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We discover cancellations upon $H^{2}(\mathbb{R}^{n})$-estimate of the Hallterm for $n \in \{2,3\}$. As its consequence, first, we derive a regularitycriterion for the 3-dimensional Hall-magnetohydrodynamics system in terms ofonly horizontal components of velocity and magnetic fields. Second, we provethe global regularity of the $2\frac{1}{2}$-dimensional electronmagnetohydrodynamics 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 $2\frac{1}{2}$-dimensionalHall-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 globalregularity result requires is $11+ \epsilon$ for any $\epsilon > 0$ while theanalogous sum for the classical $2\frac{1}{2}$-dimensionalHall-magnetohydrodynamics system is 12 considering $-\Delta u$ and $-\Delta b$.

Another remark on the global regularity issue of the Hall-magnetohydrodynamics system