An Exceptional Split Geometry

Abstract.

In view of results obtained in split decomposition theory, it is of some interest to investigate the structure of weakly compatible split systems. A particular class of such split systems — the so-called octahedral split systems — can be constructed as follows: Given a set X together with a surjective map $\phi:X\twoheadrightarrow V$ onto the six-element set V of vertices of an octahedron, form the four bipartitions $X = A_i \dot{\cup} B_i$ (i = 1, 2, 3, 4) of X obtained by first partitioning V in all four possible ways into two disjoint 3-subsets U _i and W _i (i = 1, 2, 3, 4) so that the vertices in both U _i and W _i form an equilateral triangle, and then taking their pre-images A _i : = $\phi$ ^-1(U _i ) and B _i : = $\phi$ ^-1(W _i ) (i = 1, 2, 3, 4).¶In this note, it will be shown that a weakly compatible split system ${\cal S}$ is octahedral if and only if it is not circular while, simultaneously, any two splits in ${\cal S}$ are incompatible. This result appeared originally in Martina Moeller's Ph.D. thesis. Here, we give an alternative proof based on the close relationship between weakly compatible split systems and weak hierarchies.