinese remainder theorem
for getting a density result for composition rings.
Functional Decomposition on Nearrings
Cesar L. Alonso*
Jaime Gutierrez
The functional decomposition problem has been studied by
several authors in the last fifty years.
Firstly some authors proposed the first algorithm
for solving the polynomial decomposition problem.
More later some works appeared about rational
function decomposition problem, and recently
the decomposition of algebraic functions
is being considered.
On the other hand, several
generalizations have been proposed
for multivariate polynomials.
We deal with the problem of
decomposition of univariate polynomials with
coefficients over a factorial domain
(i.e. decomposition in the nearring (D[X],+ ,o)).
We remark also that solving this problem implies,
in some sense, the solution of the decomposition problem for
multivariate polynomials over a field.
Other important and new issue concerning functional
decomposition is the polynomial decomposition problem modulo an ideal;
this is:
Given a polynomial f(X_1,...,X_n) \in K[X_1,..., X_n]/I,
where I is a principal ideal and
K is an arbitray field, to determine if there exists g(Y),
h(X_1,...,X_n)
such that f(X_1,...,X_n)=g(h(X_1,...,X_n)) modulo I.
The motivation of this general problem
arises in the particular case I=(X^2+Y^21),
the solution of this particular problem provides a method
for simplifying systems of the so called sinecosine equations.
This simplification is very important for solving the
inverse kinematic problem in robotics.
Polynomial NearRings
Scott W. Bagley
In 1987, van der Walt overcame the difficulties caused by the lack of
one distributive property in nearrings in order to create matrix nearrings
with the property that when the nearring is a ring, the usual matrix ring
and the matrix nearring agree. In this paper we develop a new polynomial
nearring with coefficients from a nearring and derive some of its
properties. Similar to van der Walt's work, when the coefficient nearring is
a ring, our ploynomial nearring coincides with the usual polynomial ring.
On Derivations in Nearrings, II
Howard E. Bell
Bell and Mason have studied commutativity in 3prime nearrings N which
admit a nonzero derivation d with constraints involving elements of N
and elements of d(N) .
Motivated by recent results on rings, we study analogous problems where
the constraints involve only U and d(U) , U being some proper subset
of N . A typical result is the following:
{\bf Theorem:} Let N be a zerosymmetric 3prime nearring and U
a nonzero subset of N such that NU \subseteq U and UN \subseteq U .
If there exists a derivation d on N such that d^2 \not= 0 and
[U,d(U)] = {0} , then N is a commutative ring.
Automorphisms of Groups and Combinatorial Structures
Gerhard Betsch
Examples and some properties of combinatorial structures constructed
from a group \Gamma and a group G of automorphisms of \Gamma .
Polynomials with Multiplication and Composition
Franz Binder
Wheras the polynomial nearring (k[x],+,o)
has been studied in some detail, the seminearring
(k[x],*,o) and the nearring (k(x)_1,*,o)
of rational functions with fixed point 1 are rather unknown.
Because (normed) polynomials fulfill both cancellation laws
(even with respect to composition), we expect that these structures
have rather few ideals and can be embedded into some fieldlike structure.
Some first results and related topics will be outlined.
Groups, Rings and Sets: Is There a Connection?
Gary F. Birkenmeier*
H.E. Heatherly
Gunter Pilz
Let S be a nonempty set, G a group disjoint from S , and K
a subgroup of G . Define M(S,G) \equiv {~f : S \cup G > G~}
(write functions to the right of their arguments). Then, M(S,G) is a left
nearring, with
M(S,G) = Ann G\oplus Ann S
(right ideal
decomposition) where Ann G = Ann M(S,G) and
M(S,G)/Ann G \simeq Ann S \simeq M(G) .
Of particular interest are the subnearrings,
{\cal H}(G,K) \equiv gp( Hom(G,K)) , and
{\cal E}(G,K) \equiv {f \in {\cal E}(G)  G f \subseteq K}
of M(S,K) , where S = G  K . We will consider various distributivity
conditions for {\cal H}(G,K) and {\cal E}(G,K) including conditions
on K which insure they are rings. Also, we will discuss how
{\cal H}(G,K) and {\cal E}(G,K) ''fit'' inside {\cal E}(G) .
Distributive Nearrings Do Exist
Gary F. Birkenmeier
H.E. Heatherly
Gunter Pilz*
We present a method to construct distributive nearrings. Take a group
G , an abelian subgroup A of G and a set S disjoint from G . Let
D(S;G,A) = {f : S \cup G > G  f/G \in
Hom(G,A)}
Then D(S;G,A) is a distributive nearring and Hom(G,A) is a ring.
Also, D(S;G,A) is a ring iff G is itself abelian. The class of all
D(S;G,A) is ''catholic'' in the sense that every distributive nearring
can be embedded into some D(S;G,A) ; this embedding is constructive.
It is easy to see that D(S;G,A) decomposes directly (as right ideals)
into Ann G and Ann S , the latter being isomorphic
to Hom (G,A) . Hence D(S;G,A) = G^{S}  Hom (G,A) .
If G is solvable of desired length m then the additive group of
D(S;G,A) is solvable of length \leq m . If A is a homomorphic
image of the finite group G then A divides  Hom (G,A)
and hence D(S;G,A) . We also study questions about nilpotency of
D(S;G,A) and permutation identities.
