On dynamic algorithms for factorization invariants in numerical monoids

TL;DR

This paper introduces dynamic algorithms for efficiently computing key factorization invariants in numerical monoids, significantly improving runtime and memory usage over existing methods.

Contribution

The paper presents novel dynamic algorithms for factorization invariants in numerical monoids, extending the computation of $oldsymbol{ extomega}$-primality to quotient groups.

Findings

Algorithms significantly reduce computation time.

Memory usage is optimized for large monoids.

Extended $oldsymbol{ extomega}$-primality to quotient groups.

Abstract

Studying the factorization theory of numerical monoids relies on understanding several important factorization invariants, including length sets, delta sets, and $\omega$-primality. While progress in this field has been accelerated by the use of computer algebra systems, many existing algorithms are computationally infeasible for numerical monoids with several irreducible elements. In this paper, we present dynamic algorithms for the factorization set, length set, delta set, and $\omega$-primality in numerical monoids and demonstrate that these algorithms give significant improvements in runtime and memory usage. In describing our dynamic approach to computing $\omega$-primality, we extend the usual definition of this invariant to the quotient group of the monoid and show that several useful results naturally extend to this broader setting.