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 $(\theta_0,\theta_1,\ldots,\theta_n)$ of $n+1$ permutations of $\Sigma$ such that $f(\theta_1(x_1),\ldots,\theta_n(x_n))\equiv \theta_0(f(x_1,\ldots,x_n))$. 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.

Cite: Gorkunov E.V. , Krotov D.S. , Potapov V.N.
On the number of autotopies of an n-ary qusigroup of order 4
Quasigroups and Related Systems. 2019. V.27. N2. P.227-250. Scopus РИНЦ

Dates:

Submitted:

Mar 3, 2019

Identifiers:

Scopus:	2-s2.0-85078449851
Elibrary:	43240256

Citing:

DB	Citing
Scopus	2
Elibrary	1