Abstracts
In case of multiple authors, the speaker is
indicated by a star^{*}.
The list is sorted by the last name of the speaker.
Bounds on the size of Ngroups that lead to
decidability results
Erhard Aichinger, Linz, Austria
Given a finite zerosymmetric nearring with identity N, we ask whether N has any of the following properties:
In particular, we investigate whether there is
an algorithm that tells whether a given N
has any of these properties.
The fact that such an algorithm exists is based
on the following theorem:
Theorem.
Let N be a finite nearring, and let m : = N.
If there is a group G such that
N is isomorphic to
I(G), then
there is a group H with H £ m^{2m+1}
such that N is isomorphic to I(H).
Using a similar result, we prove that is decidable
whether a given nearring is compatible in the sense
of S.D.Scott.
Recently, we have proved that the variety of nearrings is generated by its finite members. We explain which information this gives on the decidability of certain wordproblems for nearrings.
On prime nearrings with semiderivations
Nurcan Argaç, Izmir, Turkey
Throughout this paper N always stands for a right nearring. An additive map d:N® N is a derivation if d(xy) = xd(y)+d(x)y for all x,y Î N  or equivalently ( cf. [6] ) that d(xy) = d(x)y+ xd(y) for all x,y Î N . The study of derivations of nearrings was initiated by H.E. Bell and G. Mason in 1987 [3], but up to now only a few papers on this subject in nearrings were published ( see [1], [2] , [4] and [5] ). Now we introduce the notion of semiderivation of a nearring N as follows.
Definition . An additive mapping d:N® N is a
semiderivation if there exists a function a: N® N such that

Theorem 1
Let N be a semiprime right nearring and I be a subset of N such that 0 Î I and IN Í I. Let d be a semiderivation on N with
associated function a such that a(I) = I and a(xy) = a(x)a(y) for all x, y Î I.
(i) If d
acts as a homomorphism on I, then d(I) = {0}.
(ii) If d
acts as an antihomomorphism on I and a(0) = 0 , then
d(I) = {0}.
Theorem 2 Let N be a prime nearring , I a nonzero semigroup ideal of N and d a nonzero semiderivation of N. If d(x +y xy) = 0 for all x,y Î I, then ( N,+ ) is abelian.
Freiman Nearrings
H. Bell, St. Catharines, Ontario
A nearring is called a Freiman nearring if the square of each 2subset contains at most 3 distinct elements. We discuss sufficient conditions for a Freiman nearring to be additively or multiplicatively commutative.
PBIBDs from weakly divisible nearrings and related codes
Anna Benini, Brescia
In previous papers a class of finite nearrings called "weakly divisible" (wdnearrings) is defined, studied [1] and constructed [2] and a family of partially balanced incomplete block designs (PBIBDs) is derived [3]. Now the design parameters and the design incidence matrix are investigated in order to study the related codes: row code, column code and their linear hulls.
Radical parallelism: A geometric interpretation of the Jacobson
radical of a ring
Andrea Blunck, Hamburg
We introduce a relation called radical parallelism on the
projective line over a
ring R in a purely synthetic way, using only the notion of distant
points. It turns out that this relation is an
equivalence relation that is closely related to the Jacobson radical
rad(R). In particular, rad(R) can be identified
with a radical parallel
class in the affine trace of a fixed point at infinity.
(Joint work with Hans Havlicek, Vienna University of Technology.)
Primeness and Radicals in Nearrings of Continuous Functions
G.L. Booth, University of Port Elizabeth, Nelson Mandela Bay,
South Africa
Let (G,+) be a group. The set of all zeropreserving selfmaps of G is denoted M_{0}(G), i.e. M_{0}(G): = {a:G® G  a(0) = 0}.
Let (G,+) be a topological group. The set of all continuous zeropreserving selfmaps of G is denoted N_{0}(G), i.e. N_{0}(G): = {a:G® G  a is continuous and a(0) = 0}. Note that if G has the the discrete topology, M_{0}(G) = N_{0}(G).
All topological groups in this talk will be T0, and hence completely regular. It is clear that M_{0}(G) and N_{0}(G) are (right) zerosymmetric nearrings with respect to the operations pointwise addition and composition of functions. Various generalizations of primeness exist in nearring literature. We will be considring the classical definition, as well as 3primeness (Groenewald, 1989) and equiprimeness (Booth, Groenewald and Veldsman, 1991). We will also consider two generalisations of stronlgly prime, the first of which is known by the name, and the second is called strongly equiprime. The prime and equiprime radicals of N will be denoted P(N) and P_{e}(N), respectively.
