Algebra universalis

, 80:51 | Cite as

Every quasitrivial n-ary semigroup is reducible to a semigroup

  • Miguel CouceiroEmail author
  • Jimmy Devillet


We show that every quasitrivial n-ary semigroup is reducible to a binary semigroup, and we provide necessary and sufficient conditions for such a reduction to be unique. These results are then refined in the case of symmetric n-ary semigroups. We also explicitly determine the sizes of these classes when the semigroups are defined on finite sets. As a byproduct of these enumerations, we obtain several new integer sequences.

Mathematics Subject Classification

05A15 20N15 16B99 20M14 


Quasitrivial polyadic semigroup Reducibility Enumeration Symmetry 



Both authors would like to thank Jean-Luc Marichal and the anonymous referee for their useful comments and insightful remarks that helped improving the current paper.


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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Université de Lorraine, CNRS, Inria, LORIANancyFrance
  2. 2.Mathematics Research UnitUniversity of LuxembourgEsch-sur-AlzetteLuxembourg

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