What can we learn from the transitivity parts of a relation?
Résumé
A transitivity part of a relation on a set X is any subset of X on which the restriction of the relation is transitive. What can be recovered of a relation from the sole knowledge of its transitivity parts? In general, the relation itself cannot be recovered, because it has the same transitivity parts as its converse. In certain situations, the unordered pair formed by the relation and its converse can be recovered. This is the case for relations known to be indecomposable tournaments. The result first appeared in Boussa¨ıri, Ille, Lopez, and Thomass´e [2004]. Our proof is simpler, and at the same time conveys some interesting insight into the structure of tournaments.
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