Algebraic Quasigroups

It is well-known that the only polynomially defined group laws on an infinite field k are of the form x+y +c for some cin k. Thus any such group law on k is both linear and commutative. In this note, we generalize this result to arbitrary quasigroup structures polynomially definable over k. Also, it is well-known that a group structure which is a morphism on a projective curve is abelian. Here we show that a quasigroup morphism on such a curve always yields a (commutative) group structure. In the case of fields, we apply the technique of Hilbert Nullstellensatz to a suitable Mal'cev polynomial, and for the projective curves we invoke the powerful rigidity property of morphisms on complete varieties. The conclusion in both cases are strikingly similar: the definable Mal'cev law in both cases reduce to the familiar "group" law x - y + z !