Extensions of the noncommutative Standard Model and the weak order one condition

In the derivation of the Standard Model from the axioms of Noncommutative Geometry, the scalar sector is given by a finite Dirac operator which has to satisfy the so-called \emph{first-order condition}. However, the general solution to this constraint still has unphysical terms which must be fine-tuned to zero. Moreover, the first-order condition generally does not survive in extensions to models with gauge groups larger that $U(1)\times SU(2)\times SU(3)$. In this paper we show that in the $U(1)_{\rm B-L}$-extension one can implement a weaker form of the first-order condition which we argue is necessary in order for Noncommutative Gauge Theory to make sense at all, and that this condition reduce the amount of fine-tuning to the off-diagonal terms in the Yukawa mass matrices for the leptons and quarks. We also show that this condition eliminates the Majorana mass terms for right-handed neutrinos when it is applied to the Pati-Salam model.