  Spontaneously broken Abelian gauge symmetries can explain the fermion mass
hierarchies of the minimal supersymmetric standard model. In most cases it is
assumed that the $U(1)_H$ symmetry is anomalous. However, non-anomalous models
are also viable and yield an interesting phenomenology. Cancellation of the
gauge anomalies implies the following results: unification of leptons and
down-type quarks Yukawa couplings is allowed at most for two generations. The
$\mu$ term is necessarily somewhat below the supersymmetry breaking scale. The
superpotential has accidental $B$ and $L$ symmetries, and R-parity is
automatically conserved in the supersymmetric limit. Anomaly canncellation also
implies that the determinant of the quark mass matrix must vanish, wich is
possible only if $m_{up}=0$. This solves the strong CP problem and provides an
unambiguous low energy test of the model.
