On the fine structure of the projective line over GF(2)⊗GF(2)⊗GF(2)

The paper gives a succinct appraisal of the properties of the projective line defined over the direct product ring R▵≡GF(2)⊗GF(2)⊗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. 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.