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 and need not have an identity element. A quasigroup with an identity element is called a loop. Definitions There are at least two structurally 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. The homomorphic image of a quasigroup defined with a single binary operation, however, need not be a quasigroup. We begin with the first definition. Algebra A quasigroup is a non-empty set Q with a binary operation ∗ (that is, a magma, indicating that a quasigroup has to satisfy closure property), 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, and, in particular, finite group, is a Latin square.) The requirement that x and y be unique can be replaced by the requirement that the magma be cancellative. The unique solutions to these equations are written and . The operations '\' and '/' are called, respectively, left division and right division. With regard to the Cayley table, the first equation (left division) means that the b entry in the a row marks the x column while the second equation (right division) means that the b entry in the a column marks the y row. 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. Universal algebra Given some algebraic structure, an identity is an equation in which all variables are tacitly universally quantified, and in which all operations are among the primitive operations proper to the structure. Algebraic structures that satisfy axioms that are given solely by identities are called a variety. Many standard results in universal algebra hold only for varieties. Quasigroups form a variety if left