Note that quasigroups have the cancellation property: if a * b = a * c, then b = c. This is because x = b is certainly a solution of the equation a * b = a * x, and the solution is required to be unique. Similarly, if a * b = c * b, then a = c.
The multiplication table of a finite quasigroup is called a Latin square: an n × n table filled with n different symbols in such a way that each symbol occurs exactly once in each row and exactly once in each column.
A quasigroup group with an identity element is called a loop. It follows from the definition of a quasigroup that each element of a loop has both a left inverse and a right inverse.
A Moufang loop is a quasigroup Q in which (a * b) * (c * a) = (a * (b * c)) * a, for all a, b and c in Q. As the name suggests, Moufang loops are actually loops, and we will now prove this. Let a be any element of Q, and let e be the element such that a * e = a. Then for any x in Q, (x * a) * x = (x * (a * e)) * x = (x * a) * (e * x), and cancelling gives x = e * x. So e is a left identity element. Now let b be the element such that b * e = e. Then y * b = e * (y * b), as e is a left identity, so (y * b) * e = (e * (y * b)) * e = (e * y) * (b * e) = (e * y) * e = y * e. Cancelling gives y * b = y, so b is a right identity element. Lastly, e = e * b = b, so e is a twosided identity element.
Every group is a quasigroup, because a * x = b if and only if x = a^{1} * b, and y * a = b if and only if y = b * a^{1}. Moreover, an associative quasigroup must be a Moufang loop, and an associative loop must clearly be a group. This shows that groups are precisely the associative quasigroups. The structure theory of loops is quite analogous to that of groups.
Examples of quasigroups:
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