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On definitions of groupoids closely connected with quasigroups. (English) Zbl 1176.20065

Summary: Both “existential” and “equational” definitions of binary quasigroups and groupoids closely connected with quasigroups are given. It is proved that a groupoid \((Q,\cdot)\) is a quasigroup if and only if all middle translations of \((Q,\cdot)\) are bijective maps of the set \(Q\).

MSC:

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