Multidimensional quadrangle condition and cuboctahedra in latin hypercubes

TL;DR

This paper extends the quadrangle criterion to multidimensional latin hypercubes, linking maximal cuboctahedra counts to the structure of associative quasigroups and providing new theoretical and computational insights.

Contribution

It introduces the multidimensional quadrangle condition for latin hypercubes, relating it to cuboctahedra counts and the structure of associative quasigroups.

Findings

Most associative $d$-ary quasigroups have Cayley tables with every 2D plane isotopic to a group Cayley table.

The paper provides lower bounds on the number of cuboctahedra in latin squares and hypercubes.

Computational results support the theoretical estimates.

Abstract

The well-known quadrangle criterion states that a latin square is isotopic to the Cayley table of a group if and only if all quadrangles spanned by the same triple of symbols coincide on the fourth symbol. Gowers and Long (2020) reformulated this result in the following way: the Cayley tables of the most associative quasigroups have the maximum number of octahedra. In the present paper, we state the multidimensional quadrangle condition for $d$-dimensional latin hypercubes in terms of the reconstruction of submatrices of order $2$ from a bundle of $d+1$ entries and in terms of the maximal number of cuboctahedra. In particular, we show that the most associative $d$-ary quasigroups have Cayley tables such that every $2$-dimensional plane is isotopic to a latin square that is principally isotopic to the Cayley table of a group. We also estimate the number of cuboctahedra in latin squares and hypercubes from below and provide computational results.