arXiv:math/0508412v1 [math.RA] 22 Aug 2005

arXiv:math/0508412v1 [math.RA] 22 Aug 2005

Completions of µ-Algebras Luigi Santocanale LIF-CMI Marseille [email protected] July 17, 2011 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, µx .f ) where µx .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 µalgebras, (ii) least W prefixed points of Σ1 -operations satisfy the constructive relation µx .f = n≥0 f n (⊥). These properties imply the following statement: 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(Σ1 , Π1 ) of the fixed point alternation hierarchy.

Introduction When L is a complete lattice, the least fixed point µx .f of a monotone function f : L ✲ L enjoys a remarkable property. We like to say that the least fixed point is constructive: the equality _ µx .f = f α (⊥) (1) α∈Ord

holds and provides a method to construct µx .f from the bottom of the lattice. The expressions f α (⊥), indexed by ordinals, are commonly called the

1

approximants of µx .f . They are defined by W transfinite induction as expected: f 0 (⊥) = ⊥, f α+1 = f (f α (⊥)), and f α (⊥) = β