In 1997, following the works of Klaus W. Wagner on omega-regular sets, Olivier Carton and Dominique Perrin introduced the notions of Chains and Superchains for omega-semigroups. There is a clear correspondance between the algebraic representation of each of these operations and the automata-theoretical one. Unfortunately, chains and superchains do not suffice to describe the whole Wagner hierarchy. We introduce a third notion which completes the task undertaken in Carton O., Perrin D.: "Chains and superchains for omega-rational sets, automata and semigroups" (Int. J. Alg. Comput., vol. 7, no. 7, pp. 673-695, 1997).