Veldsman (1992) has noted that M_{0}(G) is always equiprime. The next two results shed some light on the situation for N_{0}(G).
Definition 1 A topological space X is called 0dimensional if the topology on X has a base consisting of clopen (i.e. both open and closed) sets.
Proposition 2 Let G be a topological group with more than one element which is either 0dimensional or arcwise connected. Then N_{0}(G) is equiprime.
Proposition 3 Let G be a disconnected topological group, with open components which are arcwise connected and which contain more than one element. Let H be the component of G which contains 0, I: = {a Î N_{0}(G)  a(G) Í H} and J: = {a Î N_{0}(G)  a(H) = 0}. Then P(N_{0}(G)) = P_{e}(N_{0}(G)) = IÇJ.
Remark 4 In this case P(N) ¹ 0, so N_{0}(G) is not semiprime.
Proposition 5 Let G be a 0dimensional topological group with more than one element. Then
Proposition 6 Let on G be an arcwise connected topological group whichhas a base B consisting of arcwise connected open sets (i.e. G is locally arcwise connected). Then N_{0}(G) is not strongly prime (and hence not strongly equiprime).
Remark 7 It is not known whether there exists a nondiscrete topological group G such that N_{0}(G) is strongly prime.
Let X and G be a topological space and a topological group respectively, and let q:G® X be a continuous map. The sandwich nearring N_{0}(G,X,q) is the set {a:X® G  a is continuous and aq(0) = 0}. Addition is pointwise and multiplication is defined by a·b: = aqb. If the topologies on X and G are discrete we denote the nearring by M_{0}(G,X,q). In this talk we will assume that both G and X have more than one element.
Proposition 8 (Veldsman, 1992) M_{0}(G,X,q) is equiprime if and only if q is a bijection. In this case M_{0}(G,X,q) @ M_{0}(G).
Proposition 9 Suppose that X is a 0dimensional, T_{0} space and G is a 0dimensional, T_{0} topological group. Then N_{0}(G,X,q) is equiprime if and only if q is injective and c(q(G)) = G.
Proposition 10 Let X and G be a completely regular topological space and an arcwise connected topological group, respectively, and let g:G® X be a continuous, injective map. Then the following are equivalent:
(a) cl(g(G)) = X.
(b) N_{0}(G,X,g) is equiprime.
(c) N_{0}(G,X,g) is 3semiprime.
Difference Methods and Ferrero Pairs
Tim Boykett^{*} and Peter Mayr
There are several constructions of designs from planar nearrings. In this paper, we present a generalised theory for constructing BIBdesigns from a group G and a group F of fixed point free automorphisms of this group. Using this theory we can unify several previous constructions of BIBdesigns. In particular we can apply Sun's results on lines and segments in planer nearrings to this more general situation.
Generalized Centers of Nearrings
G. Alan Cannon, Hammond, LA, U. S. A.
Let N be a right nearring. Denote by C(N) the multiplicative center of N and by N_{d} the set of left distributive elements of N. In general, C(N) need not be closed under the addition of N. However, the generalized center of N, GC(N) = {a Î N  an_{d} = n_{d} a for all n_{d} Î N_{d}}, is always a subnearring of N containing C(N). We initiate a study of the structure of GC(N) as well as when GC(N) equals C(N) for certain classes of nearrings.
Finite affine complete groups
Jürgen Ecker, Linz
A mapping on a group G which respects all congruence relations of G is called a compatible function. The compatible functions on a group G form a subnearring C(G) of the nearring M(G) of all mappings on G. The inner automorphism nearring I(G) is a prominent subnearrings of C(G). A group G is called 1affine complete if I(G) = C_{0}(G). In my talk I describe a few methods needed to find the 1affine complete Frobenius groups and the 1affine complete generalised dihedral groups. I show which quotients of a group inherit 1affine completeness, and I present an example of a 1affine complete group G such that the direct product G×G is not 1affine complete.
Jacobson Nearrings
Nico Groenewald
In this talk we construct special radicals using class pairs of nearrings. We establish necessary conditions for a class pair to be a special radical class. We then define Jacobsontype nearrings and show that most cases the class of all nearrings of this type are special radical classes. Subsequently we investigate the relationship between each Jacobsontype of nearring and the corresponding matrix nearring.
