Standard pairs for monomial ideals in semigroup rings

TL;DR

This paper generalizes the concept of standard pairs to monomial ideals in semigroup rings, offering algorithms for their computation and applications in ideal operations, especially in non-normal cases.

Contribution

It introduces a new framework for standard pairs in semigroup rings and provides effective algorithms for their computation, addressing challenges of non-normality.

Findings

Effective algorithms for computing standard pairs from generators.

Standard pairs encode monomial ideals efficiently for intersection and decomposition.

Addresses complexities arising from non-normal semigroup rings.

Abstract

We extend the notion of standard pairs to the context of monomial ideals in semigroup rings. Standard pairs can be used as a data structure to encode such monomial ideals, providing an alternative to generating sets that is well suited to computing intersections, decompositions, and multiplicities. We give algorithms to compute standard pairs from generating sets and vice versa and make all of our results effective. We assume that the underlying semigroup ring is positively graded, but not necessarily normal. The lack of normality is at the root of most challenges, subtleties, and innovations in this work.