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arXiv · 1903.00188

On the number of autotopies of an $n$-ary qusigroup of order $4$

Denis S. Krotov·Evgeny V. Gorkunov·Vladimir N. Potapov
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Abstract

An algebraic system from a finite set $\Sigma$ of cardinality $k$ and an $n$-ary operation $f$ invertible in each argument is called an $n$-ary quasigroup of order $k$. An autotopy of an $n$-ary quasigroup $(\Sigma,f)$ is a collection $(\theta_0,\theta_1,...,\theta_n)$ of $n+1$ permutations of $\Sigma$ such that $f(\theta_1(x_1),...,\theta_n(x_n))\equiv \theta_0(f(x_1,\ldots,x_n))$. We show that every $n$-ary quasigroup of order $4$ has at least $2^{[n/2]+2}$ and not more than $6\cdot 4^n$ autotopies. We characterize the $n$-ary quasigroups of order $4$ with $2^{(n+3)/2}$, $2\cdot 4^n$, and $6\cdot 4^n$ autotopies.

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Denis S. Krotov, Evgeny V. Gorkunov, Vladimir N. Potapov. 2019-03-01. On the number of autotopies of an $n$-ary qusigroup of order $4$. https://arxiv.org/abs/1903.00188

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