Let $L\subset \mathbb{ Z}^n$ be a lattice and $I_L=\langle x^{ \bf u}-x^{ \bf v}:\ { \bf u}-{ \bf v}\in L\rangle$ be the corresponding lattice ideal in $\Bbbk[x_1,\ldots, x_n]$, 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.
Categories: math.AC math.CO
