Minimal Generating Sets of Lattice Ideals

Let $L\subset {\mathbb{Z}}^n$ be a lattice and $I_L=\langle x^{\bf u}-x^{\bf v}:\ {\bf u}-{\bf v}\in L\rangle$bethecorrespondinglatticeidealin$\Bbbk[x_1,\ldots, x_n]$,where$\Bbbk$ is a field. In this paper we describe minimal binomial generating sets of $I_L$ and their invariants. We use as a main tool a graph construction on equivalence classes of fibers of $I_L$. As one application of the theory developed we characterize binomial complete intersection lattice ideals, a longstanding open problem in the case of non-positive lattices.