  In this article we study specializations of multigradings and apply them to
the problem of the computation of the arithmetical rank of a lattice ideal
$I_{L_{\mathcal{G}}} \subset K[x_{1},...,x_{n}]$. The arithmetical rank of
$I_{L_{\mathcal{G}}}$ equals the $\mathcal{F}$-homogeneous arithmetical rank of
$I_{L_{\mathcal{G}}}$, for an appropriate specialization $\mathcal{F}$ of
$\mathcal{G}$. To the lattice ideal $I_{L_{\mathcal{G}}}$ and every
specialization $\mathcal{F}$ of $\mathcal{G}$ we associate a simplicial
complex. We prove that combinatorial invariants of the simplicial complex
provide lower bounds for the $\mathcal{F}$-homogeneous arithmetical rank of
$I_{L_{\mathcal{G}}}$.
