In the Standard Model and GUT extended by the local chiral S U ( 3 ) F symmetry quarks and leptons are supposed to be S U ( 3 ) F chiral triplets, so that their left-handed (weak-doublet) components – Q L ( i ) and L L ( i ) – are taken to be the triplets of S U ( 3 ) F , while their right-handed (weak-singlet) components – U R ( i ) , D R ( i ) , N R ( i ) and E R ( i ) – are anti-triplets (or vice versa). Here i is the S U ( 3 ) F family symmetry index ( i = 1 , 2 , 3 ), rather than the index ( i ) introduced in Section 1 in order to simply number all the families involved. The spontaneous breaking of this symmetry gives some understanding to the observed hierarchy between elements of the quark-lepton mass matrices and presence of texture zeros in them. This breaking is normally provided by some set of the horizontal scalar multiplets being symmetrical and anti-symmetrical under the S U ( 3 ) F , χ χ i j ( a ) and η η [ i j ] ( b ) ( a = 1, 2,..., b = 1, 2,...). When they develop their VEVs, the up and down quark families acquire their effective Yukawa coupling constants which generally have a form