# The alternation hierarchy for the theory of $\mu$-lattices

## Luigi Santocanale

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.

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