Row-Hamiltonian Latin squares and Falconer varieties

TL;DR

This paper introduces a new family of Latin squares called row-Hamiltonian, which are linked to perfect 1-factorisations, and uses them to construct specific algebraic loop varieties, solving a longstanding open problem.

Contribution

It presents the first explicit family of Latin squares that are row-Hamiltonian but not column- or symbol-Hamiltonian, enabling the construction of novel algebraic loop varieties.

Findings

Established a family of row-Hamiltonian Latin squares with unique properties.

Constructed non-trivial, anti-associative, isotopically L-closed loop varieties.

Solved an open problem posed by Falconer in 1970.

Abstract

A \emph{Latin square} is a matrix of symbols such that each symbol occurs exactly once in each row and column. A Latin square $L$ is \emph{row-Hamiltonian} if the permutation induced by each pair of distinct rows of $L$ is a full cycle permutation. Row-Hamiltonian Latin squares are equivalent to perfect $1$-factorisations of complete bipartite graphs. For the first time, we exhibit a family of Latin squares that are row-Hamiltonian and also achieve precisely one of the related properties of being column-Hamiltonian or symbol-Hamiltonian. This family allows us to construct non-trivial, anti-associative, isotopically $L$-closed loop varieties, solving an open problem posed by Falconer in 1970.