[Paper Review] Computation of the w-primality and asymptotic w-primality with applications to numerical semigroups

J. I. García-García, M. A. Moreno-Frías+1 more|arXiv (Cornell University)|Jul 22, 2013

Mathematical and Theoretical Analysis9 references4 citations

TL;DR

This paper presents an algorithm to compute the ω-primality of elements in finitely generated atomic monoids, with a focus on numerical semigroups. It derives a closed-form formula for asymptotic ω-primality in quasi-Archimedean monoids, proving that for a numerical semigroup minimally generated by $ s_1 < \cdots < s_p $, the asymptotic ω-primality of any element $ s $ is $ \overline{\omega}(s) = s/s_1 $.

ABSTRACT

We give an algorithm to compute the $ω$-primality of finitely generated atomic monoids. Asymptotic $\w$-primality is also studied and a formula to obtain it in finitely generated quasi-Archimedean monoids is proven. The formulation is applied to numerical semigroups, obtaining an expression of this invariant in terms of its system of generators.

Motivation & Objective

• To develop an algorithm for computing the ω-primality of elements in finitely generated atomic monoids using their finite presentations.
• To study the asymptotic ω-primality in finitely generated quasi-Archimedean cancellative monoids, a class that includes numerical semigroups.
• To derive a closed-form expression for asymptotic ω-primality in numerical semigroups based on their minimal generating set.
• To implement the theoretical results in a Mathematica software package named OmegaPrimality for practical computation.
• To establish a connection between the ω-invariant and the structure of the monoid’s generators, particularly in the context of numerical semigroups.

Proposed method

• Uses finite presentations of finitely generated commutative monoids as $ \mathbb{N}^p / \sigma $, where $ \sigma $ is a congruence relation.
• Applies the concept of minimal generating sets and equivalence classes $[\gamma]_\sigma$ to represent elements in the monoid.
• Employs the notion of $ \mathrm{E}(a+S) $, the set of elements that are not minimal in the divisor ideal, to compute $ \omega(a) $ via bounds on the length of minimal elements.
• Derives a lower bound for $ \overline{\omega}(a) $ using the limit $ \lim_{n \to \infty} \omega(na)/n $, and proves equality via optimization over rational points in $ \mathbb{Q}^p_{\geq} $.
• Uses the structure of quasi-Archimedean monoids to show that $ \overline{\omega}(a) = k_1(\gamma_1/k_1 + \cdots + \gamma_p/k_p) $, where $ k_i $ are derived from the least common multiple of the generators.
• Applies the result to numerical semigroups by setting $ k_i = \mathrm{lcm}(s_1,\dots,s_p)/s_i $, leading to the final formula $ \overline{\omega}(s) = s/s_1 $.

Experimental results

Research questions

• RQ1 How can the ω-primality of an element in a finitely generated atomic monoid be computed algorithmically from its presentation?
• RQ2 What is the asymptotic ω-primality in quasi-Archimedean cancellative monoids, and can it be expressed in closed form?
• RQ3 What is the asymptotic ω-primality of elements in numerical semigroups, and how does it relate to their minimal generating set?
• RQ4 Can the theoretical results be effectively implemented in a computational tool for practical use in factorization theory?
• RQ5 What is the precise relationship between the ω-invariant and the structure of the monoid’s generators in numerical semigroups?

Key findings

• An algorithm is developed to compute $ \omega(a) $ for any element $ a $ in a finitely generated atomic monoid from its finite presentation.
• For quasi-Archimedean cancellative monoids, the asymptotic ω-primality is given by $ \overline{\omega}(a) = k_1(\gamma_1/k_1 + \cdots + \gamma_p/k_p) $, where $ \gamma $ represents the coordinate vector of $ a $ and $ k_i $ are derived from the monoid’s structure.
• In numerical semigroups minimally generated by $ \langle s_1 < \cdots < s_p \rangle $, the asymptotic ω-primality of any element $ s $ is $ \overline{\omega}(s) = s/s_1 $, a simple and explicit formula.
• The formula $ \overline{\omega}(s) = s/s_1 $ is derived by expressing $ s $ as a linear combination of the generators and using the least common multiple of the generators to define the scaling factors $ k_i $.
• The theoretical results are complemented by the implementation of the OmegaPrimality software in Mathematica, enabling computation of $ \omega $-invariants from presentations or generators.
• The paper proves that the limit $ \lim_{n \to \infty} \omega(na)/n $ exists and equals the derived expression, confirming the asymptotic behavior of the ω-invariant.