Geodesic
The Generalized Equations of Bisymmetry Associativity and Transitivity on Quasigroups
Canadian mathematical bulletin, Tome 15 (1972) no. 1, pp. 119-124

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The generalized equations of bisymmetry, associativity and transitivity are, respectively, (1) (x1y)2(z3u) = (x4z)5(y6u) (2) (x1y)2z = x3(y4z) (3) (x1z)2(y3z) = x4y. The numbers 1, 2, 3,..., 6 represent binary operations and x, y, z and u are taken freely from certain sets.We shall be concerned with the cases in which x, y, z, and u are from the same set and each operation is a quasigroup operation. Under these conditions the solution of all three equations is known [1], [2]; equations (1) and (3) having been reduced to the form of (2) and a solution of (2) being given. We wish to present a new approach to these equations which we feel has the advantages that the equations may be resolved independently, the motivation behind the proof is clear, and the method lends itself to application on algebraic structures weaker than quasigroups. (Details of these generalizations will be given elsewhere.) 
Taylor, M. A. The Generalized Equations of Bisymmetry Associativity and Transitivity on Quasigroups. Canadian mathematical bulletin, Tome 15 (1972) no. 1, pp. 119-124. doi: 10.4153/CMB-1972-021-2
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