Bi-atomic classes of positive semirings

Abstract

Abstract A subsemiring S of

$$\mathbb {R}$$

R

is called a positive semiring provided that S consists of nonnegative numbers and

$$1 \in S$$

1 ∈ S

. Here we study factorizations in both the additive monoid

$$(S,+)$$

( S , + )

and the multiplicative monoid

$$(S\backslash \{0\}, \cdot )$$

( S \ { 0 } , · )

. In particular, we investigate when, for a positive semiring S, both

$$(S,+)$$

( S , + )

and

$$(S\backslash \{0\}, \cdot )$$

( S \ { 0 } , · )

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.