Cyclic and Rotational Oriented Triple Systems

An oriented (or ordered) triple means either a Mendelsohn or a transitive triple. An oriented (or ordered) triple system of order \(v\), briefly OTS(\(v\)), is a pair \((V, B)\), where \(V\) is a \(v\)-set and \(B\) is a collection of oriented triples of elements of \(V\), such that every ordered pair of distinct elements of \(V\) belongs to exactly one member of \(B\). It is known that an OTS(\(v\)) exists if and only if \(v \equiv 0, 1 \pmod{3}\). An OTS(\(v\)) is cyclic if it admits an automorphism consisting of a single cycle of length \(v\); an OTS(\(v\)) is rotational if it admits an automorphism consisting of a single fixed point and one cycle of length \(v-1\). In this note we give some constructions of OTS(\(v\))’s which allow to determine the spectrum of cyclic and of rotational OTS(\(v\))’s.