ON SOME OPERATIONS ON FINITE AFFINE PLANES APPLIED TO THE THEORY OF REGULAR CONFIGURATIONS

摘要

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 (Bcontractions), 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)_0- configuration and research some problems of embeddability for (P-P), (H-T), and (HT)_0. 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 1 lines to the initial plane. We prove that for -1 n + 1 an arbitrary n odd, the non-existence of the symmetric configuration (~(n~-12)/2,~(n+1)/ 2 ) implies the non-existence of t he projective plane of order n. On the basis of Gropp's article [101, we solve some current problems concerning the existence of non-symmetric configurations with a natural index.