On class pairs and radicals in nearrings
Lungisile Godloza
In this talk we inestigate the properties of classes nearrings constructed usind a class pair (M_{1}:M_{2}) of nearrings. We establish the relationship between a class pair (M_{1}:M_{2}) and the radical pair (r_{1}:r_{2}) constructed using the preradicals r_{1} and r_{2} associated with the classes M_{1} and M_{2} respectively. Conditions under which (r_{1}:r_{2}) = (Sr_{2}:Sr_{1}), where Sr_{i} is the semisimple class of r_{i}, i = 1,2, and is a KuroshAmitsur radical class are established.
Classification of type0 Rgroups and the nilpotence level of
the sradical
J.F.T. Hartney, Johannesburg, South Africa
For d.g. nearrings R, Laxton and Machin [1] constructed an antiradical, which they called the critical ideal by using a classification of Rgroups of type0 into two classes H_{1} and H_{2}. Two assumptions were made with regard to H_{1} and H_{2}. In addition, H_{1} was assumed to be the maximal class for which these assumptions were valid. The Rgroups of type0 were all subfactors of a faithful Rgroup W and the critical ideal, Crit(R), was defined as the annihilator of the class H_{2}. Crit(R) is independent of W as long as H_{1} is a maximal class.
Our classification is slightly different and extends to arbitrary zerosymmetric nearrings R. For nearrings with DCCL we define C(W) as the annihilator of one of the classes obtained from a faithful Rgroup W. For W = R, C(R) = Soi(R), the socleideal of R. For nearrings with DCCR Crit(R) = C(W) = Soi(R) for any faithful Rgroup W. We then investigate how the classification is affected as we move along a nilrigid series for R. This enables one to view the nilpotence level (cf.[2]) of the sradical J_{s}(R) via faithful Rgroups.
[1] LAXTON, R.R. and MACHIN, A.W. On the decomposition of
nearrings. Abh. Math. Sem. Univ. Hamburg 38 (1972), 221230.
[2] HARTNEY J.F.T. On the decomposition of the sradical of a
nearring. Proc. Edinb. Math. Soc. 33 (1990) 1122.
The Minimal and Maximal Right Ideals in the Nearring of Polynomials
Lucyna Kabza, Hammond, LA, USA
Let R be a commutative ring with identity. Then (R[x],+,°) is a nearring of polynomials with the usual addition and function composition. The maximal ideals of this nearring were studied by J. R. Clay and D. K. Doi (with R = F, F an arbitrary field) and by H. Kautschitsch, who determined all maximal ideals over an arbitrary commutative ring R with identity with more than two elements. In this paper the minimal and maximal right ideals of R[x] are studied and some necessary properties of these ideals are established. Moreover, some maximal right ideals of Z_{2}[x] are explicitly given.
Ideals in the nearring of formal power series over local rings
Herrmann Kautschitsch
In this talk an overview about a special class of nearring ideals, which includes prime ideals, is given. It is also shown, how they are related to the ideal structure of the underlying coefficient ring and how they are connected to the ring ideals.
On Recent developments of planar nearrings
WenFong Ke, Tainan
In this talk, I shall present some ongoing researches on planar nearrings. For example, we have obtained a classification of isomorphic planar nearrings constructed from a fixed Ferrero pair, and the description of the automorphism group of some designs constructed. Some emphases will also be put on circular planar nearrings including the realization of the regular group of automorphisms, and the ``overlapping" of the graphs.
On Finite Goldie Dimension of Mn(N) Group Nn
Syam Prasad Kuncham, Manipal Institute of Technology, India
In this paper we studied the concepts: linearly independent elements and ulinearly independent elements in Ngroup G where N is a zero symmetric right nearring. We proved that the Goldie dimension of the NGroup N is equal to that of the M_{n}(N)group N_{n} where M_{n}(N) is the matrix nearring.
Radicals and Antiradicals of Nearrings and their associated
Matrix nearrings.
J.F.T. Hartney and A.M. Matlala^{*}, Johannesburg, South Africa
The socle ideal of R , Soi(R), is an antiradical in the sense that it annihilates at least one Jacobson radical, in particular J_{[1/2]}(R) . We show that Soi( M_{n} (R) ) Í ( Soi(R) )^{*} , where R is a zerosymmetric nearring with identity and M_{n} (R) the matrix nearring associated with R . It is not known whether the strict inclusion holds. The importance of the socleideal, for our purposes, is in investigating the nilpotency of the sradical, J_{s} (R) , since Soi(R) is a nonnilpotent part of a nearring. It is not known if (J_{0} (R))^{+} Í J_{0} ( M_{n} (R) ) for general nearrings, as pointed out by Meldrum and Meyer [5]. We prove that (J_{s} (R))^{+} Í J_{s} ( M_{n} (R) ) for nearrings satisfying DCCR. In fact, the example given in [5] is a special case of our result as the given nearring is such that J_{s} (R) = J_{0} (R) . By using a special nilrigid series, as defined by Scott [6], a unique minimal ideal A such that J_{s} (R) / A is nonzero nilpotent was defined under suitable chain conditions by Hartney [3]. We call this ideal A the ssocle of R . For finite nearrings R we show that A^{+} Í A , where A is the ssocle of M_{n} (R) .
