On strongly primary monoids, with a focus on Puiseux monoids

TL;DR

This paper investigates the structure and properties of Puiseux monoids, a class of additive submonoids of non-negative rationals, focusing on conditions for strong primeness and tameness, and analyzing their factorization behavior.

Contribution

It provides new conditions for Puiseux monoids to be strongly primary and characterizes when they are globally tame, advancing the understanding of their factorization properties.

Findings

Puiseux monoids are primary monoids and conditions for strong primeness are established.

Characterization of globally tame Puiseux monoids is provided.

Structure of sets of lengths in locally tame strongly primary monoids is analyzed.

Abstract

Primary and strongly primary monoids and domains play a central role in the ideal and factorization theory of commutative monoids and domains. It is well-known that primary monoids satisfying the ascending chain condition on divisorial ideals (e.g., numerical monoids) are strongly primary; and the multiplicative monoid of non-zero elements of a one-dimensional local domain is primary and it is strongly primary if the domain is Noetherian. In the present paper, we focus on the study of additive submonoids of the non-negative rationals, called Puiseux monoids. It is easy to see that Puiseux monoids are primary monoids, and we provide conditions ensuring that they are strongly primary. Then we study local and global tameness of strongly primary Puiseux monoids; most notably, we establish an algebraic characterization of when a Puiseux monoid is globally tame. Moreover, we obtain a result on the structure of sets of lengths of all locally tame strongly primary monoids.