Reconstructible and Half-Reconstructible Tournaments: Application to Their Groups of Hemimorphisms

✍ Scribed by Youssef Boudabbous

• Publisher: John Wiley and Sons
• Year: 1999
• Tongue: English
• Weight: 709 KB
• Volume: 45
• Category: Article
• ISSN: 0044-3050
• DOI: 10.1002/malq.19990450313

✦ Synopsis

Abstract

Let T and T^1^ be tournaments with n elements, E a basis for T, E′ a basis for T′, and k ≥ 3 an integer. The dual of T is the tournament T” of basis E defined by T(x, y) = T(y, x) for all x, y ε E. A hemimorphism from T onto T′ is an isomorphism from T onto T” or onto T. A k‐hemimorphism from T onto T′ is a bijection f from E to E′ such that for any subset X of E of order k the restrictions T/X and T^1^/f(X) are hemimorphic. The set of hemimorphisms of T onto itself has group structure, this group is called the group of hemimorphisms of T. In this work, we study the restrictions to n – 2 elements of a tournament with n elements. In particular, we prove: Let k ≥ 3 be an integer, T a tournament with n elements, where n ≥ k + 5. Then the following statements are equivalent: (i) All restrictions of T to subsets with n – 2 elements are k‐hemimorphic. (ii) All restrictions of T to subsets with n – 2 elements are 3‐hemimorphic. (iii) All restrictions of T to subsets with n – 2 elements are hemimorphic. (iv) All restrictions of T to subsets with n – 2 elements are isomorphic, (v) Either T is a strict total order, or the group of hemimorphisms of T is 2‐homogeneous.