Cyclically Indecomposable Cyclic \(\lambda\)-fold Triple Systems that are Decomposable for \(\lambda= 2, 3,4\)

A triple system is decomposable if the blocks can be partitioned into two sets, each of which is itself a triple system. It is cyclically decomposable if the resulting triple systems are themselves cyclic. In this paper, we prove that a cyclic two-fold triple system is cyclically indecomposable if and only if it is indecomposable. Moreover, we construct cyclic three-fold triple systems of order $v$ which are cyclically indecomposable but decomposable for all \(v \equiv 3 \mod 6\). The only known example of a cyclic three-fold triple system of order \(v \equiv 1 \mod 6\) that is cyclically indecomposable but decomposable was a triple system on 19 points. We present a construction which yields infinitely many such triple systems of order \(v \equiv 1 \mod 6\). We also give several examples of cyclically indecomposable but decomposable cyclic four-fold triple systems and few constructions that yield infinitely many such triple systems.