  The Bose-Hubbard model is extended to include nearest and far neighbor
interactions and is related to the fractional quantum Hall effect (FQHE). Both
models may be studied in optical lattices with quantum gases. The ground state
is calculated for the extended Bose-Hubbard model with strong repulsive
interactions (weak hopping). Incompressible Mott insulator states are found at
rational filling fractions compatible with the principal and secondary FQHE
filling fractions of the lowest Landau levels observed experimentally. It is
discussed to which extent these states at fractional filling survive or
undergoes a Mott insulator transition to a superfluid as hopping terms are
included.
