We solve positively a conjecture of L. P. Belluce by using the notion
of singular element of an $MV$-algebra. This concept implies a
decomposition theorem for complete $MV$-algebras, formally analogous to
that one for lattice-ordered complete groups. We also prove that
strongly stonian $MV$-algebras correspond, via the well known functor $\Gamma$,
to lattice-ordered Abelian groups with strong unit which are strongly
projectable.