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Addition Theorems in Acyclic Semigroups

In: Additive Number Theory

Author

Listed:
  • Javier Cilleruelo

    (Universidad Autónoma de Madrid, Instituto de Ciencias Matemáticas CSIC-UAM-UC3M-UAM, and Departamento de Matemáticas)

  • Yahya O. Hamidoune

    (Univ. Paris VI, UER Combinatoire)

  • Oriol Serra

    (Univ. Politècnica de Catalunya, Dept. Matemàtica Aplicada 4)

Abstract

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.

Suggested Citation

  • Javier Cilleruelo & Yahya O. Hamidoune & Oriol Serra, 2010. "Addition Theorems in Acyclic Semigroups," Springer Books, in: David Chudnovsky & Gregory Chudnovsky (ed.), Additive Number Theory, edition 1, pages 99-104, Springer.
    • Handle: RePEc:spr:sprchp:978-0-387-68361-4_6
    DOI: 10.1007/978-0-387-68361-4_6
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