  We explore the behavior of interacting bosonic atoms in an optical lattice
subject to a large artificial magnetic field. We extend earlier investigations
of this system where the number of magnetic flux quanta per unit cell alpha is
close to a simple rational number [Phys. Rev. Lett. 96, 180407 (2006)].
Interesting topological states such as the Laughlin and Read-Rezayi states can
occur even if the atoms experience a weak trapping potential in one direction.
An explicit numerical calculation near alpha = 1/2 shows that the system
exhibits a striped vortex lattice phase of one species, which is analogous to
the behavior of a two-species system for small alpha. We also investigate
methods to probe the encountered states. These include spatial correlation
functions and the measurement of noise correlations in time of flight expanded
atomic clouds. Characteristic differences arise which allow for an
identification of the respective quantum Hall states. We furthermore discuss
that a counterintuitive flow of the Hall current occurs for certain values of
alpha.
