In this article we associate to every lattice ideal I L, ⊂ K[x 1 , . . . , x m ] a cone and a simplicial complex with vertices the minimal generators of the Stanley-Reisner ideal of . We assign a simplicial subcomplex (F ) of to every polynomial F. If F 1 , . . . , F s generate I L, or they generate rad(I L, ) up to radical, then s i=1 (F i ) is a spanning subcomplex of . This result provides a lower bound for the minimal number of generators of I L, which 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 given bounds are sharp. (A. Katsabekis), Marcel.Morales@ujf-grenoble.fr (M. Morales), athoma@cc.uoi.gr (A. Thoma).