Irreducible decomposition of binomial ideals

TL;DR

This paper develops a combinatorial method for irreducible decomposition of binomial ideals, extending prior work and providing new insights into their structure and limitations.

Contribution

It introduces a combinatorial construction for irreducible decompositions of binomial ideals and related monoid congruences, addressing open questions about their decomposability.

Findings

Constructed irreducible decompositions for binomial ideals

Provided a counterexample to binomial ideals being intersections of binomial irreducible ideals

Extended coprincipal mesoprimary decomposition to a broader context

Abstract

Building on coprincipal mesoprimary decomposition [Kahle and Miller, 2014], we combinatorially construct an irreducible decomposition of any given binomial ideal. In a parallel manner, for congruences in commutative monoids we construct decompositions that are direct combinatorial analogues of binomial irreducible decompositions, and for binomial ideals we construct decompositions into ideals that are as irreducible as possible while remaining binomial. We provide an example of a binomial ideal that is not an intersection of binomial irreducible ideals, thus answering a question of Eisenbud and Sturmfels [1996].