Linear operator identities in quasigroups

Akhtar, Reza

Commentationes Mathematicae Universitatis Carolinae, Tome 63 (2022) no. 1, pp. 1-9.

Résumé

We study identities of the form $$ L_{x_0} \varphi_1 \cdots \varphi_n R_{x_{n+1}} = R_{x_{n+1}} \varphi_{\sigma(1)} \cdots \varphi_{\sigma(n)} L_{x_0} $$ in quasigroups, where $n \geq 1$, $\sigma$ is a permutation of $\{1, \ldots, n\}$, and for each $i$, $\varphi_i$ is either $L_{x_i}$ or $R_{x_i}$. We prove that in a quasigroup, every such identity implies commutativity. Moreover, if $\sigma$ is chosen randomly and uniformly, it also satisfies associativity with probability approaching $1$ as $n \rightarrow \infty$.

Détail

DOI : 10.14712/1213-7243.2022.010

Classification : 05C78
Keywords: quasigroup; linear identity; associativity; commutativity