AI 中文总结
研究关于复半环的双-UF正猜想，证明N₀通过二次代数数的简单半环扩张非双-UFS，确定两类满足猜想的复半环，扩展猜想陈述，还考虑双-HF性质，证明N₀是唯一具该性质的正有理半域并给出构造不同双-HFS复半环的方法。
AI 中文摘要
复半环是复数平面的一个子集，在复数的标准加法和乘法下封闭且包含0和1。若复半环S的加法幺半群(S,+)和乘法幺半群(S\{1},·)都是唯一分解幺半群(UFM)，则称S为双-UFS。双-UF正猜想指出N₀是实直线非负锥中唯一的双-UFS子半环。本文证明了N₀通过二次代数数的简单半环扩张不是双-UFS，确定了一类满足双-UF正猜想的复半环。还确定了另一类满足该猜想的复半环。通过一个关于加法幺半群为有限秩自由交换幺半群的半域的结构定理扩展了双-UF正猜想的陈述。最后考虑了双-HF性质，证明了N₀是唯一具有双-HF性质的正有理半域，并提供了两种构造不同于N₀的双-HFS复半环的方法。
英文摘要
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 $\mathbb{N}_0$.
Comments17 pages
