Every quasitrivial n-ary semigroup is reducible to a semigroup
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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 Classification05A15 20N15 16B99 20M14
KeywordsQuasitrivial 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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