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.
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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
- 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.
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 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 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) is 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.
Properties
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 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.
See also
- magma
- semigroup
- monoid
- J.D.H. Smith and Anna B. Romanowska (1999) Post-Modern Algebra, Wiley-Interscience ISBN 0471127388.de:Quasigruppe