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

## TL;DR

This paper investigates the autotopies of $n$-ary quasigroups of order 4, establishing bounds on their number and characterizing those with specific autotopy counts, advancing understanding of their algebraic structure.

## Contribution

It provides bounds on the number of autotopies for $n$-ary quasigroups of order 4 and characterizes quasigroups with extremal autotopy counts.

## Key findings

- Minimum autotopies: $2^{[n/2]+2}$
- Maximum autotopies: $6\,4^n$
- Characterizations for specific autotopy counts

## 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.