Zur multiplikativen Archimedizität in projektiven ebenen

We determine the local-Archimedean orderings of projective planes over ternary rings whose multiplicative loops of positive elements are Archimedean. In particular, we prove that a projective plane over an Archimedean, linear ternary ring with associative multiplication is Archimedean.     

Perspective Graves triads and triads of perspective inscribed triangles

Any Pappus configuration can be regarded (in six ways) as a triad of triangles, each inscribed in the previous one; such a triad of triangles is called aGraves triad. Graves triads in which each pair of triangles is in perspective may be calledperspective Graves triads; such triads occur in connection with various results in both projective and metrical geometry. In this paper we give a synthetic proof of the existence of perspective Graves triads; the proof is of interest because it holds in allharmonic planes and requires neither the axiom of Pappus nor the full axiom of Desargues. The paper also includes synthetic proofs of other related results.(Results involving perspective Graves triads can be found in [l, p.149] and [2]. This paper is the result of recent correspondence with Guinand concerning [2] and [3].)     

Galois Objects of Finitely Generated Projective Hopf Algebras

Let C be the category of cocommutative coalgebras over a commutative ring R and let H be a group object in C, i.e., let H be a cocommutative Hopf algebra. Assume that H is a finitely generated, projective R-module and that the integrals (of [4]) in H* ≡ Hom_R(H, R) are cocommutative elements. We will show that any Galois H-object (as defined in [3, Def. 1.2, p. 8]) is a finitely generated, projective R-module.     

THE LARGEST NON-TRIVIAL TORSION CLASS OF MODULES

ABSTRACT

In this paper we investigate the following two classes of left R-modules: N(P) ={A|A has no non-zero direct summand P ε P} and H(p) = {A} if B ? A with B ε N(P), then B = 0}, where P is a class of projective R-modules. We demonstrate that N(p) is, in general, not a torsion class but that H(P) is always a torsionfree class. We also investigate those classes P and rings R for which N(P) is the largest non-trivial torsion class of R-modules.     

Calcul differentiel exterieur de degre arbitraire

LetX and V be F-modules. For every integer t, it is developed an algebraic formalism which generates, by an analogous procedure of construction of the exterior de rivation in the usual sense, couples formed by a deriva tion of degree t of A_F(X,F), an endomorphism of degree t of the graded additive group A_F(X,V), and satisfying a rule of derivation of degree t. This formalism is the most general ifX is projective of finite type. If t is odd, the curvature, and ifX - V, the torsion, are discussed at some lenght. If t=1, the differen tial calculus associated to Lie modules, linear connexions and fields of endomorphisms, are special cases.

---

---

Ce travail a été financé par le Conselho Nacio nal de Pesquisas (Brésil), contract TC 8233.     

An optimal ternary [69, 5, 45] code and related codes

A ternary [69, 5, 45] code is constructed, thus solving the problem of finding the minimum length of a ternary code of dimension 5 and minimum distance 45. Furthermore, this code is shown to be a unique two-weight code with weight enumerator 1+210Z⁴⁵+32Z⁵⁴. It is also shown that a ternary [70, 6, 45] code, which would have been a projective two-weight code giving rise to a new strongly regular graph, does not exist. In order to prove the main results, the uniqueness of some other optimal ternary codes with specified weight enumerators is also established.     

Zur algebraischen Kennzeichnung mehrwertiger Halbordnungen

The radical R(T) of a planar ternary ring (PTR) T, which plays an important role in the theory of halforderings and orderings of PTRs, allows also to give an algebraic characterization ofJunkers' multiple-valued halforderings of projective planes. As an application we obtain some information on the group of projectivities in projective planes.

---

---

Herrn Professor G. Tallini zum 60. Geburtstag gewidmet     

Topological projective planes and uniform valuations

The valuation topology of any uniformly valued ternary field (K, T, v) can be extended to the projective plane II over (K, T) making it a topological projective plane in the sense of Salzmann. Appealing to Prieß-Crampe's celebrated fixed point theorem for ultrametric spaces, our result allows us to present a wide variety of new, totally disconnected, compact and non-compact topological projective planes.Dedicated to Professor S. Prieß-Crampe on the occasion of her 60th birthday     

TWO CLASSES OF HOMOMORPHISMS IN MODULE CATEGORIES

Abstract

In a category R-Mod a homomorphism α:A → B is called projective if α factors through every epimorphism with B as image. Injective homomorphisms are defined dually. Some properties of such homomorphisms are derived, and it is shown that the hereditariness of the ring R is equivalent to some conditions which can be simply stated in terms of projective and injective homomorphisms.     

