Row-Hamiltonian Latin squares and Falconer varieties

A Latin square is a matrix of symbols such that each symbol occurs exactly once in each row and column. A Latin square $L$ is 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.