Special Radicals in Omegagroups
Geoffrey L. Booth*
Nico J. Groenewald
Buys and Gerber studied special radicals in Andrunakievich varieties of
Omegagroups. While they obtained good results, this excluded many useful
varieties, such as zerosymmetric nearrings, where special radicals have
been studied. We extend Buys and Gerber's work to arbitrary varieties,
and obtain in particular, new characterizations of radical and semisimple
classes. These lead to new results, inter alia in the varieties of
zerosymmetric nearrings, gamma rings and rings with involution.
Biideals and Quasiideals in Categories
Suzette Botha
Quasiideals and biideals are defined and investigated in categories.
Examples of these ideals are discussed in some categories, e.g.
groups, rings, nearrings and Lie Algebras.
Seminearrings of Polynomials over Semifields:
A Note on Don Blackett's Fredericton Paper
Tim Boykett
At the 1993 Nearring conference Don Blackett presented a paper
where he looked at {\em Probability Generating Function
Polynomials}, those polynomials
PGF = {\sum_{i=0}^{n} a_ix^i  a_i >= 0, \sum_i a_i = 1}
over the reals that describe the probabilities a_i of
an event happening i times.
He showed that (PGF,*,o) is a seminearfield, that
the operations had meaning when interpreted as various
combinations of experiments, and that one could decompose
arbitrary PGFs to combinations of {\em SuccessFailure}
polynomials
SFP = {a+bx  a+b = 1, a,b >= 0}
That is, he showed that PGF is generated by SFP as a seminearfield.
The methods he used can be simply extended to apply to
polynomials over arbitrary commutative semifields (S,+,*)
with S* closed under +.
There are also some aspects of
the limits to which one can push the process of constructing
polynomials over arbitrary (2,2)algebras without losing the
intuitive results.
Couplings of Group Extensions and Formal Power Series
Division Rings
Beat Bühler
Let (G, *) be a group and End G its monoid of
endomorphisms.
A {\em generalized group coupling} \kappa : G > End G is a map
with
\kappa_{g} \kappa_{h} =
\kappa_{ g \kappa_g (h) }
for all g,h \in G .
A generalized group coupling is called {\em group coupling}, if the image of
G under \kappa is a subset of Aut G , the group of
automorphisms of G .
Let \kappa be a group coupling. With
g o h := g \kappa_{g}(h)
for all g, h \in G is G^{\kappa} := (G, o) a group.
Let ( F, + , * ) be a division ring and
F^* := F  {0} its multiplicative group.
A group coupling \kappa: F^* > Aut F^*
with
\kappa_{x} (y + z) =
\kappa_{x} (y) + \kappa_{x} (z)
for all x,y,z \in F, x \neq 0 is called {\em coupling}.
With
x o y := x \kappa_{x} (y)
and
0 o y := 0
for all x, y \in F, x \neq 0 is
F^{\kappa} := ( F, + , o) a (left)nearfield.
We describe all generalized group couplings of group extensions and
therefore all generalized group couplings of semidirect products and direct
products.
As an example we consider couplings of formal power series division rings
with nontrivial factor sets.
Their multiplicative structure is a semidirect product of the
{\em oneunits} with a group extension.
Simplicity of the Centralizer NearRing Determined
by End~ G
G. Alan Cannon
Let G be a finite group and let M_E(G) =
{f : G > G  f \sigma = \sigma f for all \sigma \in
End G} , the centralizer nearring determined by End G.
We discuss the simplicity of M_E(G) .
Recent Developments, Discoveries, and Directions
for Planar Nearrings
James R. Clay
During the past 10 years, the study of planar nearrings has
become more diversified. This has lead to a dramatic increase in
the number of applications, especially to geometry and combinatorics.
Most of the developments will be explained, together with some of
the more important results and problems. In addition, several
discoveries have been made which promise even more diversification
of the theory together with numerous challenging problems.
On Right Self Commutative Nearrings
P. Dheena
If N is subdirectly irreducible RSC nearring
then either N is simple with nonzero idempotent is a left identity or the
intersection of nonzero ideals of N is without nonzero idempotents.
It has been shown that a subdirectly irreducible RSC nearring with a
distributive element is a nearfield. Necessary and sufficient conditions
are obtained for a RSC nearring to be right self distributive.
Circles and Their Interior Points in Field Generated
Ferrero Pairs
Roland Eggetsberger
By the notion of circularity planar nearrings get a geometric meaning.
The concept of double planar nearrings intensifies the influence of
geometry. In this context we concentrate on interior points of a circle
and present their design theoretic properties.
On the Semiendomorphisms of Abelian Groups and
Transformation Nearrings
Yuen Fong
We investigate in this paper the subnearring of G
generated by the semiendomorphisms of the given abelian group
G. The semiendomorphisms on a given finite abelian group
G are described in detail.
Direct Decomposition in Group Nearrings
Roland Fray
The following results on direct decomposition in group algebras
are generalized to nearrings:

For a family {R_i  i \in I} of commutative rings and an
arbitrary group G:

If R is a subdirect product of {R_i  i \in I} ,
then RG is a subdirect product of {R_i G  i \in I} .

If R = \prod_{i \in I} R_i , then
RG = \prod_{i \in I} (R_iG) .

Let G be the direct product of its subgroups G_1 and G_2 .
Then RG = (RG_1)G_2 .
Dense Nearrings of Continuous Selfmaps in Convex Spaces
Peter Fuchs
Working in the nearring of all continuous selfmaps C(V) of a given
locally convex Hausdorff space we investigate how primitive nearrings
could be used in order to approximate functions in C(V) on compact
subsets of V.
