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
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