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

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 $f({\theta }_1(x_1),...,{\theta }_n(x_n))\equiv {\theta }_0(f(x_1,\dots ,x_n))$. 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.

English: 21 pages; Russian: 22 pages