As already known [14] , the mu-calculus [17] is as expressive as the bisimulation invariant fragment of monadic second order Logic (MSO). 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 van Benthem's result [3] , we know already that the

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... t free fragment of the mu-calculus (i.e. polymodal Logic) is as expressive as the bisimulation invariant fragment of monadic ¦ ¼ (i.e. first order logic). We show here that the -level (resp. the -level) of the mu-calculus hierarchy is as expressive as the bisimulation invariant fragment of monadic ¦ ½ (resp. monadic ¦ ¾ ) and we show that no other level ¦ for ¾ 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 ¦ of the monadic hierarchy, for some ¾, is also discussed.