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Bi-atomic classes of positive semirings

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Abstract

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.

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Notes

  1. By the Lindemann–Weierstrass Theorem, E(M) can be naturally identified with a subsemiring of the semigroup ring with coefficients in \(\mathbb {Z}\) and exponents in M.

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Acknowledgements

During the preparation of this paper, the third author was generously supported by the NSF Award DMS-1903069.

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Authors and Affiliations

  1. Department of Mathematics, Franklin and Marshall College, Lancaster, PA, 17604, USA

    Nicholas R. Baeth

  2. Department of Mathematics and Statistics, Sam Houston State University, Huntsville, TX, 77341, USA

    Scott T. Chapman

  3. Department of Mathematics, MIT, Cambridge, MA, 02139, USA

    Felix Gotti

Authors
  1. Nicholas R. Baeth
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  2. Scott T. Chapman
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  3. Felix Gotti
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Correspondence to Felix Gotti.

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Communicated by Jorge Almeida.

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Baeth, N.R., Chapman, S.T. & Gotti, F. Bi-atomic classes of positive semirings. Semigroup Forum 103, 1–23 (2021). https://doi.org/10.1007/s00233-021-10189-8

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