Boolean ideals and their varieties

We consider ideals in the ring Z2[x1,…,xn]Z2[x1,…,xn] that contain the polynomials xi2−xi for i=1,…,ni=1,…,n and give various results related to the one-to-one correspondence between these ideals and the subsets of Z2n. We also study the standard monomials with respect to the lexicographical ordering for these ideals and derive a distribution result.