Indecomposability and duality of tournaments

Let T = (V,A) be a tournament. A subset X of V is an interval of T provided that for a, b is an element of X and for x is an element of V - X, (a,x) is an element of A if and only if (b,x) is an element of A. For example, theta, {x}, where x is an element of V, and V are intervals of T, called trivial intervals. A tournament is said to be indecomposable if all of its intervals are trivial. In another respect, with each tournament T = (V,A) is associated the dual tournament T* = (V,A*) defined as: for x, y is an element of V, (x, y) is an element of A* if (y,x) is an element of A. A tournament T is said to be self-dual if T and T* are isomorphic. The paper characterizes the finite tournaments T = (V,A) fulfilling: for every proper subset X of V, if the subtournament T(X) of T is indecomposable, then T(X) is self-dual. The corollary obtained is: given a finite and indecomposable tournament T = (V,A), if T is not self-dual, then there is a subset X of V such that 6 less than or equal to \X\ less than or equal to 10 and such that T(X) is indecomposable without being self-dual. An analogous examination is made in the case of infinite tournaments. The paper concludes with an introduction of a new mode of reconstruction of tournaments from their proper and indecomposable subtournaments. (C) 2000 Elsevier Science B.V. All rights reserved.