On Involution Sets Induced by Neardomains
Christian M. Gabriel
A {\em specific involution set} J \subset Sym_M
{\em of characteristic} p \in N  where M is a set 
is defined by the four axioms:
 [E1.]
J is sharply 1transitive.
 [E2.]
For all \nu,\sigma \in J holds: \nu\sigma\nu \in J .
 [E3.]
All elements of J\backslash{{\rm id}_M} are of order p .
 [E4.]
Every element of \langle J \rangle (where \langle J \rangle is the group
generated by the elements of J ) has at most one fixpoint.
The characteristic of a specific involution set is an odd prime.
Let F be a neardomain. Then the set J of all functions
\nu : x > a  x , where a \in F , is a specific involution
set. It is well known, that F is a nearfield iff J^3 \subset J .
Some results for specific involution sets can be shown, for example:
 [1.]
If p = 3 , then J^3 \subset J .
 [2.]
If J < \infty , then J^3 \subset J .
 [3.]
For all \tau \in \langle J \rangle and given x \in M
there is exaktly one \nu \in J and exactly one
\alpha \in J \rangle with fixpoint x such that
\tau = \nu\alpha .
 [4.]
If J^3 \not\subset J , then there is a specific involution set J'
of characteristic p with (J')^3 \not\subset J' which is
generated by three of its elements \nu,\sigma,\varrho such
that \nu,\sigma,\varrho has a fixpoint.
Furthermore some propositions on relations between specific involution sets
and the centralisators of their elements will be shown.
Superprime Radical for Nearrings
Nico J. Groenewald
The concept of superprime radical is introduced for nearrings.
We show that this radical is a special radical in the class of
A nearrings. Relationships with other well known prime radicals
are determined.
Polynomial Nearrings in Several Variables
Jaime Gutierrez*
Carlos Ruiz de Velasco
The following interesting exploratory problem appears in
the recent book of J.R. Clay "Nearrings, Geneses and Applications":
Let R be an arbitrary commutative ring with identity.
It is well known
that (R[X],+, o) is a right nearring with identity and (R[X],+,.,o)
is a composition algebra. Now , we consider the set of all
polynomials over R in
a finite number of indeterminates, R[X_1,...,X_n] .
What does do with R[X_1,...,X_n]?
Certainly, (R[X_1,...,X_n],+,.) is a ring with identity.
For f(X_1,...,X_n), g(X_1,...,X_n) \in R[X_1,...,X_n] ,
we have to define f(X_1,...,X_n) o g(X_1,...,X_n)
if we want to get
the nearring (R[X_1,...,X_n],+, o) .
What are some significant problems concerning these nearrings?.
For instance, how do the
elements f \in R[X_1,...,X_n] relate to functions from R^n into R .
What about the structure ideals of the these nearrings ?...etc.
One purpose of this talk is to try to give partial answer to
this kind of questions.
Nearrings and Involutions
H. E. Heatherly
E.S.K. Lee
Richard Wiegandt*
Involution can be defined for universal algebras in a very general manner.
In particular, on a nearring (N,+,*) an involution \star may be
defined as an additive and multiplicative automorphism or antiautomorphism
of order two. If the involution \star is a multiplicative
antiautomorphism, then the nearring is distributive. However, if the
involution \star is a multiplicative automorphism, the nearring need
not be distributive. If (N,+,*,\star) is subdirectly irreducible, then
either (N,+,*) is a subdirectly irreducible nearring, or
(N,+,*) is a subdirect sum of two subdirectly irreducible nearrings
and the involution \star is the exchange involution. Examples are provided
and the subdirectly irreducible distributive nearrings are described.
Fibered Incidence Loops by Neardomains
Herbert Hotje
Let G be a sharply two transitive permutation group. To G there can
be associated a neardomain F . For (G,F) we investigate different
kinematic structures:
 i) (G,F) is a general kinematic space and moreover a kinematic group
([1]),\
 ii) if the set Z(1):={x\in F; 1+x=x+1} is additively closed then
F has the structure of a kinematik Kloop ([2]),\\
 iii) (F,+)\times (F,+) is a fibered incidence loop ([3]).
 {1} H. Hotje: Allgemeine kinematische Räume. Mitt. Hamburger Math. Ges. 12 (1991), 785791.
 {2} H. Karzel: RaumZeitWelt und hyperbolische Geometrie. Vorlesungsausarbeitung von A. Konrad, TUMBeiträge zur Geometrie und Algebra 29, 1994.
 {3} P. Tancke: Kollineationen und Anordnungen in 2Strukturen mit Rechtecksaxiom. Dissertation Hannover 1989.
Kloops and Quasidirect products
Bokhee Im
W. Kerby and H. Wefelscheid
were led to the concept of K loops by their investigations on neardomains.
H. Karzel and H. Wefelscheid discussed
K loops in the Minkowski space time world over a commutative
euclidean field \overline K .
If we replace \overline K by
an ordered commutative field K=(K, +, *, \leq) and
let L=K(i) be the quadratic extension of K with i^2=1 ,
then we can still form the future cone
{\frak H}^{++}:={ A\in GL(2, L)A=A^*, det A>0, Tr A>0 }.
But the operation
A\oplus B:=\frac1{Tr A+2\sqrt{\det A}}(\sqrt{\det A}E+A)B(\sqrt{\det A}E+A)
turns {\frak H}^{++} in a K loop if and only if K is euclidean.
