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Arithmetic of additively reduced monoid semidomains

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Abstract

The term semidomain refers to a subset S of an integral domain R, in which the pairs \((S,+)\) and \((S, \cdot )\) are semigroups with identities. If S contains no additive inverses except 0, we say that S is additively reduced. By taking polynomial expressions with coefficients in S and exponents in a torsion-free monoid M, we obtain the additively reduced monoid semidomain S[M]. In this paper, we investigate the factorization properties of such semidomains, providing necessary and sufficient conditions for them to be bounded factorization semidomains, finite factorization semidomains, and unique factorization semidomains. We also identify large classes of semidomains with full and infinite elasticity. Throughout the paper, we present examples to help elucidate the arithmetic of additively reduced monoid semidomains.

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Notes

  1. The notion of strong finite factorization was introduced by Anderson and Mullins in [1].

  2. The localization of semirings is presented in greater generality in [19, Chapter 11].

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Acknowledgements

The authors express their gratitude to an anonymous referee for their meticulous review of the manuscript and valuable feedback that significantly enhanced the quality of this paper. Additionally, the authors would like to extend their appreciation to Alfred Geroldinger and the referee for pointing out that Proposition 13 cannot be derived as a direct consequence of a similar result already established for transfer Krull monoids (see Remark 2).

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Authors and Affiliations

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

    Scott T. Chapman

  2. Department of Mathematics, University of Florida, Gainesville, 32611, FL, USA

    Harold Polo

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  1. Scott T. Chapman
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Correspondence to Harold Polo.

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Communicated by Jan Okniński.

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Chapman, S.T., Polo, H. Arithmetic of additively reduced monoid semidomains. Semigroup Forum 107, 40–59 (2023). https://doi.org/10.1007/s00233-023-10363-0

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