A subsemiring S of \(\mathbb {R}\) is called a positive semiring provided that S consists of nonnegative numbers and \(1 \in S\). Here we study factorizations in both the additive monoid \((S,+)\) and the multiplicative monoid \((S\backslash \{0\}, \cdot )\). In particular, we investigate when, for a positive semiring S, both \((S,+)\) and \((S\backslash \{0\}, \cdot )\) 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.

甲subsemiring小号的\（\ mathbb {R} \）被称为半环正提供的小号由非负数的和（1处于S \ \）\。在这里，我们在加和单项\（（S，+）\）和乘法单项\（（S \反斜杠\ {0 \}，\ cdot）\）中研究分解。具体地，我们研究时，对于正半环小号，既\（（S，+）\）和\（（S \反斜杠\ {0 \}，\ CDOT）\）具有以下属性：原子性，ACCP，有界分解属性（BFP），有限分解属性（FFP）和半分解属性（HFP）。众所周知，在单义和可交换单义词的上下文中，含义链HFP \（\ Rightarrow \） BFP和FFP \（\ Rightarrow \） BFP \（\ Rightarrow \） ACCP \（\ Rightarrow \）原子。在这里，我们构造了正半环类，其中加性和乘性结构都满足这些特性中的每一个，并且我们还给出了一些示例，以显示一般而言，前一条链中的任何含义都是不可逆的。