Border bases for lattice ideals

The main ingredient to construct an O -border basis of an ideal I ⊆ K [ x 1 , . . . , x n ] is the order ideal O , which is a basis of the K -vector space K [ x 1 , . . . , x n ]/ I. In this paper we give a procedure to find all the possible order ideals associated with a lattice ideal I M (where M is a lattice of Z n ). The construction can be applied to ideals of any dimension (not only zero-dimensional) and shows that the possible order ideals are always in a finite number. For lattice ideals of positive dimension we also show that, although a border basis is infinite, it can be defined in finite terms. Furthermore we give an example which proves that not all border bases of a lattice ideal come from Gröbner bases. Finally, we give a complete and explicit description of all the border bases for ideals I M in case M is a 2-dimensional lattice contained in Z 2 .