On non-$n$-semiabelianism polyadic groupoids of special class

A. M. Gal'mak

Problemy fiziki, matematiki i tehniki, no. 1 (2019), pp. 31-39

Résumé

The permutability of the elements in polyadic groupoids with polyadic operation $\eta_{s,\sigma,k}$ that is defined on Cartesian power of $A^k$ $n$-ary groupoid $\langle A,\eta\rangle$ by substitution $\sigma\in\mathbf{S}_k$ and $n$-ary operation $\eta$ are considered. The main result of the article is the theorem in which sufficient conditions of non-$n$-semiabelianism of $l$-ary ($l = s(n-1) + 1$, $k\geqslant 2$) groupoid $\langle A^k,\eta_{s,\sigma,k}\rangle$ are formulated. Numerous consequences of this theorem are given. In particular, it was found that if substitution $\sigma$ satisfies the conditions $\sigma^{n-1}\ne\sigma$, $\sigma^l=\sigma$, $n$-ary group $\langle A,\eta\rangle$ has no less than two elements, then polyadic groupoid $\langle A^k,\eta_{s,\sigma,k}\rangle$ is a non-$n$-semiabelian polyadic group.

Détail

Keywords: polyadic operation, $n$-ary groupoid, semiabelianism, neutral sequence.
Mots-clés : abelianism