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Levi-Civita functional equations and the status of spectral synthesis on semigroups

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Abstract

We show, contrary to some published statements, that spectral synthesis does not generally hold for commutative semigroups that are not groups. On the positive side we prove that it holds if the semigroup is a monoid with no prime ideal. For semigroups with a prime ideal, the picture is not so clear. On the negative side we provide a variety of examples illustrating the failure of spectral synthesis for many semigroups with prime ideals, but we also give examples of semigroups with prime ideals on which spectral synthesis holds.

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Acknowledgements

The authors appreciate the referee’s attention to details and are grateful for the insightful and helpful comments received.

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Authors and Affiliations

  1. Department of Mathematics, University of Louisville, Louisville, KY, 40292, USA

    Bruce Ebanks

  2. Department of Pure Mathematics, University of Waterloo, Waterloo, ON, N2L3G1, Canada

    Che Tat Ng

Authors
  1. Bruce Ebanks
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  2. Che Tat Ng
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Correspondence to Bruce Ebanks.

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Communicated by Jimmie D. Lawson.

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Ebanks, B., Ng, C.T. Levi-Civita functional equations and the status of spectral synthesis on semigroups. Semigroup Forum 103, 469–494 (2021). https://doi.org/10.1007/s00233-021-10211-z

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  • DOI: https://doi.org/10.1007/s00233-021-10211-z

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