Relating levels of the mu-calculus hierarchy and levels of the monadic hierarchy

As is already known from the work of D. Janin & I. Walukiewicz (1996), the mu-calculus is as expressive as the bisimulation-invariant fragment of monadic second-order logic. In this paper, we relate the expressiveness of levels of the fixpoint alternation depth hierarchy of the mu-calculus (the mu-calculus hierarchy) with the expressiveness of the bisimulation-invariant fragment of levels of the monadic quantifiers alternation-depth hierarchy (the monadic hierarchy). From J. van Benthem´s (1976) results, we know already that the fixpoint free fragment of the mu-calculus (i.e. polymodal logic) is as expressive as the bisimulation-invariant fragment of monadic Σ₀ (i.e. first-order logic). We show that the ν-level of the mu-calculus hierarchy is as expressive as the bisimulation-invariant fragment of monadic Σ₁ and that the νμ-level of the mu-calculus hierarchy is as expressive as the bisimulation-invariant fragment of monadic Σ₂, and we show that no other level Σ_k (for k>2) of the monadic hierarchy can be related similarly with any other level of the mu-calculus hierarchy. The possible inclusion of all the mu-calculus in some level Σ_k of the monadic hierarchy, for some k>2, is also discussed