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Chains and Superchains for ω-Rational Sets, Automata and Semigroups

We introduce several equivalent notions that generalize ones introduced by Klaus Wagner for finite Muller automata under the name of chains and superchains. We define such objects in relation to ω-rational sets, Muller automata or also ω-semigroups. We prove their equivalence and derive some basic properties of these objects. In a subsequent paper, we show how these concepts allow us to derive a new presentation of the hierarchy due to K. Wagner and W. Wadge.