Boolean ideals and their varieties

Abstract

We consider ideals in the ring ${\mathbb{Z}}_2[x_1,\dots ,x_n]$ that contain the polynomials $x_i^2-x_i$ for $i=1,\dots ,n$ and give various results related to the one-to-one correspondence between these ideals and the subsets of ${\mathbb{Z}}_2^n$. We also study the standard monomials with respect to the lexicographical ordering for these ideals and derive a distribution result.

arXiv comment22 pages. New title. The connection between the boolean polynomial and its zero set is more carefully studied than in the earlier versions of the paper. There is also an new section about the connection between lexicographical standard monomials and boolean ideals