The Geometry of Dowling Lattices

Abstract

J. Kahn and J. Kung showed how to reconstruct the group of a Dowling lattice. Here we show how to obtain the group of a Dowling lattice through techniques parallel to those used in classical geometry to obtain the field when coordinatizing a projective or affine space. The analogy between Dowling lattices and classical geometry goes deeper. From an axiomatization of Dowling lattices of rank four and greater, first investigated in [J. Bonin and K. Bogart, J. Combin. Theory Ser. A56 (1991), 195-202], we derive the restriction of Desargues′ theorem to coordinate lines. In rank three, the construction used to define Dowling lattices works just as well for quasigroups as for groups. The planar bundle theorem, which for rank four and greater is a corollary of Desargues′ theorem, distinguishes between the rank-three lattices based on groups and those based on quasigroups. Moreover, Pappus′ theorem, restricted to coordinate lines, characterizes the Dowling lattices based on abelian groups.