A Relational Approach to Split Decomposition
Given a set of objects, a split A,B (or distinctive feature) partitions the set into two complementary parts A and B. To a system of splits we associate the relation between pairs of objects that opposes a pair u, v to a pair x, y exactly when the pairs u, v and x, y are separated by at least one split from the system. Conversely, for a quaternary relation opposing pairs of objects, we consider those splits A, B for which each pair from A is opposed to each pair from B. This sets up an adjoint situation between systems of splits and relations between pairs. We characterize the closed systems and open relations. Systems of weakly compatible splits (arising in the decomposition theory of metrics) are always closed, and the corresponding relations can be characterized by a 6-point condition. Particular instances are described by 5-point conditions. A concluding example from biology illustrates this relational approach.
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