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Addition Theorems in Acyclic Semigroups

  • Javier CillerueloEmail author
  • Yahya O. Hamidoune
  • Oriol Serra
Chapter

Summary

We give a necessary and sufficient condition on a given family \(\mathcal{A}\) of finite subsets of integers for the Cauchy–Davenport inequality
$$\vert \mathcal{A} + \mathcal{B}\vert \geq \vert \mathcal{A}\vert + \vert \mathcal{B}\vert - 1,$$
to hold for any family \(\mathcal{B}\) of finite subsets of integers. We also describe the extremal families for this inequality. We prove this result in the general context of acyclic semigroups, which also contain the semigroup of sequences of elements in an ordered group.

Keywords

Addition theorems Semigroups 

References

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    Yahya Ould Hamidoune, Some additive applications of the isopermetric approach, Annales de l’institut Fourier, 58 no. 6 (2008), p. 2007–2036 arXiv:0706.0635.Google Scholar
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    O. Serra, An isoperimetric method for the small sumset problem. Surveys in combinatorics 2005, 119–152, London Math. Soc. Lecture Note Ser., 327, Cambridge University Press, Cambridge, 2005.Google Scholar
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    T. Tao and V.H. Vu, Additive Combinatorics, Cambridge Studies in Advanced Mathematics 105, Cambridge Press University, Cambridge, 2006.Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  • Javier Cilleruelo
    • 1
    Email author
  • Yahya O. Hamidoune
    • 2
  • Oriol Serra
    • 3
  1. 1.Instituto de Ciencias Matemáticas CSIC-UAM-UC3M-UAM, and Departamento de MatemáticasUniversidad Autónoma de MadridMadridSpain
  2. 2.UER CombinatoireUniv. Paris VIParisFrance
  3. 3.Dept. Matemàtica Aplicada 4Univ. Politècnica de CatalunyaBarcelonaSpain

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