On Birkhoff's quasigroup axioms
Birkhoff defined a quasigroup as an algebra ( Q , ⋅ , \ , / ) that satisfies the following six identities: x ⋅ ( x \ y ) = y , ( y / x ) ⋅ x = y , x \ ( x ⋅ y ) = y , ( y ⋅ x ) / x = y , x / ( y \ x ) = y , and ( x / y ) \ x = y . We investigate triples and tetrads of identities composed of these six, emphasizing those that axiomatize the variety of quasigroups.