The alternation hierarchy problem asks whether every
![$\mu$](Abs/img1.gif)
-term, that is a term built up using also a least fixed point
constructor as well as a greatest fixed point constructor, is equivalent to a
![$\mu$](Abs/img1.gif)
-term where the number of nested fixed point of a different type is
bounded by a fixed number.
In this paper we give a proof that the
alternation hierarchy for the theory of
-lattices is strict, meaning
that such number does not exist if
-terms are built up from the basic
lattice operations and are interpreted as expected. The proof relies on the
explicit characterization of free
-lattices by means of games and
strategies