  We obtain local well-posedness result for the generalized
Hall-magneto-hydrodynamics system in Besov spaces ${\dot
B^{-(2\alpha_1-\gamma)}_{\infty, \infty}} \times {\dot
B^{-(2\alpha_2-\beta)}_{\infty,\infty}(\mathbb R^3)}$ with suitable indexes
$\alpha_1, \alpha_2, \beta$ and $\gamma.$ As a corollary, the generalized
electron magneto-hydrodynamics system is globally well-posed in ${\dot
B^{-(2\alpha_2-2)}}_{\infty, \infty}(\mathbb R^3)$ for small initial data.