In this paper, we assume that K is pythagorean, and show that
({\frak H}^{++}, \boxplus) is a K loop with the binary
operation A\boxplus B=\sqrt{A^2 \oplus B^2}=\sqrt{AB^2 A} , where
\sqrt A= \frac{\sqrt{\det A}E+A}{\sqrt{Tr A+2\sqrt{\det A}}}
and that
({\frak H}^{++}, \boxplus) and ({\frak H}^{++}, \oplus) are isomorphic if
K is euclidean.
Moreover, for a given group G=(G, *) , we define the (internal)
quasidirect product
{\frak F} \rtimes_Q {\frak U}
of a certain K loop (\frak F, +) with {\frak F} \subset G and
a suitable subgroup \frak U of G .
For the K loop (\frak H^{++}, \boxplus) and
the group
Q_1 := { X\in GL(2,L)X^*X=E },
we obtain the quasidirect product
{\frak H}^{++} \rtimes_Q Q_1 as a subgroup of GL(2,L) .
And SL(2,L)= {\frak H}^{1+} \rtimes_Q {\frak Q}_1,
where {\frak H}^{1+}=SL(2,L) \cap { \frak H}^{++} \leq
({\frak H}^{++}, \boxplus),
{\frak Q}_1 = SL(2,L) \cap Q_1.
If K is euclidean, then GL(2,L)={\frak H}^{++} \rtimes_Q Q_1.
We also discuss quasidirect products as subgroups of
certain Lorentz groups.
On the Structure of NonZerosymmetric Nearrings
Kalle Kaarli
Given a nearring N , an N group G is said to be
{\it strictly simple} if GN\ne 0 , G is simple and has no nonzero proper
N subgroups. We call a nearring N {\it primitive} if it admits
a faithful strictly simple N group.
In the case of 0symmetric nearrings a strictly simple N group is the same
as an N group of type 2 but in general the second notion (as
defined in Pilz's book) is stronger.
Given a vector space V we denote by A(V) the nearring of all affine
transformations on V . If \rho is an equivalence relation on a group
G then M(G,\rho) is a nearring of all such transformations on G
which are constant on \rho blocks. Obviously M(G,\rho) is a special
case of sandwich nearring.
Our results are the following.
THEOREM 1. A non0symmetric and nonconstant nearring is primitive if and
only if it is
isomorphic to a dense subnearring either in some nearring A(V) or in some
nearring M(G,\rho) where the
equivalence \rho does not contain any nonzero congruence of the group G .
THEOREM 2. Every non0symmetric simple nearring is either constant or
isomorphic
to a dense subnearring of some sandwich nearring M(G,\rho) such that
\rho does not contain a nonzero congruence of the group G .
THEOREM 3. For a nearring N the following conditions are equivalent:
\begin{itemize}
(i) N is simple non0symmetric, nonconstant and satisfies the
descending chain condition for right ideals;
(ii) N is isomorphic to a sandwich nearring M(G,\rho)
where \vert G\vert =2 , G/\rho is finite and \rho does not contain
any of nonzero congruences of the group G .
\end{itemize}
Note that the implication(ii) => (i) in
Theorem 3 is due to P.Fuchs. Since 0symmetric
simple nearrings with certain minimality condition have been described
earlier, Theorem 3 completes the classification of simple nearrings
with such condition. In particular we have now a complete description of
finite simple nearrings.
The same methods have been applied to characterize the minimal ideals of
non0symmetric nearrings with minimality condition.
From Nearrings and Nearfields to Kloops
Helmut Karzel
In 1936 and 1937 H. Zassenhaus published his influential papers
``Kennzeichnung endlicher linearer Gruppen als Permutationsgruppen''
and ``Über endliche Fastkörper'' where we find the two theorems:
{\bf (1)}\quad Let (F,+,* ) be a nearfield, for a\in F , let
a^+ : F\to F; \; x\to a+x and a^{\displaystyle*} : F\to F; \;\;
x \to a * x , and let {\rm Aff} (F) := { a^+ o
b^{\displaystyle *} \mid a,b \in F, \; b \neq 0 } . Then \big(
{\rm Aff} (F) , o \big) is a permutation group which acts sharply
2transitive on the set F ; moreover F^+ := { a^+ \mid a\in F }
resp. F^{*{\displaystyle*}} := \big{ a^{\displaystyle *}
\mid a \in F^* := F \setminus { 0} \big} is a normal
subgroup resp. a subgroup of {\rm Aff} (F) which is isomorphic to
(F,+) resp. (F^*, * ) and {\rm Aff} (F,+,* ) = F^+ \rtimes
F^{*{\displaystyle *}} is a semidirect product.
{\bf (2)}\quad Let (F, \Gamma ) (with \Gamma \leq {\rm Sym} F) be a
sharply 2transitive permutation group. If the set F is finite,
then F can be turned in a nearfield such that \Gamma = {\rm Aff}
(F,+,* ) .
