  We discover cancellations upon $H^{2}(\mathbb{R}^{n})$-estimate of the Hall
term for $n \in \{2,3\}$. As its consequence, first, we derive a regularity
criterion for the 3-dimensional Hall-magnetohydrodynamics system in terms of
only horizontal components of velocity and magnetic fields. Second, we prove
the global regularity of the $2\frac{1}{2}$-dimensional electron
magnetohydrodynamics system with magnetic diffusion $(-\Delta)^{\frac{3}{2}}
(b_{1}, b_{2}, 0) + (-\Delta)^{\alpha} (0, 0, b_{3})$ for $\alpha >
\frac{1}{2}$. Lastly, we extend this result to the $2\frac{1}{2}$-dimensional
Hall-magnetohydrodynamics system with $-\Delta u$ replaced by
$(-\Delta)^{\alpha} (u_{1}, u_{2}, 0) -\Delta (0, 0, u_{3})$ for $\alpha >
\frac{1}{2}$. The sum of the derivatives in diffusion that our global
regularity result requires is $11+ \epsilon$ for any $\epsilon > 0$ while the
analogous sum for the classical $2\frac{1}{2}$-dimensional
Hall-magnetohydrodynamics system is 12 considering $-\Delta u$ and $-\Delta b$.
