Colon structure of associated primes of monomial ideals

TL;DR

This paper provides an explicit formula for the associated primes of monomial ideals as colon ideals, introduces an algorithm for computation, and explores special cases related to combinatorics and Borel type ideals.

Contribution

It offers a new explicit expression for associated primes of monomial ideals and an algorithm for their computation, with special case analyses.

Findings

Explicit colon ideal expression for associated primes

Algorithm for computing the element v using Macaulay2

Characterization of cases with unique f for Borel type ideals

Abstract

We find an explicit expression of the associated primes of monomial ideals as a colon by an element $v$, using the unique irredundant irreducible decomposition whose irreducible components are monomial ideals (Theorem 3.1). An algorithm to compute $v$ is given using Macaulay2 (Section 7). For squarefree monomial ideals the problem is related to the combinatorics of the underlying clutter or graph (Proposition 4.3). For ideals of Borel type the monomial $f$ takes a simpler form (Proposition 5.2). The authors classify when $f$ is unique (Proposition 6.2).