  We develop a theory of k-partitions of the set of infinite words recognizable
by classes of finite automata. The theory enables to complete proofs of
existing results about topological classifications of the (aperiodic)
omega-regular k-partitions and provides tools for dealing with other similar
questions. In particular, we characterize the structure of Wadge degrees of
(aperiodic) omega-regular $k$-partitions, prove the decidability of many
related problems, and discuss their complexity.
