Description
The standard monomials of a monomial ideal
I (those monomials that are not in
I) can be enumerated as follows. Given a monomial
m and a subset
F of the variables, the pair
(m,F) indexes the set of monomials of the form
mm' where the monomial
m' is supported on
F. A
List of pairs
(m, F) form
standard pairs for the monomial ideal
I if it satisfies the following three conditions:

for each pair, the monomial m is supported on the complement of F;

all of the monomials represented by a pair are standard;

the pairs index disjoint sets of monomials.
i1 : R = QQ[x,y,z];

i2 : I = monomialIdeal(x*y^3*z, x*y^2*z^2, y^3*z^2, y^2*z^3)
3 2 2 3 2 2 3
o2 = monomialIdeal (x*y z, x*y z , y z , y z )
o2 : MonomialIdeal of R

i3 : standardPairs I
2 2 2
o3 = {{1, {x, z}}, {y, {x, z}}, {1, {x, y}}, {z, {y}}, {y z, {x}}, {y z ,

{}}}
o3 : List

The standard pairs are computed with Algorithm 3.2.5 in Gröbner Deformations of Hypergeometric Differential Equations, by Mutsumi Saito, Bernd Sturmfels and Nobuki Takayama; Algorithms and Computation in Mathematics 6, SpringerVerlag, 2000. Implemented by Gregory G. Smith.
For more information, see the
Monomial ideals chapter in Computations in algebraic geometry with Macaulay2, edited by David Eisenbud, Daniel R. Grayson, Michael E. Stillman, and Bernd Sturmfels, Algorithms and Computations in Mathematics 8, SpringerVerlag, 2001.