Completions of mu-algebras

Abstract

A $μ$-algebra is a model of a first order theory that is an extension of the theory of bounded lattices, that comes with pairs of terms $(f,μ_{x}.f)$ where $μ_{x}.f$ is axiomatized as the least prefixed point of $f$, whose axioms are equations or equational implications. Standard $μ$-algebras are complete meaning that their lattice reduct is a complete lattice. We prove that any non trivial quasivariety of $μ$-algebras contains a $μ$-algebra that has no embedding into a complete $μ$-algebra. We focus then on modal $μ$-algebras, i.e. algebraic models of the propositional modal $μ$-calculus. We prove that free modal $μ$-algebras satisfy a condition -- reminiscent of Whitman's condition for free lattices -- which allows us to prove that (i) modal operators are adjoints on free modal $μ$-algebras, (ii) least prefixed points of $Σ_{1}$-operations satisfy the constructive relation $μ_{x}.f = \bigvee_{n \geq 0} f^{n}(\bot)$. These properties imply the following statement: {\em the MacNeille-Dedekind completion of a free modal $μ$-algebra is a complete modal $μ$-algebra and moreover the canonical embedding preserves all the operations in the class $Comp(Σ_{1},Π_{1})$ of the fixed point alternation hierarchy.}