Glasnik Matematicki, Vol. 41, No.1 (2006), 115-140.

CONFIGURATIONS DEFINED ON FINITE RINGS

Andrzej Kozlowski and Krzysztof Prazmowski

Abstract.   Some configurations defined as structures of orbits under families of linear maps of cyclic rings are introduced and studied. All the admissible families which yield a connected configuration with small lines (of size 3 or 4) over the ring Z_pⁿ with prime p are found and characterized. The automorphisms of rank 3 configurations of this type are determined.

2000 Mathematics Subject Classification.   51E14, 51E26.

Key words and phrases.   Cyclic ring, (quasi) difference set, partial linear space, hank of polygons.

DOI: 10.3336/gm.41.1.11

References:

1. J. André, On non-commutative geometry, Ann. Univ. Sarav. Ser. Math. 4 (1993), 93-129.
   MathSciNet

2. W. Benz, Vorlesungen über Geometrie der Algebren, Springer Verlag, Berlin Heidelberg New York, 1973.
   MathSciNet

3. Th. Beth, D. Jungnickel, H. Lenz, Design Theory, Vol.I, Encyclopedia of Mathematics and its applications 69, University Press, Cambridge, 1999.
   MathSciNet

4. A. Herzer, Chain Geometries, Handbook of Incidence Geometries, North-Holland, Amsterdam, 1995, 781-842.
   MathSciNet

5. K. Ireland, M. Rosen, A classical introduction to modern number theory, Springer Verlag, New York, 1990.
   MathSciNet

6. W. Lipski, W. Marek, Analiza kombinatoryczna (in Polish), PWN, Warszawa, 1986.
   MathSciNet

7. C. Luksch, Die Automorphismengruppe der Polynomgeometrie vom Grad n, Mitt. Math. Sem. Giessen 181 (1987), 1-56.
   MathSciNet

8. A. Matras, A. Mierzejewska, K. Prazmowski, Some chain geometries determined by transformation groups, Result. Math. 46 (2004), 251-270.
   MathSciNet

9. K. Petelczyc, Series of inscribed n-gons and rank 3 configurations, Beiträge Algebra Geom. 46 (2005), 283-300.
   MathSciNet

10. H. Wefelscheid, Über die Automorphismengruppen von Hyperbelstrukturen, Beiträge zur geometrischen Algebra (Proc. Sympos., Duisburg, 1976), Birkhäuser, Basel, 1977, 337-343.
    MathSciNet