We describe the associative multilinear polynomial functions over commutative integral domains. This extends Marichal and Mathonet’s result on infinite integral domains and provides a new proof of Andres’s classification of two-element n-semigroups.
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Andres, S.D.: Classification of all associative mono-\(n\)-ary algebras with 2 elements. Int. J. Math. Math. Sci. 2009, Article ID 678987, 16 pages (2009)
Marichal, J.-L., Mathonet, P.: A description of \(n\)-ary semigroups polynomial-derived from integral domains. Semigroup Forum 83, 241–249 (2011)
2018 Miklós Schweitzer: Art of Problem Solving. https://artofproblemsolving.com/community/c771105_2018_mikloacutes_schweitzer. Accessed 11 June 2019
The authors would like to thank Robert Baumann, Thomas Quinn-Gregson, and Nikolaas Verhulst for inspiring discussions and the anonymous reviewer for helpful comments.
Communicated by Jorge Almeida.
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Lehtonen, E., Starke, F. On associative operations on commutative integral domains. Semigroup Forum 100, 910–915 (2020). https://doi.org/10.1007/s00233-019-10044-x