Bi-atomic classes of positive semirings

TL;DR

This paper explores the factorization properties of positive semirings, examining when both their additive and multiplicative monoids exhibit properties like atomicity, ACCP, BFP, FFP, and HFP, and provides examples of their behaviors.

Contribution

It introduces classes of positive semirings where both additive and multiplicative structures satisfy various factorization properties, highlighting the non-reversibility of known implications.

Findings

Constructed classes of positive semirings with both structures satisfying factorization properties.

Provided examples showing implications between properties are not reversible.

Analyzed conditions under which additive and multiplicative monoids share factorization characteristics.

Abstract

Let $S$ be a nonnegative semiring of the real line, called here a positive semiring. We study factorizations in both the additive monoid $(S,+)$ and the multiplicative monoid $(S\setminus\{0\}, \cdot)$. In particular, we investigate when, for a positive semiring $S$, both $(S,+)$ and $(S\setminus\{0\}, \cdot)$ have the following properties: atomicity, the ACCP, the bounded factorization property (BFP), the finite factorization property (FFP), and the half-factorial property (HFP). It is well known that in the context of cancellative and commutative monoids, the chain of implications HFP $\Rightarrow$ BFP and FFP $\Rightarrow$ BFP $\Rightarrow$ ACCP $\Rightarrow$ atomicity holds. Here we construct classes of positive semirings wherein both the additive and multiplicative structures satisfy each of these properties, and we also give examples to show that, in general, none of the implications in the previous chain is reversible.