A closure condition which is equivalent to the Thomsen condition in quasigroups.
Stochastica, Tome 7 (1983) no. 1, p. 11.

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In this note it is shown that the closure condition, X1Y2 = X2Y1, X1Y4 = X2Y3, X3Y3 = X4Y1 --> X4Y2 = X3Y4, (and its dual) is equivalent to the Thomsen condition in quasigroups but not in general. Conditions are also given under which groupoids satisfying it are principal homotopes of cancellative, abelian semigroups, or abelian groups.
Classification : 20N05
Mots-clés : Grupoides, Teoría de grupos, Thomsen condition, quasigroups, groupoids, principal homotopes 
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     author = {M. A. Taylor},
     title = {A closure condition which is equivalent to the {Thomsen} condition in quasigroups.},
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     publisher = {mathdoc},
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     zbl = {0572.20058},
     language = {en},
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M. A. Taylor. A closure condition which is equivalent to the Thomsen condition in quasigroups.. Stochastica, Tome 7 (1983) no. 1, p. 11. https://geodesic-test.mathdoc.fr/item/STO_1983__7_1_38875/