It is still an open problem if (2) is valid without the finiteness
condition. But (2) holds true for arbitrary sets F if we replace the
notation nearfield be neardomain. In a neardomain (F,+) is a loop
such that for all a,b \in F^* the map d_{a,b}^{\displaystyle
*} with d_{a,b} := (a+b) + \big( a+(b+1)\big) is an
automorphism of the loop (F,+) . Some years ago this observation
motivated W.~Kerby and H.~Wefelscheid to introduce the concept
``Kloop''. That is a loop (F,+) with the main property that for
any a,b\in F the map \delta _{a,b} :=\big((a+b)^+\big)^{1} o
a^+ o b^+ is an automorphism of the loop (F,+) . To each Kloop
there is associated the following affine group {\rm Aff} (F,+) := { a^+
o \varphi \mid a\in F,\; \varphi \in {\rm Aut} (F,+)} which can be
written only as an quasidirect product {\rm Aff}
(F,+) = F^+ \raisebox{2.5mm}{ \stackrel
{\textstyle \rtimes}{\scriptstyle Q} } \, {\rm Aut} (F,+) .
In the last time there were found many examples of proper Kloops and
it turned out that the theory on Kloops has many interesting
applications in physics and in geometry.
In my talk I like to review on this development of research which was
initiated by the papers of Zassenhaus and led finally to the theory of
Kloops.
Ideals in Nearrings of Formal Power Series over
Local Rings
Hermann Kautschitsch
Still up today all ideals of (R[[x]],+,o) , where o denotes the
operation of substitution are only known in the case that R is a field with
char(R) not equal 2 . Similar it is possible to determine almost
all ideals in the case that R is a local ring with maximal ideal M and
2 not in M . In some sense, (R[[x]],+,o) has also a local structure.
On Finite Circular Ferrero Pairs
WenFong Ke
In this talk, we show that any finite circular Ferrero pairs
(N,\Phi) must have \Phi metacyclic. A characterization
of finite circular Ferrero pairs, which generalizes the one
given by Modisett, will be described. Moreover, examples of
circular Ferrero pairs (N,\Phi) with nonabelian N will be
constructed.
Commutativity and Structure of Certain Classes of
Rings and Nearrings
Moharram Ali Khan
The aim of this paper is to study the connection between rings
and nearrings. In this direction we first prove that certain rings
satisfying the polynomial identity of the form [yx^m  x^ry^sx^t,x] = 0 ,
where s = s(x,y) > 1 , and m,r,t are positive integers larger than
1 depending on the pair of ring elements x and y , must be commutative.
Secondly, we establish a decomposition theorem for nearrings satisfying
yx = x^ry^sx^t , where s = s(x,y) > 1 , and r,t are positive integers
larger than 1 depending on the pair of nearring elements x and y .
Further, we look into the commutativity of such nearrings.
Moreover, it is also proved that under some additional hypothesis
such nearrings turned out to be commutative rings.
Finally, we provide some counter examples which show that the hypothesis
of our theorems are not altogether superfluous.
The Structure Group of Certain KLoops
Hubert Kiechle*
Angelika Konrad
Let (L,\oplus ) be a loop (i.e., there is 0\in L such
that 0\oplus a=a=a\oplus 0 and there are unique x,y\in L such
that a\oplus x=b=y\oplus a for all a,b\in L ). Then the condition
a\oplus (b\oplus x)=(a\oplus b)\oplus\delta_{a,b}(x) for
a,b,x\in L defines
a bijective map \delta_{a,b}:L\to L , the {\em precessionmaps}.
L is called a {\em Kloop} if the precessionmaps are automorphisms
of the loop, if it satisfies the
{\em {\sc Bol}identity}
a\oplus (b\oplus (a\oplus c))= (a\oplus (b\oplus a))\oplus c , and the
{\em automorphic inverse property}
\ominus(a\oplus b)=(\ominus a)\oplus (\ominus b) , where \ominus a
is uniquely defined by the equation a \oplus (\ominus a)=0 . The
Bolidentity implies the equality of left and right inverse, i.e.,
(\ominus a)\oplus a=0 as well.
The {\em structure group} \Sigma of L is the subgroup of {\rm Aut}(L)
generated by the precessionmaps.
The structure groups of certain Kloops, constructed from positive
definite symmetric or hermitian n\times n matrices will be presented.
Reflection Groups and Kloops
Angelika Konrad
The notion ``reflection group'' (\Gamma , \cal{D}) was introduced in
order to give group theoretical characterizations of absolute
planes. Here we consider ``reflection groups with midpoints''. The
motion group \Gamma together with the set \cal{D} of all
reflections in points of a Euclidean or hyperbolic geometry are
examples of reflection groups with midpoints. We show, that to each
reflection group with midpoints (\Gamma , \cal{D}) there corresponds
a Kloop (\cal{D},+) : let o \in \cal{D} be distinct and for a
\in \cal{D} let a' \in \cal{D} such that a=a'oa' ( a' is the
``midpoint'' of o and a ). Then the binary operation +: \cal{D}
\to \cal{D}, (a,b) \to a+b := a'oboa' turns \cal{D} into a Kloop.
 {1} Karzel, H. and Konrad, A.: Reflection Groups and
Kloops. J. Geom. {\bf 52} (1995), 120129.
Some Examples of Semiendomorphal Modules
Jan Krempa
Let R represent a commutative domain and X a unital R module.
It is well known that the set M = M_R(X) of all homogeneous
selfmaps of X is a nearring under standard operations. This nearring
contains the ring E = {\rm End}_R(X) .
In [1,2] the module X is called endomorphal if E = M and
semiendomorphal if M is a ring. In [1,2,3] the following result
is proved:
{\bf Theorem 1} Let R be a principal ideal domain and let {\cal M}
be the set of all maximal ideals of R .
 [1.]
