  We build a class of invisible axion models with tree-level Flavor Changing
Neutral Currents completely controlled by the fermion mixing matrices. The
scalar sector of these models contains three-Higgs doublets and a complex
scalar gauge singlet, with the same fermionic content than the Standard Model.
A horizontal Peccei-Quinn symmetry provides a solution to the strong CP problem
and predicts the existence of a very light and weakly coupled pseudo-Goldstone
boson, the invisible axion or familon. A phenomenological analysis is performed
taking into account familon searches in rare kaon and muon decays,
astrophysical considerations and axion searches via axion-photon conversion.
Drastic differences are found in the axion properties of different models due
to the strong hierarchy of the CKM matrix, making some of the models considered
much more constrained than others. We also obtain that a rich variety of these
models avoid the domain wall problem. A possible mechanism to protect the
solution to the strong CP problem against gravitational effects is also
discussed.
