Abstract
We examine the Delta set of a cancellative and reduced atomic monoid S where every set of lengths of the factorizations of each element in S is bounded. In particular, we show the connection between the elements of Δ(S) and the Betti elements of S. We prove how the minimum and maximum element of Δ(S) can be determined using the Betti elements of S. This leads to a determination of when Δ(S) is a singleton. We then apply these results to the particular case where S is a numerical monoid that requires three generators. Conclusions are drawn in the cases where S has a unique minimal presentation, or has multiplicity three.
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Acknowledgments
The authors would like to thank the referees for their suggestions, and A. Geroldinger and A. Philipp for fruitful discussions on the BF condition. P. A. García-Sánchez and D. Llena are supported by the project MTM2010–15595, the research group FQM-343 and FEDER funds. P. A. García-Sánchez is also supported by the project FQM-5849.
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Chapman, S.T., García-Sánchez, P.A., Llena, D. et al. On the delta set and the Betti elements of a BF-monoid. Arab. J. Math. 1, 53–61 (2012). https://doi.org/10.1007/s40065-012-0019-0
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DOI: https://doi.org/10.1007/s40065-012-0019-0