On the number of n-ary quasigroups of finite order

TL;DR

This paper investigates the number of n-ary quasigroups of finite order, deriving formulas and bounds that improve understanding of their asymptotic growth for various values of k and n.

Contribution

It introduces a recurrent formula for Q(n,4) and establishes improved bounds for Q(n,k) when k ≥ 5, advancing the theoretical understanding of n-ary quasigroups.

Findings

Derived a recurrent formula for Q(n,4).

Established bounds for Q(n,k) for k ≥ 5.

Improved asymptotic bounds for the number of n-ary quasigroups.

Abstract

Let $Q(n,k)$ be the number of $n$-ary quasigroups of order $k$. We derive a recurrent formula for Q(n,4). We prove that for all $n\geq 2$ and $k\geq 5$ the following inequalities hold: $({k-3}/2)^{n/2}(\frac{k-1}2)^{n/2} < log_2 Q(n,k) \leq {c_k(k-2)^{n}} $, where $c_k$ does not depend on $n$. So, the upper asymptotic bound for $Q(n,k)$ is improved for any $k\geq 5$ and the lower bound is improved for odd $k\geq 7$. Keywords: n-ary quasigroup, latin cube, loop, asymptotic estimate, component, latin trade.