Free $\mu$-lattices

Luigi Santocanale

November 2000


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 ${\cal J}_{P}$ whose elements are equivalence classes of games in a preordered class ${\cal J}(P)$. We prove that the $\mu$-lattice ${\cal J}_{P}$ is free over the ordered set $P$ and that the order relation of $\mathcal{J}_{P}$ 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

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