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On some operations on finite affine planes applied to the theory of regular configurations

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Abstrakty
EN
We consider some operations on affine planes which resemble the construction of a derived affine plane at a point of the Benz plane. We call them Benz-contractions (B-contractions) , distinguishing between chain contractions and generator contractions. We prove that the Pappos-Pascal configuration is the B-contraction of the affine plane of order 4 and we relate it to the Havlicek-Tietze configuration. We present a new (HT)o-configuration and research some problems of embeddability for (P-P), (H-T), and (HT)o. We propose a method of finding (n - 2) regular configurations on an arbitrary affine plane of order n. Among them are pairs of configurations with dual type and each such a pair can be completed with one point and n -f 1 lines to the initial plane. We prove that for an arbitrary n odd, the non-existence of the symmetric configuration ( n2-1/2, n+1/2) the non-existence of the projective plane of order n. On the basis of Gropp's article flOj, we solve some current problems concerning the existence of non-symmetric configurations with a natural index.
Wydawca
Rocznik
Strony
167--185
Opis fizyczny
Bibliogr. 29 poz., rys.
Twórcy
  • Faculty of Mathematics and Computer Science Nicolaus Copernicus University Chopina 12/18 87-100 Toruń, Poland, hannam@mat.uni.torun.pl
Bibliografia
  • [1] L. M. Batten, A. Beutelspacher, The Theory of Finite Linear Spaces. Combinatorics of Points and Lines, Cambridge University Press, Cambridge, 1993.
  • [2] L. M. Batten, Combinatories of Finite Geometries, 2nd ed., Cambridge University Press, Cambridge, 1997.
  • [3] W. Benz, Vorlesungen über Geometrie der Algebren, Springer-Verlag, Berlin-Heidelberg-New York, 1973.
  • [4] N. L. Biggs, A. T. White, Permutation Groups and Combinational Structures, Cambridge University Press, Cambridge, 1979.
  • [5] Y. Chen, A characterization of some geometries of chains, J. Math. 26 (1974).
  • [6] P. Dembowski, Finite Geometries, Springer-Verlag, New York, 1968.
  • [7] W. L. Edge, Some implications of the geometry of the 21-point plane, Math. Zeitschrieft 87 (1965).
  • [8] D. G. Glynn, A note on Nk configurations and theorems in projective space, Bull. Austral. Math. Soc. 76 (2007), 15-31.
  • [9] K. Gozdalska, On affine planes non extensible to Laguerre planes and some related problems, Math. Bohem. 116 (1991), 2-11.
  • [10] H. Gropp, Nonsymmetric configurations with natural index, Discrete Math. 124 (1994), 87-98.
  • [11] H. Gropp, Nonsymmetric configurations with line size k=4. 1st Cologne-Twente Workshop on Graphs and Combinationar Optimization, Electron Notes Discrete Math., Elsevier, Amsterdam 8 (2001).
  • [12] R. Hartshorne, Foundations of Projective Geometry, Lecture Notes Harvard University, New York, 1967.
  • [13] K. Havliček, J. Tietze, Zur Geometrie der endlichen ebene der ordnung n=4, Czechoslovak Math. J. 21 (1971), 157-164.
  • [14] D. Hilbert, S. Cohn Vossen, Anschauliche Geometrie, Berlin, 1932.
  • [15] J. Jakóbowski, Theorems on products of central collineations with distinct centres of axes applied to the Benz planes, Demonstratio Math. 37 (2004), 639-653.
  • [16] J. Jakóbowski, A new construction for Minkowski planes, Geom. Dedicata 69 (1988), 179-188.
  • [17] L. M. Kelly, Multiply Perspective Simplices, Desmic Triads and the Edel stein Theorem, vol. 4, Series in Discrete Mathematics and Theoretical Computer Science, 1991.
  • [18] H. J. Kroll, Anordnungsfragen in Benz-Ebenen, Abh. Math. Sem. Univ. Hamburg 46 (1977), 217-255.
  • [19] C. Lam, G. Kolesova, L. Thiel, A computer search for finite projective planes of order 9, Discrete Math. 92 (1991), 187-195.
  • [20] A. Lewandowski, On extensibility of affine plane to the Möbius plane, Goemetria, Scient. Chapt. of Institute of Technology in Poznań 14 (1984), 17-19.
  • [21] A. Lewandowski, H. Makowiecka, Some remarks on Havliček-Tietze configuration, Časopis pěst. mat. 104 (1979), 180-184
  • [22] A. Lewandowski, H. Makowiecka, A geometrical characterizations of the projective plane of order 4, Časopis pěst. mat. 104 (1979), 185-187.
  • [23] A. Lewandowski, H. Makowiecka, Six-fold perspective triangles in n-dimensional projective space, Nieuw. Arch. Wisk. 16 (1998), 135-141.
  • [24] H. Makowiecka, On mixed m-extensions of linear designs, Simon Stevin, J. Pure Appl. Math. 66 (1992), no. 1-2.
  • [25] A. Matraś, On Havliček-Tietze configuration in some non-desarguessia n planes, Časopis pěst. mat. 114 (1989), 133-137.
  • [26] A. Matraś, A construction of a skewaffine structure in Laguerre geometry, Bull Polish Acad. Sci. Math. 54 (2006), 277-289.
  • [27] M. Prażmowska, K. Prażmowski, Combinatorial Veronese structures, their geometry and problems of embeddability, Results Math. 51 (2008), 275-308.
  • [28] G. P. Steinke, A remark on Benz planes of order 9, Ars Combin. 34 (1992), 257-267.
  • [29] K. Witczyński, Perspective case of the Pappus theorem in the n-dimensional projective space, Demonstratio Math. 40 (2007), 925-928.
Typ dokumentu
Bibliografia
Identyfikator YADDA
bwmeta1.element.baztech-article-PWA5-0027-0036
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