  Whether or not the solution to the $2\frac{1}{2}$-dimensional
Hall-magnetohydrodynamics system starting from smooth initial data preserves
its regularity for all time remains a challenging open problem. Although the
research direction on component reduction of regularity criterion for
Navier-Stokes equations and magnetohydrodynamics system have caught much
attention recently, the Hall term has presented much difficulty. In this
manuscript we discover a certain cancellation within the Hall term and obtain
various new regularity criterion: first, in terms of a gradient of only the
third component of the magnetic field; second, in terms of only the third
component of the current density; third, in terms of only the third component
of the velocity field; fourth, in terms of only the first and second components
of the velocity field. As another consequence of the cancellation that we
discovered, we are able to prove the global well-posedness of the
$2\frac{1}{2}$-dimensional Hall-magnetohydrodynamics system with
hyper-diffusion only for the magnetic field in the horizontal direction; we
also obtained an analogous result in the 3-dimensional case via discovery of
additional cancellations. These results extend and improve various previous
works.
