文摘
The main ingredient to construct an O-border basis of an ideal 6e352731543e0fb53e1af4cf" title="Click to view the MathML source">I⊆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 46b69f7e3c704fe12ebda2f43557f76" title="Click to view the MathML source">IM (where M is a lattice of Zn). 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 46b69f7e3c704fe12ebda2f43557f76" title="Click to view the MathML source">IM in case M is a 2-dimensional lattice contained in 4693006a31649fae144c10b" title="Click to view the MathML source">Z2.