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In mathematics, especially in abstract algebra, a quasigroup is an algebraic structure resembling a group in the sense that "division" is always possible. Quasigroups differ from groups mainly in that they need not be associative. A quasigroup with an identity element is called a loop. == Definitions == There are at least two equivalent formal definitions of quasigroup. One defines a quasigroup as a set with one binary operation, and the other, from universal algebra, defines a quasigroup as having three primitive operations. We begin with the first definition. A quasigroup is a set, ''Q'', with a binary operation, ∗, (that is, a magma), obeying the Latin square property. This states that, for each ''a'' and ''b'' in ''Q'', there exist unique elements ''x'' and ''y'' in ''Q'' such that both #''a'' ∗ ''x'' = ''b'' #''y'' ∗ ''a'' = ''b'' 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 is a Latin square.) The unique solutions to these equations are written and . The operations '\' and '/' are called, respectively, left and right division. The empty set equipped with the empty binary operation satisfies this definition of a quasigroup. Some authors accept the empty quasigroup but others explicitly exclude it. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Quasigroup」の詳細全文を読む スポンサード リンク
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