On the Fine Structure of the Projective Line Over GF(2) x GF(2) x GF(2)

Abstract

The paper gives a succinct appraisal of the properties of the projective line defined over the direct product ring $R\_{\triangle} \equiv$ GF(2)$\otimesGF(2)\otimes$GF(2). The ring is remarkable in that except for unity, all the remaining seven elements are zero-divisors, the non-trivial ones forming two distinct sets of three; elementary ('slim') and composite ('fat'). Due to this fact, the line in question is endowed with a very intricate structure. It contains twenty-seven points, every point has eighteen neighbour points, the neighbourhoods of two distant points share twelve points and those of three pairwise distant points have six points in common -- namely those with coordinates having both the entries `fat' zero-divisors. Algebraically, the points of the line can be partitioned into three groups: a) the group comprising three distinguished points of the ordinary projective line of order two (the 'nucleus'), b) the group composed of twelve points whose coordinates feature both the unit(y) and a zero-divisor (the 'inner shell') and c) the group of twelve points whose coordinates have both the entries zero-divisors (the 'outer shell'). The points of the last two groups can further be split into two subgroups of six points each; while in the former case there is a perfect symmetry between the two subsets, in the latter case the subgroups have a different footing, reflecting the existence of the two kinds of a zero-divisor. The structure of the two shells, the way how they are interconnected and their link with the nucleus are all fully revealed and illustrated in terms of the neighbour/distant relation. Possible applications of this finite ring geometry are also mentioned.