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Concept
Quasigroup
Summary
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.
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 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, 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 x = a \ b and y = b / a. 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.
Official source
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