Binomial generation of the radical of a lattice ideal

Let $I_{L, \rho}$ be a lattice ideal. We provide a necessary and sufficient criterion under which a set of binomials in $I_{L, \rho}$ generate the radical of $I_{L, \rho}$ up to radical. We apply our results to the problem of determining the minimal number of generators of $I_{L, \rho}$ or of the $rad(I_{L, \rho})$ up to radical.