Cyclically Indecomposable Triple Systems that are Decomposable

In this paper, we investigate exhaustively the cyclically indecomposable triple systems \( TS_\lambda(v) \) for \( \lambda = 2, v \leq 33 \) and \( \lambda = 3, v \leq 21 \), and we identify the decomposable ones. We also construct, by using Skolem-type and Rosa-type sequences, cyclically indecomposable two-fold triple systems \( TS_2(v) \) for all admissible orders. Further, we investigate exhaustively all cyclic \( TS_2(v) \) that are constructed by Skolem-type and Rosa-type sequences up to \( v \leq 45 \) for indecomposability.