Addition Theorems in Acyclic Semigroups

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


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.


Addition theorems Semigroups 


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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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