Abstract: 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.
Keywords: quasigroup; linear identity; associativity; commutativity
DOI: DOI 10.14712/1213-7243.2022.010
AMS Subject Classification: 05C78