Free mu-lattices

A mu-lattice is a lattice with the property that every unary polynomial has both a least and a greatest fix-point. In this paper we define the quasivariety of mu-lattices and, for a given partially ordered set P, we construct a mu-lattice JP whose elements are equivalence classes of games in a preordered class J (P). We prove that the mu-lattice JP is free over the ordered set P and that the order relation of JP is decidable if the order relation of P is decidable. By means of this characterization of free mu-lattices we infer that the class of complete lattices generates the quasivariety of mu-lattices. Keywords: mu-lattices, free mu-lattices, free lattices, bicompletion of categories, models of computation, least and greatest fix-points, mu-calculus, Rabin chain games.