Bicyclic Antiautomorphisms of Mendelsohn Triple Systems with \(0\) or \(1\) Fixed Points

A cyclic triple, \((a, b, c)\), is defined to be the set \(\{(a, b), (b, c), (c, a)\}\) of ordered pairs. A Mendelsohn triple system of order \(v\), MTS\((v)\), is a pair \((M, \beta)\), where \(M\) is a set of \(v\) points and \(\beta\) 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 \(\beta\). An antiautomorphism of a Mendelsohn triple system, \((M, \beta)\), is a permutation of \(M\) which maps \(\beta\) to \(\beta^{-1}\), where \(\beta^{-1} = \{(c, b, a) \mid (a, b, c) \in \beta\}\). 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 two cycles of equal length and having \(0\) or \(1\) fixed points.