A Fine Hierarchy of ω-Regular k-Partitions

We develop the theory of ω-regular k-partitions (for arbitrary k ≥ 2) that extends the theory around the Wagner hierarchy of regular ω-languages. In particular, we characterize the structure of Wadge degrees of ω-regular k-partitions, prove the decidability of any level of the corresponding hierarchy, establish coincidence of the reducibilities by continuous functions and by functions computed by finite automata on the ω-regular k-partitions.