On the local k-elasticities of Puiseux monoids

TL;DR

This paper investigates the local k-elasticities of Puiseux monoids, identifying conditions for finiteness and infinity, and provides specific results for primary Puiseux monoids, advancing the understanding of their factorization properties.

Contribution

It extends the study of elasticities in Puiseux monoids by analyzing local k-elasticities and establishing conditions for their finiteness or infiniteness, especially in primary Puiseux monoids.

Findings

Puiseux monoids can have finite or infinite local k-elasticities depending on certain conditions.

Primary Puiseux monoids have finite local k-elasticities if they are boundedly generated without stable atoms or lack 0 as a limit point.

The paper characterizes when Puiseux monoids exhibit finite or infinite local elasticities based on their structural properties.

Abstract

If $M$ is an atomic monoid and $x$ is a nonzero non-unit element of $M$, then the set of lengths $\mathsf{L}(x)$ of $x$ is the set of all possible lengths of factorizations of $x$, where the length of a factorization is the number of irreducible factors (counting repetitions). In a recent paper, F. Gotti and C. O'Neil studied the sets of elasticities $\mathcal{R}(P) := \{\sup \mathsf{L}(x)/\inf \mathsf{L}(x) : x \in P\}$ of Puiseux monoids $P$. Here we take this study a step forward and explore the local $k$-elasticities of the same class of monoids. We find conditions under which Puiseux monoids have all their local elasticities finite as well as conditions under which they have infinite local $k$-elasticities for sufficiently large $k$. Finally, we focus our study of the $k$-elasticities on the class of primary Puiseux monoids, proving that they have finite local $k$-elasticities if either they are boundedly generated and do not have any stable atoms or if they do not contain $0$ as a limit point.