Geodesic
Axiomatization of quasigroups
Discussiones Mathematicae. General Algebra and Applications, Tome 27 (2007) no. 1

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Quasigroups were originally described combinatorially, in terms of existence and uniqueness conditions on the solutions to certain equations. Evans introduced a universal-algebraic characterization, as algebras with three binary operations satisfying four identities. Now, quasigroups are redefined as heterogeneous algebras, satisfying just two conditions respectively known as hypercommutativity and hypercancellativity.
Keywords: quasigroup, heterogeneous algebra, hyperidentity 
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Smith, Jonathan. Axiomatization of quasigroups. Discussiones Mathematicae. General Algebra and Applications, Tome 27 (2007) no. 1. http://geodesic.mathdoc.fr/item/DMGAA_2007_27_1_a8/