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 .
