Geodesic
The alternation hierarchy for the theory of $\mu$-lattices
Theory and applications of categories, CT2000, Tome 9 (2001), pp. 166-197.

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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.
Classification : 03D55, 06B25, 91A43.
Keywords: free lattices, free �-lattices, fixed points, parity games. 
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     author = {Luigi Santocanale},
     title = {The alternation hierarchy for the theory of $\mu$-lattices},
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     volume = {9},
     year = {2001},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/TAC_2001_9_a8/}
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Luigi Santocanale. The alternation hierarchy for the theory of $\mu$-lattices. Theory and applications of categories, CT2000, Tome 9 (2001), pp. 166-197. http://geodesic.mathdoc.fr/item/TAC_2001_9_a8/