On Segre's product of partial line spaces and spaces of pencils

Naumowicz, Adam; Prażmowski, Krzysztof

doi:10.1007/s00022-001-8557-1

Cited by 20 publications

(52 citation statements)

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“…However, in sharp contrast to Theorem 1, this is a rather trivial task, and the transformations of this kind do not deserve any interest. Then, using a result of A. Naumowicz and K. Prazmowski [7], we also determine all A-transformations of Gk x Gn-k in Theorem 4. Such mappings are closely related with collineations of the underlying partial linear space, and in general they can be described in terms of two semilinear bijections, but not in terms of a single semilinear bijection.…”

Section: Introductionmentioning

confidence: 99%

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Transformations on the Product of Grassmann Spaces

Havlicek¹,

Pankov²

2005

Demonstratio Mathematica

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TRANSFORMATIONS ON THE PRODUCTOF GRASSMANN SPACES IntroductionLet Gk denote the set of all fc-dimensional subspaces of an n-dimensional vector space. We recall that two elements of Gk are called adjacent if their intersection has dimension k -1. The set Gk is point set of a partial linear space, namely a Grassmann space for 1 < k < η -1 (see Section 5) and a projective space for k 6 {1 , π -1}. Two adjacent subspaces are-in the language of partial linear spaces-two distinct collinear points.W.L. Chow [4] determined all bijections of Gk that preserve adjacency in both directions in the year 1949. In this paper we call such a mapping, for short, an A-transformation. Disregarding the trivial cases A; = 1 and k = η-1, every A-transformation of Gk is induced by a semilinear transformation V -* V or (only when k = 2n) by a semilinear transformation of V onto its dual space V*. There is a wealth of related results, and we refer to [2], [6], and [9] for further references.In the present note, we aim at generalizing Chow's result to products of Grassmann spaces. However, we consider only products of the form Gk x Gn-ki where Gk and G n -k stem from the same vector space V. Furthermore, for a fixed k we restrict our attention to a certain subset of Gk

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Section: Introductionmentioning

confidence: 99%

“…Before we close this section, it is worthwhile to mention that the results from [7] could be used to describe the A-transformations of arbitrary finite products of Grassmann spaces, but this is not the topic of the present article.…”

Section: Introductionmentioning

confidence: 99%

Transformations on the Product of Grassmann Spaces

Havlicek¹,

Pankov²

2005

Demonstratio Mathematica

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“…Clearly, S, L 1 ∪ L 2 is the Segre product of M 1 and M 2 (cf. [5,9]), and the substructure S, L 0 =: M 1 ⊗ M 2 of M 1 ⊗ M 2 is a partial Steiner triple system. Consider the field G F(3) as a single-line structure {0, 1, 2}, {{0, 1, 2}} ; then…”

Section: Andmentioning

confidence: 99%

Semi-Pappus configurations; combinatorial generalizations of the Pappus configuration

Prażmowska

Prażmowski

2010

Des. Codes Cryptogr.

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“…4.3, [9]), and the closure of M determined by is not even Shultian (following [5] Shultian means that any two points on sides of a triangle are collinear). Nevertheless, with the help of the relation in M only, we could define affine lines on the natural horizon of M and the natural parallelity of these lines (cf.…”

Section: Introductionmentioning

confidence: 99%