A local structure theorem for stable, \(\mathcal {J}\)-simple semigroup biacts

Abstract

We describe a class of semigroup biacts that is analogous to the class of completely simple semigroups, and prove a structure theorem for those biacts that is analogous to the Rees–Sushkevitch Theorem. Precisely, we describe stable, \(\mathcal {J}\)-simple biacts in terms of wreath products, translations of completely simple semigroups, biacts over endomorphism monoids of free G-acts, tensor products and matrix biacts. Applications to coproducts and left acts are given.

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Acknowledgements

This research has been conducted as part of the project Labex MME-DII (ANR11-LBX-0023-01). It also benefited from very fruitful conversations with members of the Department of Mathematics, University of York, while the present author was visiting the University.

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Correspondence to Xavier Mary.

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Communicated by Victoria Gould.

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Mary, X. A local structure theorem for stable, \(\mathcal {J}\)-simple semigroup biacts. Semigroup Forum 99, 724–753 (2019). https://doi.org/10.1007/s00233-018-9986-6

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Keywords

  • Semigroup acts
  • Green’s relations
  • Stability
  • Wreath products
  • Endomorphism monoid of free G-acts
  • Completely simple semigroups
  • Rees matrix semigroups