Loops with exponent three in all isotopes

TL;DR

This paper investigates loops with exponent three, called van Rees loops, showing their structural properties, conditions for achieving a specific bound on latin subsquares, and their relation to Steiner quasigroups.

Contribution

It characterizes van Rees loops, proves they form an equational variety, and explores their structural properties and classifications.

Findings

Van Rees loops achieve the bound only if n ≡ 3 mod 6.

In a van Rees loop, subloops of index 3 are normal.

There are exactly 6 nonassociative van Rees loops of order 27.

Abstract

It was shown by van Rees \cite{vR} that a latin square of order $n$ has at most $n^2(n-1)/18$ latin subsquares of order $3$. He conjectured that this bound is only achieved if $n$ is a power of $3$. We show that it can only be achieved if $n\equiv3\bmod6$. We also state several conditions that are equivalent to achieving the van Rees bound. One of these is that the Cayley table of a loop achieves the van Rees bound if and only if every loop isotope has exponent $3$. We call such loops \emph{van Rees loops} and show that they form an equationally defined variety. We also show that (1) In a van Rees loop, any subloop of index 3 is normal, (2) There are exactly 6 nonassociative van Rees loops of order $27$ with a non-trivial nucleus and at least 1 with all nuclei trivial, (3) Every commutative van Rees loop has the weak inverse property and (4) For each van Rees loop there is an associated family of Steiner quasigroups.