On the factorization invariants of the additive structure of exponential Puiseux semirings

TL;DR

This paper studies the factorization properties of exponential Puiseux semirings, revealing their set of lengths, catenary degrees, and conditions for finite omega functions, thus advancing understanding of their algebraic structure.

Contribution

It provides new formulas and characterizations for factorization invariants of exponential Puiseux semirings, connecting these properties with semigroup invariants.

Findings

Sets of lengths are almost arithmetic progressions.

Unions of sets of lengths form arithmetic progressions.

Exact formulas for catenary degrees and characterizations of finite omega functions.

Abstract

Exponential Puiseux semirings are additive submonoids of $\qq_{\geq 0}$ generated by almost all of the nonnegative powers of a positive rational number, and they are natural generalizations of rational cyclic semirings. In this paper, we investigate some of the factorization invariants of exponential Puiseux semirings and briefly explore the connections of these properties with semigroup-theoretical invariants. Specifically, we prove that sets of lengths of atomic exponential Puiseux semirings are almost arithmetic progressions with a common bound, while unions of sets of lengths are arithmetic progressions. Additionally, we provide exact formulas to compute the catenary degrees of these monoids and show that minima and maxima of their sets of distances are always attained at Betti elements. We conclude by providing various characterizations of the atomic exponential Puiseux semirings with finite omega functions; in particular, we completely describe them in terms of their presentations.