A quasi group is an aggregate or combination, which lacks structure or organisation, and whose members may be unaware, or less aware, of the existence of groupings. Social classes, status groups, age and gender groups, crowds can be seen as examples of quasi groups. Quasigroup. ... 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 are not necessarily associative. A quasigroup with an identity element is called a loop. 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. A quasigroup (Q, ∗) is a non-empty 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, and, in particular, finite group, is a Latin square.) The uniqueness requirement can be replaced by the requirement that the magma be cancellative. The unique solutions to these equations are written x = a b and y = b / a. 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.