  The main ingredient to construct an O-border basis of an ideal I $\subseteq$
K[x1,. .., xn] is the order ideal O, which is a basis of the K-vector space
K[x1,. .., xn]/I. In this paper we give a procedure to find all the possible
order ideals associated with a lattice ideal IM (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 Gr\"obner bases. Finally, we give a complete and
explicit description of all the border bases for ideals IM in case M is a
2-dimensional lattice contained in Z 2 .
