# 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 near-rings 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 near--ring (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^2-1), the solution of this particular problem provides a method for simplifying systems of the so called sine-cosine equations. This simplification is very important for solving the inverse kinematic problem in robotics.

## Polynomial Near-Rings

### Scott W. Bagley

In 1987, van der Walt overcame the difficulties caused by the lack of one distributive property in near-rings in order to create matrix near-rings with the property that when the near-ring is a ring, the usual matrix ring and the matrix near-ring agree. In this paper we develop a new polynomial near-ring with coefficients from a near-ring and derive some of its properties. Similar to van der Walt's work, when the coefficient near-ring is a ring, our ploynomial near-ring coincides with the usual polynomial ring.

## On Derivations in Nearrings, II

### Howard E. Bell

Bell and Mason have studied commutativity in 3-prime 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 zero-symmetric 3-prime 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 semi-nearring (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 field-like 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 near-ring, 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 subnear-rings,

{\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.

### Geoffrey L. Booth* Nico J. Groenewald

Buys and Gerber studied special radicals in Andrunakievich varieties of Omega-groups. While they obtained good results, this excluded many useful varieties, such as zero-symmetric near-rings, 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 zero-symmetric near-rings, gamma rings and rings with involution.

## Bi-ideals and Quasi-ideals in Categories

### Suzette Botha

Quasi-ideals and bi-ideals 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 Success--Failure} 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 non-trivial factor sets. Their multiplicative structure is a semidirect product of the {\em one-units} with a group extension.

## Simplicity of the Centralizer Near-Ring 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 near-ring 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 near-ring then either N is simple with non-zero idempotent is a left identity or the intersection of non-zero ideals of N is without non-zero idempotents. It has been shown that a subdirectly irreducible RSC near-ring with a distributive element is a near-field. Necessary and sufficient conditions are obtained for a RSC near-ring 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 Semi-endomorphisms of Abelian Groups and Transformation Nearrings

### Yuen Fong

We investigate in this paper the subnearring of G generated by the semi-endomorphisms of the given abelian group G. The semi-endomorphisms 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 1-transitive.
• [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.

### 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 "Near-rings, Geneses and Applications":

Let R be an arbitrary commutative ring with identity. It is well known that (R[X],+, o) is a right near-ring 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 near-ring (R[X_1,...,X_n],+, o) . What are some significant problems concerning these near-rings?. 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 near-rings ?...etc.

One purpose of this talk is to try to give partial answer to this kind of questions.

## Nearrings and Involutions

### H. E. HeatherlyE.S.K. Lee Richard Wiegandt*

Involution can be defined for universal algebras in a very general manner. In particular, on a near-ring (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 near-ring is distributive. However, if the involution \star is a multiplicative automorphism, the near-ring need not be distributive. If (N,+,*,\star) is subdirectly irreducible, then either (N,+,*) is a subdirectly irreducible near-ring, or (N,+,*) is a subdirect sum of two subdirectly irreducible near-rings and the involution \star is the exchange involution. Examples are provided and the subdirectly irreducible distributive near-rings 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 K-loop ([2]),\\
• iii) (F,+)\times (F,+) is a fibered incidence loop ([3]).

• {1} H. Hotje: Allgemeine kinematische Räume. Mitt. Hamburger Math. Ges. 12 (1991), 785-791.
• {2} H. Karzel: Raum-Zeit-Welt und hyperbolische Geometrie. Vorlesungsausarbeitung von A. Konrad, TUM-Beiträge zur Geometrie und Algebra 29, 1994.
• {3} P. Tancke: Kollineationen und Anordnungen in 2-Strukturen mit Rechtecksaxiom. Dissertation Hannover 1989.

## K--loops 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 Non-Zerosymmetric Nearrings

### Kalle Kaarli

Given a near-ring 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 near-ring N {\it primitive} if it admits a faithful strictly simple N -group.

In the case of 0-symmetric near-rings 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 near-ring of all affine transformations on V . If \rho is an equivalence relation on a group G then M(G,\rho) is a near-ring of all such transformations on G which are constant on \rho -blocks. Obviously M(G,\rho) is a special case of sandwich near-ring.

Our results are the following.

THEOREM 1. A non-0-symmetric and nonconstant near-ring is primitive if and only if it is isomorphic to a dense subnear-ring either in some near-ring A(V) or in some near-ring M(G,\rho) where the equivalence \rho does not contain any nonzero congruence of the group G .

