Decompositions of commutative monoid congruences and binomial ideals

TL;DR

This paper introduces mesoprimary decomposition for commutative monoid congruences and binomial ideals, providing a refined, computationally efficient method that generalizes primary decomposition in rings.

Contribution

It develops the theory of mesoprimary decomposition, including witnesses and associated primes, and extends it to binomial ideals in monoid algebras, offering a new intersection decomposition method.

Findings

Mesoprimary decomposition captures features of primary decomposition in monoids.

The new decomposition is computationally efficient and field-independent.

Binomial primary decompositions are derived from mesoprimary decomposition.

Abstract

Primary decomposition of commutative monoid congruences is insensitive to certain features of primary decomposition in commutative rings. These features are captured by the more refined theory of mesoprimary decomposition of congruences, introduced here complete with witnesses and associated prime objects. The combinatorial theory of mesoprimary decomposition lifts to arbitrary binomial ideals in monoid algebras. The resulting binomial mesoprimary decomposition is a new type of intersection decomposition for binomial ideals that enjoys computational efficiency and independence from ground field hypotheses. Binomial primary decompositions are easily recovered from mesoprimary decomposition.