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Selfdistributive groupoids of small orders. (English) Zbl 0901.20051
By a groupoid is meant a nonempty set together with a binary, multiplicatively denoted operation. A groupoid is called left distributive if it satisfies the identity \(x(yz)=(xy)(xz)\). Right distributive groupoids are defined dually, and distributive groupoids are groupoids that are both left and right distributive. A groupoid is called medial if it satisfies the identity \((xy)(uv)=(xu)(yv)\).
The authors prove the following theorem. If \(G\) is a non-medial distributive groupoid such that every proper subgroupoid of \(G\) is medial, then \(G\) is a quasigroup. As a consequence it follows that every non-medial distributive groupoid contains at least 81 elements.
Reviewer: M.Novotný (Brno)

20N02 Sets with a single binary operation (groupoids)
20N05 Loops, quasigroups
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