The Bi-UF Positive Conjecture for quadratic monogenic semirings and related progress

A complex semiring is a subset of the complex plane that is closed under the standard addition and multiplication of complex numbers and contains both $0$ and $1$. A complex semiring $S$ is called a bi-UFS if both its additive monoid $(S,+)$ and its multiplicative monoid $(S\setminus \{1\}, \cdot)$ are unique factorization monoids (UFM). The Bi-UF Positive Conjecture states that $\mathbb{N}_0$ is the only subsemiring of the nonnegative cone of the real line that is a bi-UFS. In this paper, we prove that no simple semiring extension of $\mathbb{N}_0$ by a quadratic algebraic number is a bi-UFS, identifying a natural class of complex semirings satisfying the statement of the Bi-UF Positive Conjecture. We also identify another class of complex semirings satisfying the statement of the Bi-UF Positive Conjecture. Then we extend the statement of the Bi-UF Positive Conjecture by motivated by a structural theorem we established for semidomains whose additive monoid are finite-rank free commutative monoids. Finally, we consider the bi-HF property, which is a relaxed version of the bi-UF property. We prove that $\mathbb{N}_0$ is the only positive rational semidomain having the bi-HF property, and we provide two methods to construct bi-HFS complex semirings that are distinct from $\mathbb{N}_0$.