Existenzbedingungen für Dualitäten projektiver Ebenen

In a projective plane a permutation of the set of all points and lines is constructed, using only the operations join and meet. Under certain conditions (identities in a coordinatizing ternary field; special cases of Desargues- and Pappos-theorem) this permutation is a duality. For a topological projective plane this duality proves, that point space and line space of the plane are homeomorphic.     

Coordinate rings of topological Klingenberg planes II: the algebraic foundation for a projective theory

Ternary fields are the coordinate rings of affine and projective planes; however, the planes constructed over topological ternary fields are not necessarily topological. Surprisingly, the explanation of this phenomenon becomes evident in the more general theory of topological Klingenberg planes as we exhibited in [3] for the affine case. However, in the projective setting, we have a more formidable task. We must develop a new coordinate ring that admits a topological structure suitable for coordinatizing topological PK-planes. We accomplish this in two stages. In this paper, we revisit the standard coordinate rings [1, 11], discuss and resolve their deficiencies by developing a new coordinate ring as a unique extension of these refined standard rings. In a subsequent paper [4], we show that this new ring can be suitably topologized to coordinatize a topological PK-plane. This last result can then be used to explain why topological ternary fields do not necessarily coordinatize topological projective planes. Received 17 February 2000; revised 10 June 2000.     

Relations between two perfect ternary sequence constructions

Both R. Games [4] and V.P. Ipatov [8] have given constructions for perfect ternary sequences. Games uses difference sets and quadrics in projective space, while Ipatov uses q-ary m-sequences. We show that the Ipatov sequences are a subset of the Games sequences. Further, we show that a conjecture of Games relating to quadrics in projective spaces does not hold in general.     

On the singularities of weakly normal varieties

Among the weakly normal varieties (in the sense of Andreotti and Bombieri, [1]) are of particular interest those varieties such that the normalization morphism is unramified outside a subvariety of codimension not less than 2. We describe the singularities of these varieties (called here WN1) by means of analytic equations, tangent cones, analytic branches and we show that any irreducible projective variety is birationally equivalent to a WN1 hypersur face and that a Gorenstein variety is weakly normal if and only if it is WN1.This research was done when the authors were members of G.N.S.A.G.A. of the C.N.R.     

Polaritäten Ebener projektiver Ebenen

A projective plane is called flat if the spaces of points and lines are locally compact and 2-dimensional and the joining of points and the intersecting of lines are continuous. H. Salzmann studied planes of this type in [11]–[21]. Here polarities of such planes are considered. In II general properties of polarities of flat planes are discussed. For example, a polarity with absolute points has always an oval of absolute points. A flat projective plane with a cartesian ternary field K admits a polarity iff multiplication in K is commutative. In III the polarities of flat projective planes with a 3-dimensional collineation group are determined.     

Über lokalarchimedische Anordnungen projektiver Ebenen

The question, whether the Archimedean ordering of only one of the ternary rings of a projective plane implies that is Archimedean, i.e. that every ternary ring of is Archimedean, is answered in the negative by the construction of local-Archimedean orderings of free planes. There exists even Archimedean affine planes with non-Archimedean associated projective planes.     

A Note on the Place Topology of Projective Planes

It is shown that the place topology induced by a proper epimorphism of a projective plane , which is known to make a Lenz-topological plane, makes even a topological projective plane, if the extended radical of some underlying ternary field is bounded.     

The Correspondence Between Projective Codes and 2-weight Codes

We show how to get a 1-1 correspondence between projective linear codes and 2-weight linear codes. A generalization of the construction gives rise to several new ternary linear codes of dimension six.     

Minimal topological projective planes

The atoms of the lattice of compatible topologies on a given projective plane are examined. The notion of a field of type V is generalized to ternary fields of type V. These are always minimal, and they arise for example as coordinate structure of strictly uniformizable or of orderable topological planes. Hence, such planes are minimal.Dedicated to Professor Helmut Karzel on his 60^th birthday     

Isotopy and geotopy for ternary rings of projective planes

We investigate the question of when two ternary rings coordinatize the same finite projective plane. Necessary and sufficient conditions are obtained for right quasifields and for right distributive linear rings whose multiplicative loop is a group.