Self-Duality of Modules and Reconstruction of Tournaments Up to Duality
Youssef Boudabbous, Abderrahim Boussaïri, Abdelhak Chaïchaâ and Nadia El Amri
For each non-negative integer k ≤ 6 we give a complete description of the tournaments which are (≤ k)-reconstructible up to duality and of those which are (≤ k)-hereditarily-reconstructible up to duality. We recall that a tournament T is (≤ k)-reconstructible up to duality (resp. (≤ k)-hereditarily-reconstructible up to duality) if for every tournament T’ on the same finite set V of vertices, T’ is isomorphic to T or to its dual T*, (resp. the tournament T’[X] induced on every subset X of V is isomorphic to T [X] or to its dual) provided that the induced tournament T‘[K] is isomorphic to T [K] or to its dual for every subset K of V of size at most k.
Keywords: Tournament, module, autonomous, hereditarily self-dual module, reconstruction up to duality, hereditary reconstruction up to duality.