Completeness for concrete nearrings
Dragan Masulovi\'c, Novi Sad (Serbia and Montenegro)
In this talk a completeness criterion for nearrings over a finite group is derived using techniques from clone theory. The relationship between nearrings and clones containing the group operations of the underlying group shows that the unary parts of such clones correspond precisely to nearrings containing the identity function. Rosenberg's characterization of maximal clones is then applied to describe maximal nearrings containing the identity map, while maximal nearrings not containing the identity are described using typical nearring methods. This finally provides us with a completeness criterion. We apply this criterion to show that if the order of G is large then with high probability the set containing a single bijection is complete.
This is a joint work with E. Aichinger (Linz), R. Pöschel (Dresden) and J. S. Wilson (Birmingham).
NearRings of Mappings
Carl J. Maxson
In this talk we discuss some old and some new avenues of investigation related to nearrings of mappings.
Polynomial functions on linear groups
Peter Mayr, Linz, Austria
For the nonsolvable linear, unitary, symplectic, and orthogonal groups over finite vectorspaces and related groups G, we consider the following problems: Determine the size of the inner automorphism nearring I(G). Do we have I(G) = A(G)? Do we have I(G) = E(G)?
Our results are based on a description (jointly done with E. Aichinger) of the unary polynomial functions on those nonabelian finite groups G that satisfy the following conditions: G¢ = G¢¢, G/Z(G) is centerless, and there is no normal subgroup N of G with G¢ÇZ(G) < N < G¢.
Some recent developments in group nearrings
J. D. P. Meldrum, Edinburgh
Following the definition of matrix nearrings in 1986 by Meldrum and van der Walt, which has proved of interest, a definition of group nearrings was presented by Le Riche, Meldrum and van der Walt in 1989 using similar ideas. Here Group Nearrings are discussed emphasizing the parallels between them and matrix nearrings. The use of these ideas to develop a theory of polynomial nearrings is presented as developed by Bagley and Farag.
The main part is an account of work with J. H. Meyer on ideals in group
nearrings. As with matrix nearrings, the relation between the ideals in
the underlying nearring and those in the group nearring is much more
complicated than in the case of rings. Added to that the situation for group
rings is itself more complicated than for matrix rings and this is reflected
in the nearring case. To each ideal in the underlying nearring corresponds
a lattice of ideals in the group nearring and some information about these
lattices is given. There are also ideals in the group nearring, the
exceptional ideals, which do not fit into these lattices.
One of the most interesting of these exceptional ideals is the augmentation
ideal. This is characterised and in all but a few exceptional cases,
necessary and sufficient conditions are obtained for the augmentation ideal
to be nilpotent. These involve generalised distributivity, the arithmetic
structure of the group and the characteristic of the nearring. The proofs
involve a variety of techniques.
Modules over group nearrings
JH Meyer, Bloemfontein, South Africa
The notion of a module over a group nearring is discussed. Interesting results concerning simplicity are given. This leads to unexpected interplay between the Jacobson radicals of the nearring R and the group nearring R[G].
Seminearrings of bivariate polynomials
Kent M. Neuerburg, Hammond, Lousiana, USA
The idea of the composed product of univariate polynomials may be generalized to bivariate polynomials. Using these rootbased compositions, we define operations on bivariate polynomials analogous to addition and composition of univariate polynomials. Using these operations we investigate the seminearring of bivariate polynomials looking at its properties and internal algebraic structures.
Automorphism groups emitting local endomorphism nearrings. II
Gary L. Peterson, Harrisonburg, Virginia USA
A (left) nearring R with identity is local if the set of elements of R that do not have right inverses is a right Rsubgroup of R. The study of local endomorphism nearrings of groups was initiated by Carter Lyons and the author [Proc. Edinburgh Math. Soc., 31(1988), 409414; MR 89m:16077]. Several other papers dealing with local endomorphism nearrings by the author appeared thereafter. The primary focus of this note is to extend the results of one of these papers of the same title [Proc. Amer. Math. Soc., 105(1989), 840843; MR 89k:20051].
