A
![$\mu$](Abs/img1.gif)
-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
![$\mu$](Abs/img1.gif)
-lattices and, for a given partially ordered set
![$P$](Abs/img2.gif)
, we construct a
![$\mu$](Abs/img1.gif)
-lattice
![${\cal J}_{P}$](Abs/img3.gif)
whose elements are
equivalence classes of games in a preordered class
![${\cal J}(P)$](Abs/img4.gif)
. We prove
that the
![$\mu$](Abs/img1.gif)
-lattice
![${\cal J}_{P}$](Abs/img3.gif)
is free over the ordered set
![$P$](Abs/img2.gif)
and
that the order relation of
![$\mathcal{J}_{P}$](Abs/img5.gif)
is decidable if the order
relation of
![$P$](Abs/img2.gif)
is decidable. By means of this characterization of free
![$\mu$](Abs/img1.gif)
-lattices we infer that the class of complete lattices generates the
quasivariety of
![$\mu$](Abs/img1.gif)
-lattices