Irreducible Polyadic Semigroups Admitting the Adjunction of a Neutral Element

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.