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Israel Journal of Mathematics

, Volume 220, Issue 1, pp 479–508 | Cite as

Chains of subsemigroups

  • Peter J. Cameron
  • Maximilien Gadouleau
  • James D. Mitchell
  • Yann Peresse
Article

Abstract

We investigate the maximum length of a chain of subsemigroups in various classes of semigroups, such as the full transformation semigroups, the general linear semigroups, and the semigroups of order-preserving transformations of finite chains. In some cases, we give lower bounds for the total number of subsemigroups of these semigroups. We give general results for finite completely regular and finite inverse semigroups. Wherever possible, we state our results in the greatest generality; in particular, we include infinite semigroups where the result is true for these.

The length of a subgroup chain in a group is bounded by the logarithm of the group order. This fails for semigroups, but it is perhaps surprising that there is a lower bound for the length of a subsemigroup chain in the full transformation semigroup which is a constant multiple of the semigroup order.

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Copyright information

© Hebrew University of Jerusalem 2017

Authors and Affiliations

  • Peter J. Cameron
    • 1
  • Maximilien Gadouleau
    • 2
  • James D. Mitchell
    • 1
  • Yann Peresse
    • 3
  1. 1.Mathematics InstituteUniversity of St Andrews North HaughSt AndrewsUK
  2. 2.School of Engineering and Computing SciencesUniversity of DurhamDurhamUK
  3. 3.School of Physics, Astronomy and MathematicsUniversity of Hertfordshire HatfieldHertsUK

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