An Exceptional Split Geometry

Andreas W.M. Dress^{1}, K.T. Huber^{2}, V. Moulton^{3}

dress@mathematik.uni-bielefeld.de

Annals of Combinatorics 4 (1) p. 1-11 March, 2000

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 onto
the six-element set *V* of vertices of an octahedron, form the four
bipartitions
(*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
and
(*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*.

In this note, it will be shown that a weakly compatible split system

References:

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3. H.-J. Bandelt and A. Dress, A canonical decomposition theory for metrics on a finite set , Adv. Math. 92 (1992) 47每105.

4. A. Dress, K. Huber, and V. Moulton, Some variations on a theme by Buneman , Ann. Combin. 1 (1997) 339每352.

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.