正半环的双原子类
Bi-atomic classes of positive semirings
摘要：
设$S$是实直线的非负半环，这里称为正半环。我们研究了加法幺半群$(S,+)$和乘法幺半群$(S\setminus\{0\}, \cdot)$的因子分解。特别地，我们研究了当正半环$S$，$(S,+)$和$(S\setminus\{0\}, \cdot)$都具有以下性质：原子性，ACCP，有界因子分解性质（BFP），有限因子分解性质（FFP）和半因子性质（HFP）。众所周知，在可消幺半群和交换幺半群的上下文中，蕴涵链HFP $\Rightarrow$ BFP和FFP $\Rightarrow$ BFP $\Rightarrow$ ACCP $\Rightarrow$原子性成立。在这里，我们构建类的正半环，其中的加法和乘法结构满足这些性质，我们也给出例子来表明，在一般情况下，没有在前面的链的影响是可逆的。
英文摘要：
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.
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