The alternation hierarchy problem asks whether every $\mu$-term $\phi$, that is, a term built up also using a least fixed point constructor as well as a greatest fixed point constructor, is equivalent to a $\mu$-term where the number of nested fixed points of a different type is bounded by a constant independent of $\phi$.
In this paper we give a proof that the alternation hierarchy for the theory of $\mu$-lattices is strict, meaning that such a constant does not exist if $\mu$-terms are built up from the basic lattice operations and are interpreted as expected. The proof relies on the explicit characterization of free $\mu$-lattices by means of games and strategies.
Keywords: free lattices, free µ-lattices, fixed points, parity games.
2000 MSC: 03D55, 06B25, 91A43.
Theory and Applications of Categories, Vol. 9, 2001, No. 9, pp 166-197.