Abstract
In [8] valuations were introduced and it was shown that these were important objects for classifying near 2n-gons. Several classes were given including one arising from so-called distance-2j-ovoids. Here we introduce pseudo-valutions and explain why these objects can be important for classifying near (2n+1)-gons. Every valuation of a near polygon gives rise to pseudo-valuations and almost all known examples of pseudo-valuations arise in this way. We show that every distance-(2j+1)-ovoid gives rise to a pseudo-valuation which does not come from a valuation. Subsequently, we study distance-j-ovoids in regular near polygons. We are able to calculate the number of elements of a distance-j-ovoid in two ways, yielding a relation between the parameters of the regular near polygon. We will discuss some cases where this relation can be solved.
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References
Brouwer, A. E.: The uniqueness of the near hexagon on 759 points. In ``Finite Geometries'' (N. L. Johnson, M. J. Kallaher and C. T. Long, eds.) Lecture Notes in Pure and Appl. Math.82, 47–60 (Marcel Dekker, New York, Basel, 1982)
Brouwer, A.E., Cohen A. M.: Local recognition of Tits geometries of classical type. Geom. Dedicata 20, 181–199 (1986)
Brouwer, A.E., Cohen, A.M., Hall, J.I., Wilbrink, H.A.: Near polygons and Fischer spaces. Geom. Dedicata 49, 349–368 (1994)
Brouwer, A.E., Cohen, A.M., Neumaier, A.: Distance-Regular Graphs (Springer, Berlin, 1989)
Brouwer, A.E., Wilbrink, H.A.: The structure of near polygons with quads. Geom. Dedicata 14, 145–176 (1983)
De Bruyn, B.: Near hexagons with four points on a line. Adv. Geom. 1, 211–228 (2001)
De Bruyn, B.: Near polygons with two types of quads and three types of hexes. J. Combin. Math. Combin. Comput., to appear
De Bruyn, B., Vandecasteele, P.: Valuations of near polygons. Glasgow Math. J. 47, 347–361 (2005)
De Bruyn, B., Vandecasteele, P.: The classification of the slim dense near octagons. European J. Combin., to appear
Cameron P.J.: Dual polar spaces. Geom. Dedicata 12, 75–86 (1982)
Feit, W., Higman, G.: The nonexistence of certain generalized polygons. J. Algebra 1, 114–131 (1964)
Offer, A.: On the order of a generalized hexagon admitting an ovoid or spread. J. Combin. Theory A 97, 184–186 (2002)
Offer, A., Van Maldeghem H.: Distance j-ovoids and related structures in generalized polygons. Discrete Math. 294, 147–160 (2005)
Payne, S.E., Thas, J.A.: Finite Generalized Quadrangles. Research Notes in Mathematics 110 (Pitman, Boston, 1984)
Shult, E.E., Yanushka, A.: Near n-gons and line systems. Geom. Dedicata 9, 1–72 (1980)
Tits, J.: Classification of buildings of spherical type and Moufang polygons: a survey. pp. 229–246 in ``Teorie Combinatorie'', Proc. Intern. Colloq. (Roma 1973), volume I (Accad. Naz. Lincei, 1976)
Van Maldeghem, H.: Generalized Polygons. Monographs in Mathematics 93 (Birkhäuser, Basel, Boston, Berlin, 1998)
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Postdoctoral Fellow of the Research Foundation - Flanders
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Bruyn, B. Distance-j-Ovoids in Regular Near 2d-Gons. Graphs and Combinatorics 22, 203–216 (2006). https://doi.org/10.1007/s00373-005-0654-8
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DOI: https://doi.org/10.1007/s00373-005-0654-8