  A $\mu$-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 $(f,\mu_{x}.f)$
where $\mu_{x}.f$ is axiomatized as the least prefixed point of $f$, whose
axioms are equations or equational implications.
  Standard $\mu$-algebras are complete meaning that their lattice reduct is a
complete lattice. We prove that any non trivial quasivariety of $\mu$-algebras
contains a $\mu$-algebra that has no embedding into a complete $\mu$-algebra.
  We focus then on modal $\mu$-algebras, i.e. algebraic models of the
propositional modal $\mu$-calculus. We prove that free modal $\mu$-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
$\mu$-algebras, (ii) least prefixed points of $\Sigma_{1}$-operations satisfy
the constructive relation $\mu_{x}.f = \bigvee_{n \geq 0} f^{n}(\bot)$. These
properties imply the following statement: {\em the MacNeille-Dedekind
completion of a free modal $\mu$-algebra is a complete modal $\mu$-algebra and
moreover the canonical embedding preserves all the operations in the class
$Comp(\Sigma_{1},\Pi_{1})$ of the fixed point alternation hierarchy.}
