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Factorization properties of Leamer monoids

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Abstract

The Huneke-Wiegand conjecture has prompted much recent research in Commutative Algebra. In studying this conjecture for certain classes of rings, García-Sánchez and Leamer construct a monoid \(S_{\varGamma}^{s}\) whose elements correspond to arithmetic sequences in a numerical monoid Γ of step size s. These monoids, which we call Leamer monoids, possess a very interesting factorization theory that is significantly different from the numerical monoids from which they are derived. In this paper, we offer much of the foundational theory of Leamer monoids, including an analysis of their atomic structure, and investigate certain factorization invariants. Furthermore, when \(S_{\varGamma}^{s}\) is an arithmetical Leamer monoid, we give an exact description of its atoms and use this to provide explicit formulae for its Delta set and catenary degree.

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Acknowledgements

Much of this work was completed during the Pacific Undergraduate Research Experience in Mathematics (PURE Math), which was funded by National Science Foundation grants DMS-1035147 and DMS-1045082 and a supplementary grant from the National Security Agency. The authors would like to thank Scott Chapman, Pedro García-Sánchez, and Micah Leamer for their numerous helpful conversations, as well as the anonymous referee for their very helpful comments.

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Authors and Affiliations

  1. Eastern Illinois University, Charleston, IL, 61920, USA

    Jason Haarmann

  2. Mathematics Department, University of Hawai‘i at Hilo, Hilo, HI, 96720, USA

    Ashlee Kalauli & Roberto Pelayo

  3. McKendree University, Lebanon, IL, 62254, USA

    Aleesha Moran

  4. Mathematics Department, Duke University, Durham, NC, 27708, USA

    Christopher O’Neill

Authors
  1. Jason Haarmann
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  2. Ashlee Kalauli
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  3. Aleesha Moran
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  4. Christopher O’Neill
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  5. Roberto Pelayo
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Corresponding author

Correspondence to Roberto Pelayo.

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Communicated by Fernando Torres.

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Haarmann, J., Kalauli, A., Moran, A. et al. Factorization properties of Leamer monoids. Semigroup Forum 89, 409–421 (2014). https://doi.org/10.1007/s00233-014-9578-z

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  • DOI: https://doi.org/10.1007/s00233-014-9578-z

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