Boolean ideals and their varieties

TL;DR

This paper explores the correspondence between Boolean ideals in a polynomial ring over Z2 and subsets of Z2^n, analyzing their structure, standard monomials, and distribution properties.

Contribution

It establishes a detailed correspondence between Boolean ideals and subsets of Z2^n and investigates their standard monomials and distribution characteristics.

Findings

One-to-one correspondence between Boolean ideals and subsets of Z2^n.

Characterization of standard monomials with respect to lex order.

Distribution results related to these ideals.

Abstract

We consider ideals in the ring $\mathbb{Z}_2[x_1,\ldots, x_n]$ that contain the polynomials $x_i^2 - x_i$ for $i = 1, \ldots, n$ 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.