Row‐Hamiltonian Latin squares and Falconer varieties

Abstract

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