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A  Latin square  is a matrix of symbols such that each symbol occurs exactly once in each row and column. A Latin square    L  $L$    is  row‐Hamiltonian  if the permutation induced by each pair of distinct rows of    L  $L$    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  $L$   ‐closed loop varieties, solving an open problem posed by Falconer in 1970.

Row‐Hamiltonian Latin squares and Falconer varieties