  In this paper, axion models supplemented by leptoquarks and diquarks are
systematically analyzed. Turning on some couplings to and among these latter
states permits to unify the Peccei-Quinn symmetry with baryon (B) and lepton
(L) numbers, such that the axion becomes associated to the spontaneous breaking
of the three $U(1)$ symmetries. All possible four- and six-fermion patterns of
B and L violation are discussed, including those inducing proton decay, with
$\Delta B = 1$ and $\Delta L = \pm 1, \pm 3$, neutron-antineutron oscillations
with $\Delta B = 2$, and Majorana neutrino masses with $\Delta L = 2$.
Scenarios in which one or two axion fields necessarily appear in any B and/or L
violating operators are also constructed. Nucleon decays would then necessarily
involve an axion in the final state, while neutron-antineutron oscillations
would only happen in an axionic background. This could have implications for
the neutron lifetime puzzle, and more generally, opens the door to new
phenomenological and cosmological applications.
