Forgetful Polygons
by
Eline Govaert
Ghent University, Belgium (Research Assistant of the Fund for Scientific Research-Flanders (Belgium))
Coauthors: Hendrik Van Maldeghem, (Ghent University, Belgium)

Let (P, L,  I ) be an incidence structure, and ~ an equivalence relation on the point set P such that collinear points are never equivalent. An ordinary forgetful k-gon, k >= 3, is a set of k points a₁, a₂, ... , a_k, a_k+1=a₁ of P such that a_i and a_i+1 are collinear or equivalent, and a_i and a_j are neither collinear nor equivalent whenever |i-j| >= 2, for all 1 <= i, j <= k.

Now \Gamma = (P, L,  I , ~ ) is a forgetful n-gon, n >= 3, if the following three axioms are satisfied :

(FP1) there are no ordinary forgetful k-gons in \Gamma, for k < n,
(FP2) every two elements of P \cup L of \Gamma are contained in an ordinary forgetful n-gon,
(FP3) Every line is incident with at least three points; every point is incident with at least two lines, and if the point is only equivalent with itself, then it is incident with at least three lines.

If every equivalence class contains exactly one point, then \Gamma is a so called generalized n-gon.

Example. Let \Delta = (P, L',  I ) be a generalized n-gon, and D a set of disjoint lines of \Delta. Two points p and p' of \Delta are said to be equivalent if and only if they lie on a line of D. Then (P, L,  I , ~ ), with L =L'\D, is a forgetful n-gon.

A forgetful 3-gon is exactly a dual semi affine plane, as introduced by Dembowski in [1]. In [1], it is shown that in the finite case, these structures all arise from projective planes (= generalized 3-gons). The main question now reads : does every finite forgetful polygon arise from a generalized polygon, by replacing (pieces of) lines by equivalence classes ?

We show that this is indeed the case for all finite forgetful n-gons with n odd, and for finite forgetful n-gons with n even satisfying the assumption that at least one line and one equivalence class have the same number of points. Moreover, we give further information on (and examples of) finite forgetful quadrangles that do not satisfy this assumption.

References

[1]  Dembowski, P., Semiaffine Ebenen, Arch. Math. 10 (1962), 120 - 131.

Date received: February 20, 2001

Copyright © 2001 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cafv-29.