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Algebraic quasigroups. (English) Zbl 0815.20068

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.

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
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