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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 thelarge and small hierarchies among the quark and active neutrino massesrespectively, the quark and lepton families are not treated on the same footingunder the assignment of the discrete group. As a consequence, the NearestNeighbor Interactions (NNI) textures appear in the quark sector, leading to theCKM mixing matrix, whereas in the lepton sector, a soft breaking of the $\mu\leftrightarrow \tau$ symmetry in the effective neutrino mass that comes fromtype I see-saw mechanism, provides a non-maximal atmospheric angle and anon-zero reactor angle.

B-L Model with ${\bf S}_{3}$ Symmetry: Nearest Neighbor Interaction Textures and Broken $\mu\leftrightarrow\tau$ Symmetry