## Paper: Completions of µ-algebras (at LICS 2005)

**Luigi Santocanale**

### Abstract

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(?1,?1) 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.

### BibTeX

@InProceedings{Santocanale-Completionsofalgebr, author = {Luigi Santocanale}, title = {Completions of µ-algebras}, booktitle = {Proceedings of the Twentieth Annual IEEE Symposium on Logic in Computer Science (LICS 2005)}, year = {2005}, month = {June}, pages = {219--228}, location = {Chicago, USA}, publisher = {IEEE Computer Society Press} }