The theory of μ-lattices

The concept of a μ-algebra for an equational theory T = 〈Ω, E〉 was introduced in [10]. If the underlying set of a T-algebra is a complete lattice and if the operations from Ω preserve the partial order, the existence of least and greatest fix-points makes it possible to interpret μ-terms, which are derived from the primitive ones in Ω by substitution and by two formal operations of taking fixpoints. More generally, if a partial order is definable by means of the operations and relations of T, as it is the case for the theory of lattices, the least and greatest fix-points can be axiomatised by means of implications between equations; in this abstract way we define a theory μ-T and a corresponding quasivariety of μ-T-algebras. It is well known fact that the investigation of free μ-T-algebras is equivalent to the investigation of the theory, at least if we are concerned with the equations which are satisfied in every model.