Completions of [formula omitted]-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 nontrivial quasivariety of μ -algebras contains a μ -algebra that has no embedding into a complete μ -algebra. We then focus 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 = ⋁ 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 ( Σ 1 , Π 1 ) of the fixed point alternation hierarchy.