Computing generating sets of lattice ideals and Markov bases of lattices

In this article, we present an algorithm for computing generating sets of lattice ideals or equivalently for computing Markov bases of lattices. Generating sets of lattice ideals and Markov bases of lattices are essentially equivalent concepts. In contrast to other existing methods, the algorithm in this article computes with projections of lattices. This algorithm clearly outperforms other algorithms in our computational experience. Two areas of application for generating sets of lattice ideals and Markov bases lattices are algebraic statistics and integer programming.