Distributive and trimedial quasigroups of order 243

TL;DR

This paper classifies and enumerates various non-medial quasigroups of order 243, extending previous work and utilizing affine representations, automorphism properties, and computational methods.

Contribution

It provides the first complete enumeration of non-medial trimedial, distributive, and Mendelsohn quasigroups of order 243, using advanced algebraic and computational techniques.

Findings

17004 non-medial trimedial quasigroups of order 243

92 non-medial distributive quasigroups of order 243

6 non-medial distributive Mendelsohn quasigroups of order 243

Abstract

We enumerate three classes of non-medial quasigroups of order $243=3^5$ up to isomorphism. There are $17004$ non-medial trimedial quasigroups of order $243$ (extending the work of Kepka, B\'en\'eteau and Lacaze), $92$ non-medial distributive quasigroups of order $243$ (extending the work of Kepka and N\v{e}mec), and $6$ non-medial distributive Mendelsohn quasigroups of order $243$ (extending the work of Donovan, Griggs, McCourt, Opr\v{s}al and Stanovsk\'y). The enumeration technique is based on affine representations over commutative Moufang loops, on properties of automorphism groups of commutative Moufang loops, and on computer calculations with the \texttt{LOOPS} package in \texttt{GAP}.