Rings of geometries II

David G. Glynn

doi:10.1016/0097-3165(88)90027-1

Rings of geometries II

David G. Glynn

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16 Citations (Scopus)

Abstract

Linear spaces are investigated using the general theory of "Rings of Geometries I." By defining geometries and ring structures in several different ways, formulae for linear spaces embedded in finite projective and affine planes are obtained. Several "fundamental theorems" of counting in finite projective planes are proved which show why configurations with at least three points per line and at least three lines through every point are important. These theorems are illustrated by finding the formulae for the number of k-arcs in a projective plane of order q for all k ≤ 8 and also by finding a formula for the number of blocking sets. A quick proof that a projective plane of order 6 does not exist follows from the formula for the number of 7-arcs in such a plane.

Original language	English
Pages (from-to)	26-66
Number of pages	41
Journal	Journal of Combinatorial Theory, Series A
Volume	49
Issue number	1
DOIs	https://doi.org/10.1016/0097-3165(88)90027-1
Publication status	Published - Sept 1988
Externally published	Yes

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Rings of geometries II

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