  We investigate to what extent a generic, generation-dependent $U(1)$ symmetry
acting on the quark Yukawa operators can reduce the number of free parameters
by forcing some entries in the Yukawa matrices to vanish. The maximal reduction
compatible with CP violation yields nine real parameters and one phase, which
matches the number of physical observables, implying that such models have no
free parameters. We derive a set of results: (i) the only possible structures
have the form $M_4 \oplus M_5$, where the subscripts indicate the number of
real parameters in the Yukawa matrices, (ii) there are only two inequivalent
Yukawa structures, each one giving rise to six different models depending on
quark flavour assignments, (iii) the $U(1)$ symmetries that generate these
textures all have a QCD anomaly, and hence are Peccei-Quinn symmetries,
reinforcing the idea of a possible connection between the quark flavour puzzle
and the axion solution to the strong CP problem, (iv) in some cases the
contributions to the QCD anomaly of two generations cancels out, and this opens
the possibility that the axion coupling to nucleons could be strongly
suppressed. Flavour-violating axion couplings to quarks are completely fixed,
up to the axion decay constant, providing a non-trivial complementarity between
low-energy flavour-violating processes and standard axion searches.
