Complete mappings of semigroups

Let A be a magma, a set with a binary operation. A complete mapping for A is a bijection θ on A such that the map ψ defined by ψ(a) = a·θ(a) is also a bijection, where · is the operation. It defines a transversal of the Cayley table of the magma, a set of cells meeting each row, column and symbol in a unique element. The celebrated Hall–Paige conjecture asserted that a finite group has a complete mapping if and only if either it has odd order or its Sylow 2-subgroups are non-cyclic. They proved the necessity of the conditions, but sufficiency had to wait for the work of Wilcox, Evans and Bray in 2009. (Note that, for groups, the Cayley table is a Latin square; and the existence of a complete mapping, or transversal to the Latin square, implies the existence of an orthogonal mate to the square. So there is a tenouous link with the last thing I posted. Little can be said about the existence of complete mappings for arbitrary finite magmas. One simple necessary condition is that every…