If R > \cal{M} then any semiendomorphal and torsion free
R module is endomorphal;
 [2.]
If R = \cal{M} then there exists a torsion free R module of
rank 2 which is semiendomorphal but not endomorphal.
In the talk we are going to extend this result to some other classes of
domains. For example we show that there exists a local domain R and a
torsion free semiendomorphal R module which is not endomorphal.
 {1}
J. Hausen, {\it Abelian groups whose semiendomorphisms form a ring},
Proc. Curacao Conference, Marcel Dekker 1993.
 {2}
J. Hausen, J. A. Johnson {\it Centralizer nearrings that are rings},
J. Austral. Math. Soc.
 {3}
D. Niewieczerzal, {\it On semiendomorphal modules over Ore domains},
Proc. of Fredericton Nearring Conference.
Automorphisms of Loops
Alexander Kreuzer
In every loop (L,+) the equation a+(b+x) = (a+b) + \delta
_{a,b}(x) defines a bijection \delta _{a,b} : L \to L . Some
properties of (L,+) imply that \delta _{a,b} is an automorphism,
for example the equation a+\Big(b+(a+c)\Big)= \Big(a +(b+a)\Big) + c\; or \;\delta
_{a,b+a} = \delta ^{1}_{b,a} .
Finally the investigation shows that Kloops and Bruck loops are the same.
Topological Ngroups Where the Nearrings
are Real Nearrings
Kenneth D. Magill, Jr.
A {\em real nearring} is any topological nearring whose additive group
is the additive topological group of real numbers. For any topological
nearring N , a topological N group is a pair (G,\mu) where G
is a topological group and \mu is a continuous map from N \times G
into G such that the following two conditions are satisfied for all
a,b \in N and every c \in G :

\mu(a+b,c) = \mu(a,c) + \mu(b,c) .

\mu(ab,c) = \mu\bigl(a,\mu(b,c)\bigr) .
The topological groups under consideration here are the Euclidean
n groups R^n . We describe all the topological N groups (R^n,\mu)
where N is a real nearring. Specifically, we show that each such function
\mu is induced by continuous selfmap of R^n which must have certain
properties and it will become apparent that there are many such maps
\mu for which (R^n,\mu is a topological N group whenever n > 1 .
The situation is quite different in the case where n = 1 . It is shown
that for all real nearrings N which are not rings, there are
precisely three topological N groups (R,\mu) . In addition to the
two {\em standard} topological N groups where \mu is defined,
respectively, by \mu(x,v) = 0 and \mu(x,v) = xv , it is shown that
there is exactly one other and that there exist two real numbers a and b
with a \leq 0 , b \geq 0 and a^2 + b^2 \not= 0 such that, in this
remainding case, \mu is defined by \mu(x,v) = bxv for v \leq 0
and \mu(x,v) = axv for v > 0 .
Bicentralizer Nearrings and Entire Functions
Kenneth D. Magill, Jr.
Prabudh Misra*
Let A be any 2\times 2 matrix of real numbers and let N_A(R^2) be the
nearring of all continuous selfmaps of R^2 under pointwise addition and
composition which commute with A . By N_A(R^2) is meant the
{\it bicentralizer nearring of the group R^2 which is induced by A .}
We denote by sq
the function of a complex variable defined by sq(z) = z^2 and
the nearring of all entire functions of a complex variable under pointwise
addition and composition will be denoted by N_{ent} .
{\bf Theorem.} {\it Let A be a nonsingular 2\times 2 matrix of real
numbers which differs from the identity matrix. Then the following
statements are equivalent:}
 (2.3.2)
N_A(R^2)\cap N_{ent} consists exactly of those entire functions
whose Maclaurin expansions have only real coefficients,

We mention two immediate consequences of this theorem, the second one being
essentially an analytical result.
 {\bf Corollary 1.} {\it Suppose N_A(R^2) contains all the entire functions.
Then A is the identity matrix and N_A(R^2) = N(R^2) , the nearring
of all continuous selfmaps of R^2 .}

{\bf Corollary 2.} {\it Let f(z) = u(x,y)+iv(x,y) be an entire function
of a complex variable. In order for f^{(n)}(0) to be a real number for
each integer n\ge 0 , it is both necessary and sufficient that u(x,y)
be an even function in y and v(x,y) be an odd function in y , that
is u(x,y) = u(x,y) and v(x,y) = v(x,y) for all x,y\in R .}
Boolean Orthogonalities for Nearrings
Gordon Mason
Cornish has developed a theory of Boolean orthogonalities for
sets with an associated algebraic closure system of "ideals", and has
used it to derive results for reduced rings and for semiprime rings. We
apply these ideas to the nearring setting and in particular to
3semiprime nearrings. As a consequence we identify some nearrings
whose 3semiprime ideals are intersections of 3prime ideals. We also
discuss local ideals and normality conditions for nearrings with a
Boolean orthogonality.
Nearrings of Homogeneous Functions
Carl J. Maxson
Let V be a module over a ring R such that the maximal
submodules form an Ecover. We investigate the problem of determining
when homogeneous functions on V , linear on the cells of the cover, are
endomorphisms of V .