Loopnearrings
Silvia Pianta, Brescia
A loopnearring is a triple (L,+,·) where (L,+) is a loop, with neutral element 0, (L^{*},·) is a semigroup (L^{*}: = L\{0}) and the left (right) distributive law holds.
This notion arises naturally from the study of semigroups of endomorphisms of loops or, more generally, considering the set M(L) of all mappings from L to L. The interest of this kind of structure lies not just in the possibility of extending a number of wellknown results already valid for nearrings to the case where the addition is not associative, but also, and expecially, in the discovery of a wide range of choices for such generalization.
The beauty of it is that, at least for some extent, each different choice fits with some suitable interesting class of examples, coming from the families of Aloops, Kloops or Moufang loops which take most of their significance from the foundation of geometry and the theory of permutation groups.
A right radical for right d.g. nearrings
J.F.T. Hartney and D.S. Rusznyak^{*}, Johannesburg, South Africa
We discuss a Jacobsontype radical, ^{r}J_{0}(R)), for right d.g. nearrings. ^{r}J_{0}(R) is defined using annihilators of certain d.g. right Rgroups, which are the equivalent of type0 Rgroups from left representation. We know for d.g. nearrings R satisfying the DCC for left Rsubgroups of R that ^{r}J_{0}(R) contains the left 0radical J_{0}(R). For such d.g. nearrings we have J_{0}(R) Í ^{r}J_{0}(R) Í J_{2}(R). Since J_{0}(R) Í J_{s}(R) Í J_{2}(R), our main focus is connections between J_{s}(R) and ^{r}J_{0}(R).
Linearly Independent Elements in Ngroups with Finite
Goldie Dimension
Bhavanari Satyanarayana^{*} (Nagarjuna
University, India) and Syam Prasad Kuncham (Manipal Institute of
Technology, India)
The concepts ``linearly independent elements'' and ``ulinearly independent elements'' in Ngroup G where N is a nearring, were introduced and studied. Few important results that exist in the theory of Vector Spaces were obtained for Ngroups. An Ngroup G is said to have Finite Goldie Dimension if it contains no direct sum of infinite number of nonzero ideals. We proved that (i) if G, G_{1} are two Ngroups, f : G® G_{1} is an isomorphism, and u_{i} belongs to G (for i = 1 to n), then u_{i}'s are ulinearly independent in G if and only if f(u_{i})'s are ulinearly independent in G_{1}; (ii) if G has finite Goldie dimension, then K is a complement ideal of G if and only if there exist ulinearly independent elements u_{i} < n and u_{i}'s (for i = 1 to k) are ulinearly independent elements in G, then there exists u_{k+1},..., u_{n} in G such that u_{i}'s (for i = 1 to n) are ulinearly independent elements of G which spans G essentially.
A.M.S. Subject Classification: 16A55, 16A66, 16A76 Key Words: Goldie dimension, uniform ideal, complement, linearly independent elements, essentially spanning subset, ulinearly independent elements.
The Zconstrained Conjecture
S. D. Scott
The talk starts by introducing compatibility. It is asked if, there is some reasonable condition for faithful compatible Ngroups to be unique. A fairly well known theorem is stated. It uses rconstraint. This result can be made very much stronger by using Zconstraint. That is what this talk is about (the Zconstrained conjecture). Zconstraint yields nine equivalent conditions. In attempting to prove the Zconstrained conjecture, faithful hulls arise. These tend to tell us what the real nature of the theorem must be. How this is proved is outlined. Fitting factors are involved. They are subdirect sums. Proving the uniqueness of their components is a big step toward establishing the Zconstrained conjecture. The other big step is proving that, in the Zconstrained case, subdirect sums are unique.
Centralizer nearrings, matrix nearrings and cyclic pgroups
Kirby C. Smith, College Station
If G is a finite group and A is a group of automorphisms of G, then it is known that the matrix nearring \mathbbM_{m}(M_{A}(G); G) is a subnearring of the centralizer nearring M_{A}(G^{m}) for every m ³ 2. Conditions are known under which \mathbbM_{m}(M_{A}(G); G) is a proper subnearring of M_{A}(G^{m}), and if A and G are abelian, then conditions are known which imply the equality \mathbbM_{m}(M_{A}(G); G) = M_{A}(G^{m}). In this paper we characterize the groups A of automorphisms of a cyclic pgroup G for which this equality holds. We also show that, for every group A of automorphisms of a cyclic pgroup G, either all the nonzero orbits of G are of unique type or none of the orbits of G is of unique type if p is odd, and there is a third possibility if p = 2, namely precisely one of the orbits of G is of unique type.
