  In this article we associate to every lattice ideal $I_{L,\rho}\subset
K[x_1,..., x_m]$ a cone $\sigma $ and a graph $G_{\sigma}$ with vertices the
minimal generators of the Stanley-Reisner ideal of $\sigma $. To every
polynomial $F$ we assign a subgraph $G_{\sigma}(F)$ of the graph $G_{\sigma}$.
Every expression of the radical of $I_{L,\rho}$, as a radical of an ideal
generated by some polynomials $F_1,..., F_s$ gives a spanning subgraph of
$G_{\sigma}$, the $\cup_{i=1}^s G_{\sigma}(F_i)$. This result provides a lower
bound for the minimal number of generators of $I_{L,\rho}$ and therefore
improves the generalized Krull's principal ideal theorem for lattice ideals.
But mainly it provides lower bounds for the binomial arithmetical rank and the
$A$-homogeneous arithmetical rank of a lattice ideal. Finally we show, by a
family of examples, that the bounds given are sharp.
