Editing Quasigroup (section)

=== Algebra ===
A '''quasigroup''' {{math|(''Q'', ∗)}} is a non-empty [[Set (mathematics)|set]] {{mvar|Q}} with a binary operation {{math|∗}} (that is, a [[magma (algebra)|magma]], indicating that a quasigroup has to satisfy the closure property), obeying the '''Latin square property'''. This states that, for each {{mvar|a}} and {{mvar|b}} in {{mvar|Q}}, there exist unique elements {{mvar|x}} and {{mvar|y}} in {{mvar|Q}} such that both
<math display=block>a \ast x = b</math>
<math display=block>y \ast a = b</math> 
hold. (In other words: Each element of the set occurs exactly once in each row and exactly once in each column of the quasigroup's multiplication table, or [[Cayley table]]. This property ensures that the Cayley table of a finite quasigroup, and, in particular, a finite group, is a [[Latin square]].)  The requirement that {{mvar|x}} and {{mvar|y}} be unique can be replaced by the requirement that the magma be [[Cancellation property|cancellative]].{{sfn|ps=|Rubin|Rubin|1985|p=[https://archive.org/details/equivalentsofaxi0000rubi/page/109 109]}}{{efn|For clarity, cancellativity alone is insufficient: the requirement for existence of a solution must be retained.}}

The unique solutions to these equations are written {{math|''x'' {{=}} ''a'' \ ''b''}} and {{math|''y'' {{=}} ''b'' / ''a''}}.  The operations '{{math|\}}' and '{{math|/}}' are called, respectively, [[left division]] and [[right division]].  With regard to the Cayley table, the first equation (left division) means that the {{mvar|b}} entry in the {{mvar|a}} row is in the {{mvar|x}} column while the second equation (right division) means that the {{mvar|b}} entry in the {{mvar|a}} column is in the {{mvar|y}} row.

The [[empty set]] equipped with the [[Function_(mathematics)#Standard_functions|empty binary operation]] satisfies this definition of a quasigroup.  Some authors accept the empty quasigroup but others explicitly exclude it.{{sfn|ps=|Pflugfelder|1990|p=2}}{{sfn|ps=|Bruck|1971|p=1}}