On the nearring counterpart of the matrix ring
isomorphism M_{mn}(R)\cong M_n(M_m(R))
Johan H. Meyer
When R is a ring with identity, it is wellknown (and easy
to show) that the n\times n matrix ring over the m\times m
matrix ring over R is isomorphic to the mn\times mn matrix ring over
R . The nearring situation is somewhat different to handle, because
matrices over nearrings are defined in a functional way (
Meldrum and van der Walt [1]) which has very little, if any,
resemblance to the traditional way of portraying matrices. However,
when the nearring happens to be a ring, the matrix nearring is
isomorphic to the familiar matrix ring.
The introduction of the concept of a matrix nearring (in 1984)
was soon followed by a series of papers covering a variety of
basic results on this concept. Notably absent
among these results is an answer to the natural question on the
existence or nonexistence of the matrix nearring isomorphism
under consideration. This problem has to date withstood a
considerable amount of effort  at one stage it was even strongly
believed that the two matrix nearrings are isomorphic if and
only if R is a ring. The question is finally settled in this
paper.
 [1] J.D.P. Meldrum and A.P.J. van der Walt, {\it Matrix nearrings},
Arch. Math., {\bf 47} (1986), 312  319.
Extensions of Hoehnkeradicals of Multioperatorgroups.
Rainer Mlitz
Given a Hoehnkeradical r with a property P on some class C of
multioperatorgroups, the question of the existence of extensions of r
with property P on a larger (universal) class U is investigated, with
special emphasis on minimal resp. maximal extensions. The properties P
considered are: idempotence, completeness, s heredity, r heredity,
heredity of the radical class, idealheredity, ADS. For many of them the
answer is affirmative concerning minimal extensions, but negative concerning
maximal ones.
Distributively Generated Subrings of Homogeneous Maps
Dorota Niewieczerzal
Throughout R represents an associative ring with 1\neq 0 and X a unital
R module. It is well known that the set M=M_R(X) of all homogeneous
selfmaps of X is a nearring under standard operations.
This nearring contains the ring E=End_R(X) of all R endomorphisms
of X.
If D is the set of all
distributive elements of M then D is a subring of M and
E\subseteq D \subseteq M.
>From [1,4] we know, in fact, that the following conditions are equivalent:
 E=M for every R module X;
 D=M for every R module X.
Under our notation a module X is endomorphal if E=D=M
and semiendomorphal if D=M (see [2,3]).
It is known that in general D=M does not imply E=D .
In the talk we are going to exhibite some other connections between
nearrings E,D and M determined by specific properties of R and X.
 {1} P. Fuchs C.J. Maxson G. Pilz, {\em On rings which homogeneous
maps are linear,} Proc. AMS, 112(1991).
 {2} J. Hausen, {\em Abelian groups whose semiendomorphisms form
a ring,} Proc. Curacao Conference, Marcel Dekker 1993.
 {3} J. Hausen J.A. Johnson, {\em Centralizer nearrings that are
rings,} J. Austral. Math. Soc.
 {4} J.Krempa D. Niewieczerza\l, {\em On homogeneous mappings of
modules,} Contributions to General Algebra 8(1992).
Nearrings Implemented in GAP
Christof Noebauer
GAP is a free computer algebra system, designed for
computations on (finite) groups.
We want to suggest an extension for GAP which provides functions
for doing computations on (finite) nearrings.
An algorithm by Yearby (dissertation 1973), implemented in GAP,
made it possible to provide a GAP library of small nearrings.
Complementary Radicals and Direct Sum Decompositions
of Nearrings
Werner A. Olivier
Sufficient conditions are given for a general regularity for nearrings
to sponsor the construction of an idempotent radical and its
complementary radical. In particular, we show that the radical F
(determined by the F regular property of {\sc Blair}) has the 0prime
radical Po as its complement. We discuss various conditions which
will ensure that the direct sum of F(R) and Po(R) is an essential
ideal in a nearring R . Examples of nearrings which essentially decompose
in this way are also given.
Composition Nearrings
Quentin Petersen
Not much work has been done on composition nearrings. Here we initiate
such investigations. Amongst others we give construction techniques
for double composition nearrings and we give two nonisomorphic Peirce
decompositions for a composition nearring (using both the multiplication
and composition).
Endomorphism nearrings of extraspecial p groups
generated by the inner automorphism and automorphism group
Gerhard Saad
Let p be a prime. A finite pgroup G is called {\it extraspecial} if
the center Z and the commutator subgroup
G' of G both have order p .
Since an extraspecial pgroup G is nilpotent of class 2 the nearring
I(G) generated by the inner automorphisms of G is a ring. We investigate
the ideal N = { \alpha \in {\rm I}(G)\,\,G\alpha \subseteq Z\;,\;Z\alpha
= 1 } of I (G) and determine the order of I (G) . Furthermore
we consider the nearring A (G) generated by the automorphisms of G
in the case that G is of odd exponent p .
Let K = A(G) and M = Ann_K(G/Z) .
We proof the following result: The quotient nearring K/M is isomorphic
to the endomorphism ring of the vector space G/Z .
Different Types of Uniform Ideals in NGroups
Bhavanari Satyanarayana
The aim of the present paper is to introduce and study different types
of uniform ideals (or N subgroups) in an N group G where N
is a zerosymmetric right nearring.
r uniform ideals (for r = 0,1,2,3,4 ) ideals (or N subgroups)
were introduced. Some examples and results were obtained on these concepts.
Compatibility
Stuart D. Scott
The study of compatible nearrings and N groups impinges
on many branches of mathematics.
First DCCR is used to outline theorems which are old and new, known
and not well known. The very much weaker conditions of DCCI and ACCI
also yield substantial results. Nearrings of polynomials are then
used to outline a far reaching structure theory for Omegagroups.
