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.
@article{bwmeta1.element.bwnjournal-article-doi-10_7151_dmgaa_1116,
author = {Jonathan D.H. Smith},
title = {Axiomatization of quasigroups},
journal = {Discussiones Mathematicae - General Algebra and Applications},
volume = {26},
year = {2006},
zbl = {1135.20048},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgaa_1116}
}
Jonathan D.H. Smith. Axiomatization of quasigroups. Discussiones Mathematicae - General Algebra and Applications, Tome 26 (2006) . http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgaa_1116/
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