[Paper Review] On three families of dense Puiseux monoids

Chapman, Scott. T., Marly Gotti+2 more|arXiv (Cornell University)|Dec 31, 2016

Rings, Modules, and Algebras4 references3 citations

TL;DR

This paper investigates the atomic structure of three families of dense Puiseux monoids—k-primary, p-adic, and multiplicatively cyclic monoids—proving that atomic multiplicatively cyclic Puiseux monoids are FF-monoids and hereditarily atomic. It further constructs infinitely many non-isomorphic atomic Puiseux monoids whose sets of atoms are dense in ℝ≥0.

ABSTRACT

A positive monoid is a submonoid of the nonnegative cone of a linearly ordered abelian group. The positive monoids of rank $1$ are called Puiseux monoids, and their atomicity, arithmetic of length, and factorization have been systematically investigated for about ten years. Each Puiseux monoid can be realized as an additive submonoid of the nonnegative cone of $\mathbb{Q}$. We say that a Puiseux monoid is dense if it is isomorphic to an additive submonoid of $\mathbb{Q}_{\ge 0}$ that is dense in $\mathbb{R}_{\ge 0}$ with respect to the Euclidean topology. Every non-dense Puiseux monoid is known to be a bounded factorization monoid. However, the atomic structure as well as the arithmetic and factorization properties of dense Puiseux monoids turn out to be quite interesting. In this paper, we study the atomic structure and some arithmetic and factorization aspects of three families of dense Puiseux monoids.

Motivation & Objective

• To understand the complex atomic structure of dense Puiseux monoids, which can be antimatter, atomic, or have finitely or countably many atoms.
• To construct infinitely many non-isomorphic atomic Puiseux monoids whose sets of atoms are dense in ℝ≥0.
• To characterize antimatter and hereditarily atomic subfamilies within k-primary and multiplicatively cyclic Puiseux monoids.
• To establish necessary and sufficient conditions for atomicity in p-adic Puiseux monoids and extend results to broader families.

Proposed method

• Constructing infinite families of atomic Puiseux monoids by selecting generators with specific rational forms to ensure density of their atom sets in ℝ≥0.
• Generalizing k-primary Puiseux monoids and analyzing their atomicity via valuation-theoretic arguments on prime powers.
• Using p-adic valuation techniques to determine when a Puiseux monoid with denominators as powers of a fixed prime is atomic.
• Proving that every atomic multiplicatively cyclic Puiseux monoid is an FF-monoid by bounding the number of atoms in factorizations.
• Applying the result that every BF-monoid in an ordered field is hereditarily atomic, and extending it to dense Puiseux monoids.
• Using submonoid embeddings and gcd conditions on numerators to prove hereditary atomicity in broader families of Puiseux monoids.

Experimental results

Research questions

• RQ1 Can we construct infinitely many non-isomorphic atomic Puiseux monoids whose sets of atoms are dense in ℝ≥0?
• RQ2 Which k-primary Puiseux monoids are antimatter, and what conditions characterize them?
• RQ3 What are the necessary and sufficient conditions for a p-adic Puiseux monoid to be atomic?
• RQ4 Are all atomic multiplicatively cyclic Puiseux monoids hereditarily atomic, and do they satisfy the FF-monoid property?
• RQ5 Under what conditions is a Puiseux monoid generated by powers of rational numbers with common prime factors in numerators hereditarily atomic?

Key findings

• The paper constructs infinitely many non-isomorphic atomic Puiseux monoids whose sets of atoms are dense in ℝ≥0.
• A subfamily of k-primary Puiseux monoids is characterized as antimatter when the greatest common divisor of the numerators of the generators is greater than 1.
• Every atomic multiplicatively cyclic Puiseux monoid is an FF-monoid, meaning that every element has only finitely many factorizations into atoms.
• Every atomic multiplicatively cyclic Puiseux monoid is hereditarily atomic, meaning all its submonoids are atomic.
• The condition that the numerators of the generators share a common prime factor is necessary for hereditary atomicity; without it, non-hereditarily atomic monoids can arise.
• The Puiseux monoid ⟨(2/77)^n, (3/77)^m | n,m∈ℕ⟩ is not hereditarily atomic because it contains the antimatter monoid ⟨1/7^k | k∈ℕ⟩ as a submonoid.