Completions of μ-algebras

We define the class of algebraic models of μ-calculi and study whether every such model can be embedded into a model which is a complete lattice. We show that this is false in the general case and focus then on free modal μ-algebras, i.e. Lindenbaum algebras of the propositional modal μ-calculus. We prove the following fact: the MacNeille-Dedekind completion of a free modal μ-algebra is a complete modal algebra, hence a modal μ-algebra (i.e. an algebraic model of the propositional modal μ-calculus). The canonical embedding of the free modal μ-algebra into its Dedekind-MacNeille completion preserves the interpretation of all the terms in the class Comp(Σ₁Π₁) of the alternation-depth hierarchy. The proof uses algebraic techniques only and does not directly rely on previous work on the completeness of the modal μ-calculus.