The Bi-UF Positive Conjecture Holds

Abstract

A subsemiring $S$ of the nonnegative cone of the real line containing $1$ is called a positive semiring. The Bi-UF Positive Conjecture, posed by Baeth, Chapman, and Gotti in 2021, states that the prototypical semiring ${\mathbb{N}}_0$ is the only positive semiring whose additive monoid $(S,+)$ and nonzero multiplicative monoid $(S^{\bullet },\cdot )$, where $S^{\bullet }=S\setminus \{0\}$, are both factorial. In this paper, we prove their conjecture.