Polarities in n-uniform projective Hjelmslev planes

In [9] the author has studied polarities in finite 2-uniform projective Hjelmslev planes. The present paper deals with polarities in finite n-uniform projective Hjelmslev planes (n 2).The author's research was supported by IWONL grant no. 840037.     

Order and topology in projective Hjelmslev planes

The concept of an ordered projective Hjelmslev plane was intuitively introduced by Hjelmslev in Einleitung in die allgemeine Kongruenglehre ([9], [10]).This paper is concerned with formalizing and examing preorderings and orderings for projective Hjelmslev planes. In addition we show that orderings generated topologies of the point and line sets which render the plane a topological Hjelmslev plane ([19], [13]). These planes — unlike the ordinary ordered planes ([18]) — are, due to the existence of infinitesimals, non-archimedian, non-compact and disconnected with the neighbour classes as certain quasi-components.The authors gratefully acknowledge the support of the Natural Sciences and Engineering Research Council of Canada.     

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     

Polarities in finite 2-uniform projective Hjelmslev planes

In this paper we deal with polarities in finite projective Hjelmslev planes. A polarity is an involutory one-to-one mapping of the points onto the lines and of the lines onto the points such that the incidence relation and the neighbor relation are preserved. Most of the results are obtained in the case of 2-uniform PH-planes. In a forthcoming paper [5], we shall deal with the case of n-uniform (n > 2) PH-planes.The author's research was supported by I.W.O.N.L. grant No. 840037.     

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.     

A class of pbib-designs obtained from projective geometries

In this paper, a new class of partially balanced incomplete block designs is constructed over an association scheme obtained from a finite projective geometry. Further, a general method for deriving a balanced incomplete block design from another one is given.     

Nonbinary quantum codes derived from finite geometries

The paper gives explicit parameters for several infinite families of q-ary quantum stabilizer codes. These codes are derived from combinatorial designs which arise from finite projective and affine geometries.     

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.     

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.     

On the rank of incidence matrices in projective Hjelmslev spaces

Let \(R\) be a finite chain ring with \(|R|=q^m\) , \(R/{{\mathrm{Rad}}}R\cong \mathbb {F}_q\) , and let \(\Omega ={{\mathrm{PHG}}}({}_RR^n)\) . Let \(\tau =(\tau _1,\ldots ,\tau _n)\) be an integer sequence satisfying \(m=\tau _1\ge \tau _2\ge \cdots \ge \tau _n\ge 0\) . We consider the incidence matrix of all shape \(\varvec{m}^s=(\underbrace{m,\ldots ,m}_s)\) versus all shape \(\tau \) subspaces of \(\Omega \) with \(\varvec{m}^s\preceq \tau \preceq \varvec{m}^{n-s}\) . We prove that the rank of \(M_{\varvec{m}^s,\tau }(\Omega )\) over \(\mathbb {Q}\) is equal to the number of shape \(\varvec{m}^s\) subspaces. This is a partial analog of Kantor’s result about the rank of the incidence matrix of all \(s\) dimensional versus all \(t\) dimensional subspaces of \({{\mathrm{PG}}}(n,q)\) . We construct an example for shapes \(\sigma \) and \(\tau \) for which the rank of \(M_{\sigma ,\tau }(\Omega )\) is not maximal.     

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.     

Some low-density parity-check codes derived from finite geometries

We look at low-density parity-check codes over a finite field \mathbbK{\mathbb{K}} associated with finite geometries T₂^*(K){T_2^*(\mathcal{K})}, where K{\mathcal{K}} is a sufficiently large k-arc in PG(2, q), with q = p ^h . The code words of minimum weight are known. With exception of some choices of the characteristic of \mathbbK{\mathbb{K}} we compute the dimension of the code and show that the code is generated completely by its code words of minimum weight.     

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     

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.     

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.