  We study interacting bosons on a lattice in a magnetic field. When the number
of flux quanta per plaquette is close to a rational fraction, the low energy
physics is mapped to a multi-species continuum model: bosons in the lowest
Landau level where each boson is given an internal degree of freedom, or
pseudospin. We find that the interaction potential between the bosons involves
terms that do not conserve pseudospin, corresponding to umklapp processes,
which in some cases can also be seen as BCS-type pairing terms. We argue that
in experimentally realistic regimes for bosonic atoms in optical lattices with
synthetic magnetic fields, these terms are crucial for determining the nature
of allowed ground states. In particular, we show numerically that certain
paired wavefunctions related to the Moore-Read Pfaffian state are stabilized by
these terms, whereas certain other wavefunctions can be destabilized when
umklapp processes become strong.
