Tree representations of betweenness relations defined by intersection and inclusion

ABSTRACT: On a family of sets, a ternary relation may be defined by stating that, for U,V,W members of the family, V is “between” U and W if and only if V includes the intersection of U and W. The relation is called “intersection-betweenness” and may be understood as the description of proximities between objects associated with sets in the family. The problem of using a tree graph for representing such a relation is discussed. Characterisations are proven both for full tree representation (there is a tree-betweenness identical to the given intersection-betweenness: Section 2) and for partial tree representation (there is a tree-betweenness included in the given intersection-betweenness: Section 3). Procedures for actually finding solutions to full and partial tree representation problems are illustrated in Section 4. In Section 5 some related paradigms of modern psychometrics are mentioned, to highlight the peculiar aspects of the proposed approach.