Cyclic antiautomorphisms of Mendelsohn triple systems

✍ Scribed by Neil P. Carnes

• Publisher: Elsevier Science
• Year: 1994
• Tongue: English
• Weight: 1016 KB
• Volume: 126
• Category: Article
• ISSN: 0012-365X
• DOI: 10.1016/0012-365x(94)90251-8

✦ Synopsis

A cyclic triple (a, b, c) is defined to be set { (a, b) ,(b,c),(c,a)} of ordered pairs. A Mendelsohn triple system of order v, M(2,3, u), is a pair (M, fi), w h ere M is a set of u points and fi is a collection of cyclic triples of pairwise distinct points of M such that any ordered pair of distinct points of M is contained in precisely one cyclic triple of fi, An antiautomorphism of a Mendelsohn triple system (M, fl) is a permutation of M which maps fi to /I -I, where j _ 1 = {(c, b, a); (a, b, c)E~}. In this paper we give necessary and sufficient conditions for the existence of a Mendelsohn triple system of order v admitting an antiautomorphism consisting of a single cycle of length d and having v-d fixed points. Further, we give a more general result for partial Mendelsohn triple systems in which the missing ordered pairs are precisely those containing two fixed points.