Padmanabhan, R.; Venkataraman, R. Algebraic quasigroups. (English) Zbl 0815.20068 J. Ramanujan Math. Soc. 8, No. 1-2, 61-80 (1993). From the authors’ abstract: 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 \(c\) in \(k\). 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. Reviewer: M.Csikós (Gödöllö) MSC: 20N05 Loops, quasigroups 08A40 Operations and polynomials in algebraic structures, primal algebras 14L05 Formal groups, \(p\)-divisible groups 03C05 Equational classes, universal algebra in model theory 20A05 Axiomatics and elementary properties of groups Keywords:polynomially defined group laws; quasigroup structures; group structures; projective curves PDFBibTeX XMLCite \textit{R. Padmanabhan} and \textit{R. Venkataraman}, J. Ramanujan Math. Soc. 8, No. 1--2, 61--80 (1993; Zbl 0815.20068)