Quasigroup

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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.

Contents

Definitions

Formally, a quasigroup (Q, *) is a set Q with a binary operation * : Q × QQ (that is, it is a groupoid or magma), such that for all a and b in Q there are unique elements x and y in Q such that

  • a * x = b
  • y * a = b

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.

Two quasigroups Q and R are said to have the same order if there is a one-to-one correspondence between them. Such quasigropus can be regarded as consisting of the same elements and thus differing only in their multiplications.

Now let Q and R be quasigroups of the same order and denote the set of their elements by M. Q and R are said to be isotopic if there exist permutations A, B, C on M such that

  • (x, y) = [xA, yB]C

where ( , ) and [ , ] are the products in Q and R respectively.

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 (named after Ruth Moufang) 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).

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 (clearly, since they form a group) or loop or Latin square.
  • Any real vector space forms an idempotent, commutative quasigroup under the operation x * y = (x + y) / 2. (The vector space can actually be over any field of characteristic not equal to 2).
  • Every Steiner triple system defines an idempotent, commutative quasigroup: a * b is the third element of the triple containing a and b.
  • The nonzero octonions form a Moufang loop under multiplication.However, unlike the quaternions, they are not associative, and do not form a group. The subset of unit octonions (i.e. those with norm 1) is closed under multiplication and therefore gives 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.

Properties

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.

Each quasigroup is isotopic to a loop, and if a loop is isotopic to a group, then it is isomorphic to that group and thus is itself a group.

The multiplication table of a finite quasigroup is 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. Conversely, every Latin square can be taken as the multiplication table of a quasigroup.

We stated earlier that Moufang loops are loops, which is to say that they have a unique identity element.

Proof. Let a be any element of M, 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 two-sided identity element. □

Any 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.

Although Moufang loops are not generally associative, they do satisfy weaker forms of associativity. One can show that the defining Moufang identity (multiplication denoted by juxtaposition)

  • (ab)(ca) = (a(bc))a

is equivalent to each of:

  • a(b(ac)) = ((ab)a)c
  • a(b(cb)) = ((ab)c)b

All three of these are called Moufang identities. Any one of them can serve to define a Moufang loop. By setting various elements to the identity one can show that these laws imply

  • a(ab) = (aa)b
  • (ab)b = a(bb)
  • a(ba) = (ab)a

Thus all Moufang loops are alternative. Moufang showed moreover that the subloop generated by any two elements of a Moufang loop is associative (and therefore a group). In particular, Moufang loops are power associative. When working with Moufang loops, it is common to drop the parenthesis in expressions with only two distinct elements.

Multary quasigroups

An n-ary quasigroup is a set with an n-ary operation, (Q, f) with f: QnQ, such that the equation f(x1,...,xn) = y has a unique solution for any one variable if all the other n variables are specified arbitrarily. Multary means n-ary for some nonnegative n.

An example of a multary quasigroup is an iterated group operation, y = x1 · x2 ··· xn; then it is not necessary to use parentheses because the group is associative. One can also carry out a sequence of same or different group or quasigroup operations, if the order of operations is specified. There exist multary quasigroups that cannot be represented in any of these ways.

See also

References

it:Quasigruppo fr:Quasigroupe ru:Квазигруппа

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