In this area very meaningful material on Omegagroups solubility
or nilpotency can be obtained.
A powerful tool available to study important classes of compatible
nearrings (e. g. prime compatible nearrings) is topology.
It is indicated how this arises.
This talk is aimed at initiating much greater interest in this
very large subject.
The right nearring of a right nearring
Kirby C. Smith
Let N be a right nearring with 1 . Then N enjoys the right
distributive law. So a right multiplication map on the set N determined
by an element of N is a group endomorphism of (N,+) . Let S be the set
of all right multiplication maps on N . Let R be the subnearring
of M(N) generated by S . The structure of R will be discussed
in various situations.
Sandwich Nearrings of Homogeneous Functions
Aletta Speegle
A study of the sandwich nearring of homogeneous functions,
N := M_F(D,W,\Psi) where F is a field, D is an F set,
W a vector space over F and \Psi : W > D
a homogeneous map, is initiated. The structure of N is related
to properties of the components W , D and \Psi . For example:
{\bf Theorem:} If dim_F W < \infty , then M_F(D,W,\Psi) is
simple if and only if \Psi is surjective and {\cal U} =
{W_1 \leq W  \Psi(W_1+y) = \Psi(y) for every y \in W}
= {{0}} .
Commutativity of Nearrings of Homogeneous Maps
Brink Van der Merwe
We will consider the commutativity of {\cal M}_R(V) :=
{ f: V > V  f(rv)=rf(v) for all v\in V and
r\in R }, where R is a commutative ring, V an R module,
and {\cal M}_R(V) is endowed with pointwise addition and function
composition as multiplication. For finitely generated modules we
will see that this is equivalent to {\cal M}_R(V) being a ring.
The General Radical Theory of Nearrings 
Answers to Some Open Problems
Stefan Veldsman
It is shown that in the variety of all, not necessarily
0symmetric nearrings, there are no nontrivial classes of nearrings
which satisfies condition (F), no nontrivial radical classes with
the ADSproperty and consequently no nontrivial idealhereditary
radical classes. It is also shown that any hereditary semisimple class
contains only 0symmetric nearrings.
Ordered Nearfields
Heinz Wähling
A subset P of a (left) nearfield F is called an order of F ,
if F^* := F  {0} = P \cup (P), P+P \subset P and
P* P \subset P . The induced total order relation <
( a < b \Leftrightarrow b  a \in P ) has the usual properties:
x < y => a + x < a + y for every a \in F and
ax < ay for all a \in P . But '' a \in P, x < y \Rightarrow
xa < ya '' cannot be proved. So we distinguish between {\em weak} and
{\em full} orders (W. Kerby, D. Gr??ger).
Every order < induces an interval topology T_< in the usual way.
We call < (and P ) a {\em V order}, if \nu : x \rightarrow x^{1}
is continuous at \infty :
x > \infty => x^{1} > 0
Let (F,P) an ordered skewfield and \kappa : F^* \rightarrow
{\rm Aut}~F , a > \kappa_a be a coupling of F
( a,b \in F^\ast => \kappa_a\kappa_b =
\kappa_{a* \kappa_a(b)} ) with \kappa_a(P) = P for every
a \in F^* . Then P is an order of the Dickson nearfield
F^\kappa ( F^\kappa = (F,+,o) ),
where ao x = a* \kappa_a(x) ). This method yields a lot of
weakly and fully ordered nearfields. We list some results:
 [1.]
There exists nearfield orders, which are not V orders and
consequently weak.
 [2.]
If (F,<) is a fully ordered nearfield, then T_< is a nearfield
topology.
 [3.]
There exists weakly ordered nearfields (F,<) with nearfield
topologies and such with nonnearring topologies T_< .
 [4.]
If < is a V order of F , then T_< is a V topology:
(F,+,T_<) is a topological group, the multiplication is continuous
in (0,0) , every left translation x \rightarrow ax is continuous,
and for given neighbourhoods U , V of 0 there exists a neighbourhood
W of 0 such that W* (F \ U)^{1} \subset V .
 [5.]
Different V topologies T_1, ..., T_n of a nearfield are
always independent: \emptyset \not= A_i \in T_i (i = 1, ...,n) \
=> A_1 \cap ... \cap A_n \not= \emptyset .
 [6.]
Every fully ordered nearfield posesses a fully ordered continuous closure.
 [7.]
Let (F,<) be a nonarchimedean ordered nearfield. Then
R := {x \in F  n < x < n for a suitable n \in N}
is a convex valuation nearring of F ( x \in F \ R =>
x^{1} \in R ), and v : x > xR is a nontrivial valuation
(in the sense of T. Rado and K. Mathiak), which is comparable
with < ( 0 <= a <= b ==> v(a) <= v(b) ).
 [8.]
Let (F,<) be a fully ordered nearfield.
Then the solution of ax  bx = c depends  if it exists 
continuously on a , b , c ( a \not= b ).
Special Quasigroups and Steiner Systems
Herbert Zeitler
Idempotent quasigroups IQG(v) such that any two elements
''span up'' a subquasigroup IQG(k), k <= v are denoted
here by kIQG(v) .
Two constructions for kIQG(k) are given, discussed and compared
with one another (automorphism groups, isomorphism, ...)
Using these kIQG(k) and Steiner systems S(2,k,v) some
kIQG(v), k < v are constructed.
July 20, 1995