Factorizations in reciprocal Puiseux monoids

TL;DR

This paper studies reciprocal Puiseux monoids, a class of additive submonoids of rationals generated by reciprocals of pairwise coprime integers, focusing on their atomicity, ideal structure, and element factorizations.

Contribution

It identifies and investigates generalized classes of atomic reciprocal Puiseux monoids, analyzing their factorization properties and ideal conditions.

Findings

Certain classes of reciprocal Puiseux monoids are atomic.

These monoids satisfy the ascending chain condition on principal ideals.

The sets of lengths of elements in these monoids are characterized.

Abstract

A Puiseux monoid is an additive submonoid of the real line consisting of rationals. We say that a Puiseux monoid is reciprocal if it can be generated by the reciprocals of the terms of a strictly increasing sequence of pairwise relatively primes positive integers. We say that a commutative and cancellative (additive) monoid is atomic if every non-invertible element $x$ can be written as a sum of irreducibles. The number of irreducibles in this sum is called a length of $x$. In this paper, we identify and investigate generalized classes of reciprocal Puiseux monoids that are atomic. Moreover, for the atomic monoids in the identified classes, we study the ascending chain condition on principal ideals and also the sets of lengths of their elements.