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An Exceptional Split Geometry
Andreas W.M. Dress1, K.T. Huber2, V. Moulton3
1FSPM-Strukturbildungsprozesse, University of Bielefeld, Bielefeld, D-33501, Germany
dress@mathematik.uni-bielefeld.de
2Institute of Fundamental Sciences, Massey University, Private Bag 11 222, Palmerston North, New Zealand
3FMI, Mid Sweden University, Sundsvall, S 851-70, Sweden
Annals of Combinatorics 4 (1) p. 1-11 March, 2000
AMS Subject Classification: 04A03,04A20, 05C99, 52B99, 92B99
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 $\phi:X \surjarrow V$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 Ui and Wi (i =1,2,3,4) so that the vertices in both Ui and Wi 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 S is octahedral if and only if it is not circular while, simultaneously, any two splits in 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.
Keywords: split system, incompatible split system, incompatibility, weakly compatible split system, weak compatibility, T-theory, tight span, Buneman complex, metrics, finite metric spaces

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5. A. Dress, K.T. Huber, and V. Moulton, A comparison between two distinct continuous models in projective cluster theory : The median and the tight-span construction , Ann. Combin. 2 (1998) 299每311.

6. A. Dress, K.T. Huber, and V. Moulton, An explicit computation of the injective hull of certain finite metric spaces in terms of their associated Buneman complex , submitted.

7. A. Dress, K.T. Huber, and V. Moulton, Split decomposable antipodal metrics , in preparation.

8. M. Moeller, Mengen schwach verträglicher Splits , Ph.D. Dissertation, Universitaet Bielefeld, 1990.