Topological projective geometries

A concept of topological projective geometry is defined, which in contrast to the definitions given in [Mi] and [SÖ] does not contain any dimensional restrictions. Besides elementary properties it is shown in this paper that these topological geometries always possess a coordinatization over a uniquely determined topological division ring if the dimension is finite.Dedicated to Prof. Dr. Hanfried Lenz on his 70^th birthday     

A unified approach to projective lattice geometries

The interest in pursuing projective geometry on modules has led to several lattice theoretic generalizations of the classical synthetic concept of projective geometry on vector spaces.Introduced in this paper is an approach that is capable of unifying various attempts within a new conceptual frame. This approach reflects algebraic properties from a lattice-geometric point of view. Together with new results we are presenting results from previous publications which have been improved in the frame of this work.     

Hermitian geometries in projective space

Using a Hermitian form on a vector space over GF (l), we produce a geometry on the associated projective space and prove that this geometry is characterized by its plane sections.     

Dualities for infinite-dimensional projective geometries

As main theorem we show that any duality in the projective geometry associated to a vector space is induced by a non-degenerate sesquilinear form on this space. In particular, any polarity is induced by a non-degenerate orthosymmetric sesquilinear form.Supported by a grant from the Swiss National Founds for Scientific Research.     

Isospectrality and galois projective geometries

We construct a series of pairs of domains in the plane and pairs of surfaces with boundary that are isospectral but not isometric. The construction is based on the existence of finite transformation groups that are spectrally equivalent but not isomorphic. Translated fromMatematicheskie Zametki, Vol. 63, No. 5, pp. 660–664, May, 1998. This research was partially supported by the Russian Foundation for Basic Research and by INTAS under joint grant No. 96-01-00043.     

On point-irreducible projective lattice geometries

Within the conceptual frame of projective lattice geometry (as introduced in [5]) we are considering the class of all point-irreducible geometries. In the algebraic context these geometries are closely connected with unitary modules over local rings. Besides several synthetic investigations we obtain a lattice-geometric characterization of free left modules over right chain rings which allows a purely lattice-theoretic version in the Artinian case.This paper results from a joint work of the authors at the Hungarian Academy of Sciences (Budapest) in the fall of 1991, supported by the DAAD.     

A construction of integral standard generalized table algebras from parameters of projective geometries

We construct new integral standard generalized table algebras from parameters of projective geometries. The algebras are noncommutative, imprimitive, and six dimensional.     

New ring-linear codes from dualization in projective Hjelmslev geometries

In this article, several new constructions for ring-linear codes are given. The class of base rings are the Galois rings of characteristic 4, which include ${\mathbb {Z}_4}$ as its smallest and most important member. Associated with these rings are the Hjelmslev geometries, and the central tool for the construction is geometric dualization. Applying it to the ${\mathbb {Z}_4}$ -preimages of the Kerdock codes and a related family of codes we will call Teichmüller codes, we get two new infinite series of codes and compute their symmetrized weight enumerators. In some cases, residuals of the original code give further interesting codes. The generalized Gray map translates our codes into ordinary, generally non-linear codes in the Hamming space. The obtained parameters include (58, 2⁷, 28)₂, (60, 2⁸, 28)₂, (114, 2⁸, 56)₂, (372, 2¹⁰, 184)₂ and (1988, 2¹², 992)₂ which provably have higher minimum distance than any linear code of equal length and cardinality over an alphabet of the same size (better-than-linear, BTL), as well as (180, 2⁹, 88)₂, (244, 2⁹, 120)₂, (484, 2¹⁰, 240)₂ and (504, 4⁶, 376)₄ where no comparable (in the above sense) linear code is known (better-than-known-linear, BTKL).     

On the representations of projective geometries in algebraic combinatorial geometries

In this paper, we show that the full algebraic combinatorial geometry is not a projective geometry, it is only semimodular, but the p-polynomial points give a projective subgeometry. Also, we show that the subgeometry can be coordinatized by a skew field, which is quotient ring of an Ore domain. As a corollary, we prove the existence of algebraic representations over fields of prime characteristic of the non-Pappus matroid and its dual matroid. Regarding the existence of algebraic representations of the non-Pappus matroid, this result was earlier proved by Lindström [7] for finite fields.     

Partitioning projective geometries into Segre varieties

Using suitable subgroups of Singer cyclic groups we prove some properties of regular spreads and Segre varieties, which in turn yield a necessary and sufficient condition for partitioning a finite projective space into such varieties.In ricordo di Giuseppe TalliniResearch performed within the activity of G.N.S.A.G.A. of C.N.R. with the support of the Italian Ministry for Research and Technology     

Projective mappings between projective lattice geometries

The concept of projective lattice geometry generalizes the classical synthetic concept of projective geometry, including projective geometry of modules.In this article we introduce and investigate certain structure preserving mappings between projective lattice geometries. Examples of these so-calledprojective mappings are given by isomorphisms and projections; furthermore all linear mappings between modules induce projective mappings between the corresponding projective geometries.     

Morphisms of projective geometries and semilinear maps

Morphisms between projective geometries are introduced; they are partially defined maps satisfying natural geometric conditions. It is shown that in the arguesian case the morphisms are exactly those maps which in terms of homogeneous coordinates are described by semilinear maps. If one restricts the considerations to automorphisms (collineations) one recovers the so-called fundamental theorem of projective geometry, cf. Theorem 2.26 in [2].Supported by a grant from the Fonds National Suisse de la Recherche Scientifique.     

On a projective representation of chain geometries

We define a distance d on the set of r-spaces of an n-space. By the transfer of d to the GraßmannianG=G(n, r) we obtain a distinguished class of normal rational curves of order 1, the 1-distance lines, 1=1,..., r, which are in 1–1-correspondence to the so-called generalized reguli of type (r, 1).To every chain geometry there are subspaces T and Z of the surrounding space ofG, such that forV=GT andW = VZ we have a projective representation of on V\W as pointset, where the chains of are exactly the r-distance lines on V\W.Dedicated to Prof.A. Barlotti on occasion of his 60 birthday     

A characterization of truncated projective geometries as flag-transitive PG *.PG-geometries

We prove that every flag-transitive locally finite (PG ^*.PG)-geometry is a truncated projective geometry. Dedicated to Daniel Hughes on the occasion of his 80th birthday.     

On Barbilian spaces in projective lattice geometries

We introduce the notion of a Barbilian space of a projective lattice geometry in order to investigate the relationship between lattice-geometric properties and the properties of point-hyperplane structures associated with. We obtain a characterization of those projective lattice geometries, the Barbilian space of which is a Veldkamp space.