Anwendungen der Computer Graphik in der Geometrie
Grozio Stanilov
Mit jedem Dreieck verbinde ich 6 merkwuerdige Punkte, so dass bekommt man zwei neue Dreiecke, die homethetisch zu dem vorgegebenen sind. Auch 6 Ellipsen werden eingefuehrt  3 davon eingeschriebene und 3 umgeschriebene sind, 6 Hyperbeln begleiten das Dreieck und noch 6 Parabeln. Alle besondere Punkten und Geraden dieser Kurven zweiter Ordnung sind neue merkwuerdige Objekten in der Dreiecksgeoemtrie. Alle Objekten werden mittels Maple 8 visualisiert.
Cohomology of near rings and applications
Mirela Stefanescu, Ovidius University, Constantza, Romania
The cohomological methods in studying algebraic structures give the possibility of constructing new structures and to deduce some properties of them from properties of associated groups. This is what we have done in the paper for near rings, taking into account the previous results in the field. We study also the pseudohomomorphisms of groups and the applications of near rings to group cohomology.
Identify the elements of a finite dimentional center algebra
over field through WedderburnArtin's theorem
Yohanes Sukestiyarno, Semarang State University, Indonesia
The paper is concerned with the WedderburnArtin's theorem (Passman, 1991). This theorem states that ``if R is a simple Artinian ring, then there exists a unique division ring D and a positive integer number n such that R @ D_{n}''. The Change problem of R in this theorem into other ring is discussed. The Changing of the simple Artinian R into A a finite dimensional center simple algebra over a field F will be presented so that the isomorphic of A and D_{n} can still be fulfiled.
The next presentation will be shown that a finite dimentional center algebra D over a field real number \mathbb R and \mathbb R itself, or D and integer C, or D and a quaternion Q over \mathbb R are isomorphic. Based on this result, we have D_{n} and M_{n×n}(\mathbb R), D_{n} and M_{n×n}(C), or D_{n} and M_{n×n}(Q) are isomorphic. So, we regard the elements of M_{n×n}(\mathbb R), or M_{n×n}(C), or M_{n×n}(Q) as the elements of a finite dimentional central simple algebra D over a field \mathbb R.
Key words: WedderburnArtin's theorem, finite dimentional center simple algebra
Characterization of Riemannian manifolds by products of some
curvature operators
Yulian Tsankov
We characterize the conformal flat or Einstein or foliated manifolds using the following curvature operators :
1. The classical Jacobi operator, 2. The skewsymmetric curvature operator and 3. The Stanilov curvature operator
Small Moufang 2loops
Petr Vojtechovský, University of Denver, USA
Let M be a finite Moufang loop with normal subloop S such that M/S is cyclic of even order, or dihedral of doubly even order. Then it is possible to modify the multiplication table of M in such a way that the resulting loop (M,*) is again Moufang and, in general, with different center than that of M.
The construction is subtle enough to preserve the associator (as a map, up to equivalence), yet powerful enough to produce all nonassociative Moufang loops of order 16 (5 of them) and of order 32 (71 of them). For n=64, thousands of Moufang loops are obtained. Since the classification of Moufang loops of order 64 is not known, this promises to be a nice tool in describing most (perhaps all) of these 2loops.
Results on planar nearrings and related nearrings
Gerhard Wendt, Joh. Kepler Univ. Linz
We study planar nearrings and nearrings which have similar properties and give structure results. In particular, we describe planar nearrings (and nearrings with similar properties) as certain centralizer sandwich nearrings and we will consider possible links to the general structure theory of (primitive) nearrings.
When are homogeneous maps linear? A lattice point of view
Marcel Wild, Stellenbosch, South Africa
Call a Rmodule V FuchsMaxsonPilz (FMP) if for every Rmodule W every homogeneous map f from V to W is automatically additive. Let L(V) be the modular lattice of all submodules of V. We find necessary respectively sufficient latticetheoretic properties for being FMP. For instance, each FMP module V has each join irreducible element of L(V) strictly contained in a cyclic submodule. If one restricts attention to b i j e c t i v e homogeneous maps then interesting connections with projective geometry arise.