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arXiv · 2509.14947

Irreducible Polyadic Semigroups Admitting the Adjunction of a Neutral Element

Jean-Luc Marichal·Pierre Mathonet·Tamás Waldhauser
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Abstract

It was claimed in [4] that for any integer $n\geqslant 2$, a neutral element can be adjoined to an $n$-ary semigroup if and only if the $n$-ary semigroup is reducible to a binary semigroup. We show that the `only if' direction of this statement is incorrect when $n$ is odd. Moreover, we offer a comprehensive characterization of the class of irreducible $n$-ary semigroups, for odd $n$, that admit the adjunction of a neutral element.

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Jean-Luc Marichal, Pierre Mathonet, Tamás Waldhauser. 2025-09-18. Irreducible Polyadic Semigroups Admitting the Adjunction of a Neutral Element. https://arxiv.org/abs/2509.14947

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