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Keywords:
quasigroup; linear identity; associativity; commutativity
quasigroup; linear identity; associativity; commutativity
Summary:
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 $n \rightarrow \infty$.
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