Bi-atomic classes of positive semirings

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

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Bibliographic Details
Published in:Semigroup forum 2021-08, Vol.103 (1), p.1-23
Main Authors: Baeth, Nicholas R., Chapman, Scott T., Gotti, Felix
Format: Article
Language:English
Subjects:
Algebra
Chains
Factorization
Mathematics
Mathematics and Statistics
Monoids
Research Article
Rings (mathematics)
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Summary:A subsemiring S of R is called a positive semiring provided that S consists of nonnegative numbers and 1 ∈ S . Here we study factorizations in both the additive monoid ( S , + ) and the multiplicative monoid ( S \ { 0 } , · ) . In particular, we investigate when, for a positive semiring S , both ( S , + ) and ( 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 ⇒ BFP and FFP ⇒ BFP ⇒ ACCP ⇒ 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.
ISSN:0037-1912
1432-2137
DOI:10.1007/s00233-021-10189-8