THEOREM 2. Every non-0-symmetric simple near-ring is either constant or isomorphic to a dense subnear-ring of some sandwich near-ring M(G,\rho) such that \rho does not contain a nonzero congruence of the group G .

THEOREM 3. For a near-ring N the following conditions are equivalent: \begin{itemize}

• (i) N is simple non-0-symmetric, nonconstant and satisfies the descending chain condition for right ideals;
• (ii) N is isomorphic to a sandwich near-ring 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 0-symmetric simple near-rings with certain minimality condition have been described earlier, Theorem 3 completes the classification of simple near-rings with such condition. In particular we have now a complete description of finite simple near-rings.

The same methods have been applied to characterize the minimal ideals of non-0-symmetric nearrings with minimality condition.

## From Nearrings and Nearfields to K--loops

### 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 2--transitive 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 2--transitive 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 K--loop''. 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 K--loop 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 K--loops and it turned out that the theory on K--loops 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 K--loops.

## 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

### Wen-Fong 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 K--Loops

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 precession-maps}.

L is called a {\em K-loop} if the precession-maps 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 Bol-identity 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 precession-maps.

The structure groups of certain K-loops, constructed from positive definite symmetric or hermitian n\times n -matrices will be presented.

## Reflection Groups and K--loops

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 K--loop (\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 K--loop.
• {1} Karzel, H. and Konrad, A.: Reflection Groups and K--loops. J. Geom. {\bf 52} (1995), 120--129.

## Some Examples of Semi-endomorphal 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 near-ring under standard operations. This near-ring contains the ring E = {\rm End}_R(X) . In [1,2] the module X is called endomorphal if E = M and semi-endomorphal 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 semi-endomorphal 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 semi-endomorphal 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 semi-endomorphal R -module which is not endomorphal.

• {1} J. Hausen, {\it Abelian groups whose semi-endomorphisms form a ring}, Proc. Curacao Conference, Marcel Dekker 1993.
• {2} J. Hausen, J. A. Johnson {\it Centralizer near-rings that are rings}, J. Austral. Math. Soc.
• {3} D. Niewieczerzal, {\it On semi-endomorphal modules over Ore domains}, Proc. of Fredericton Near-ring 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 K--loops and Bruck loops are the same.

## Topological N-groups 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 near-ring setting and in particular to 3-semiprime near-rings. As a consequence we identify some near-rings whose 3-semiprime ideals are intersections of 3-prime ideals. We also discuss local ideals and normality conditions for near-rings 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 E-cover. 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 well-known (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 near-ring situation is somewhat different to handle, because matrices over near-rings 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 near-ring happens to be a ring, the matrix near-ring is isomorphic to the familiar matrix ring.

The introduction of the concept of a matrix near-ring (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 non-existence of the matrix near-ring 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 near-rings 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 near-rings}, Arch. Math., {\bf 47} (1986), 312 -- 319.

## Extensions of Hoehnke-radicals of Multioperator-groups.

### Rainer Mlitz

Given a Hoehnke-radical r with a property P on some class C of multioperator-groups, 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, ideal-heredity, 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 near-ring under standard operations. This near-ring 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 semi-endomorphal 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 near-rings 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 semi-endomorphisms form a ring,} Proc. Curacao Conference, Marcel Dekker 1993.
• {3} J. Hausen J.A. Johnson, {\em Centralizer near-rings 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 0-prime 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 non-isomorphic 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

Let p be a prime. A finite p-group G is called {\it extraspecial} if the center Z and the commutator subgroup G' of G both have order p .

Since an extraspecial p-group 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 N--Groups

### 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 zero-symmetric 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 Omega-groups. In this area very meaningful material on Omega-groups 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 near-ring 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 subnear-ring 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 0-symmetric near-rings, there are no non-trivial classes of near-rings which satisfies condition (F), no non-trivial radical classes with the ADS-property and consequently no non-trivial ideal-hereditary radical classes. It is also shown that any hereditary semisimple class contains only 0-symmetric near-rings.

## 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 non-nearring 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 non-archimedean 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 k-IQG(v) .

Two constructions for k-IQG(k) are given, discussed and compared with one another (automorphism groups, isomorphism, ...)

Using these k-IQG(k) and Steiner systems S(2,k,v) some k-IQG(v), k < v are constructed. July 20, 1995