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In abstract algebra, a quasigroup is a 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.1 Definitions
Formally, a quasigroup is a magma (Q, *), i.e. a set Q with a binary operation * : Q × Q → Q, such that for all a and b in Q there are unique elements x and y in Q such that
The unique solutions to these equations are often written x = a \ b and y = b / a. The operations \ and / are called left and right division. In this encyclopedia, it will be assumed that a quasigroup is nonempty.
A loop is a quasigroup with an identity element. It follows that each element of a loop has both a unique left inverse and a unique right inverse.
A Moufang loop is a quasigroup (L, *) satisfying
- (a*b)*(c*a) = (a*(b*c))*a
for all a, b and c in L. As the name suggests, Moufang loops are actually loops (a proof is given below).
2 Examples
- Every group is a quasigroup, because a * x = b iff x = a−1 * b, and y * a = b iff y = b * a−1. Since groups are associative, they are also Moufang loops.
- The integers Z with subtraction (−) form a quasigroup.
- The nonzero rationals Q (or the reals R) with division (÷) form a quasigroup.
- The set {±1, ±i, ±j, ±k} where ii = jj = kk = +1 and all other products as in the quaternions forms a quasigroup or loop or latin square.
- Any real vector spaceThe fundamental concept in linear algebra is that of a vector space or linear space . This is a generalization of the set of all geometrical vectors and is used throughout modern mathematics. Formal definition A set V is a vector space over a field F (for forms an idempotentIn mathematics, an idempotent element (or simply an idempotent is something that when multiplied by (for a function, composed with) itself, gives itself as a result. For example, the only two real numbers which are idempotent under multiplication are 0 an, commutative quasigroup under the operation x * y = (x + y) / 2. (The vector space can actually be over any fieldIn abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication, and division (except division by zero) may be performed and the associative, commutative, and distributive rules hold, which are famil of characteristic not equal to 2).
- Every Steiner triple system is an idempotent, commutative quasigroup.
- The nonzero octonions form a Moufang loop under multiplication. The subset of unit octonions (i.e. those with norm 1) are closed under multiplication and therefore give the 7- sphere the structure of a Moufang loop.
- More generally, the set of nonzero elements of any finite-dimensional algebra with no zero divisors forms a quasigroup.
