  We make a scalar extension of the B-L gauge model where the ${\bf S}_{3}$
non-abelian discrete group drives mainly the Yukawa sector. Motived by the
large and small hierarchies among the quark and active neutrino masses
respectively, the quark and lepton families are not treated on the same footing
under the assignment of the discrete group. As a consequence, the Nearest
Neighbor Interactions (NNI) textures appear in the quark sector, leading to the
CKM mixing matrix, whereas in the lepton sector, a soft breaking of the $\mu
\leftrightarrow \tau$ symmetry in the effective neutrino mass that comes from
type I see-saw mechanism, provides a non-maximal atmospheric angle and a
non-zero reactor angle.
