Linearly independent split systems

An important procedure in the mathematics of phylogenetic analysis is to associate, to any collection of weighted splits, the metric given by the corresponding linear combination of split metrics. In this note, we study necessary and sufficient conditions for a collection of splits of a given finite set XX to give rise to a linearly independent collection of split metrics. In addition, we study collections of splits called affine split systems   induced by a configurations of lines and points in the plane. These systems not only satisfy the linear-independence condition, but also provide a ZZ-basis of the ZZ-lattice Deven(X∣Z) consisting of all integer-valued symmetric maps D:X×X→ZD:X×X→Z defined on XX that vanish on the diagonal and for which, in addition, D(x,y)+D(y,z)+D(z,x)≡0mod2 holds for all x,y,z∈Xx,y,z∈X. This ZZ-lattice is generated by all split metrics considered as vectors in the real vectorspace D(X∣R)D(X∣R) consisting of all real-valued symmetric maps defined on XX that vanish on the diagonal — and, hence, is also an RR-basis of that vectorspace.