Moufang loop

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In mathematics, a Moufang loop is a special kind of algebraic structure. It is similar to a group in many ways but may fail to be associative. Moufang loops were introduced by Ruth Moufang.

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Definition

A Moufang loop is a loop Q that satisfies any one of the following identities (the binary operation in Q is denoted by juxtaposition):

  1. z(x(zy)) = ((zx)z)y
  2. x(z(yz)) = ((xz)y)z
  3. (zx)(yz) = (z(xy))z

for all x, y, z in Q. These identities—known as Moufang identities—are, in fact, equivalent in any loop. Therefore if Q satisfies one of them it satisfies all of them.

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Examples

  • Any group is associative loop and therefore a Moufang loop.
  • The nonzero octonions form a nonassociative Moufang loop under octonion multiplication.
  • The subset of unit norm octonions (forming a 7-sphere in O) is closed under multiplication and therefore forms a Moufang loop.
  • The basis octonions and their additive inverses form a finite Moufang loop of order 16.
  • The set of invertible split-octonions form a nonassociative Moufang loop, as do the set of unit norm split-octonions. More generally, the set of invertible elements in any octonion algebra over a field F form a Moufang loop, as do the subset of unit norm elements.
  • For any field F let M(F) denote the Moufang loop of unit norm elements in the (unique) split-octonion algebra over F. Let Z denote the center of M(F). If the characteristic of F is 2 then Z = {e}, otherwise Z = {±e}. The Paige loop over F is the loop M*(F) = M(F)/Z. Paige loops are nonassociative simple Moufang loops. All finite nonassociative simple Moufang loops are Paige loops over finite fields. The smallest Paige loop M*(2) has order 120.
  • A large class of nonassociative Moufang loops can be constructed as follows. Let G be an arbitrary group. Define a new element u not in G and let M(G,2) = G ∪ (G u). The product in M(G,2) is given by the usual product of elements in G together with
    <math>(gu)h = (gh^{-1})u</math>
    <math>g(hu) = (hg)u</math>
    <math>(gu)(hu) = h^{-1}g</math>
It follows that <math>u^2 = 1</math> and <math>ug = g^{-1}u</math>. With the above product M(G,2) is a Moufang loop. It is associative iff G is abelian.
  • The smallest nonassociative Moufang loop is M(S3,2) which has order 12.
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Properties

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Associativity

Moufang loops differ from groups in that they need not be associative. A Moufang loop that is associative is a group. The Moufang identities may be viewed as weaker forms of associativity.

By setting various elements to the identity, the Moufang identities imply

Moufang's theorem states that when three elements x, y, and z in a Moufang loop obey the associative law: (xy)z = x(yz) then they generate an associative subloop; that is, a group. A corollary of this is that all Moufang loops are di-associative (i.e. the subloop generated by any two elements of a Moufang loop is associative and therefore a group). In particular, Moufang loops are power associative, so that exponents xn are well-defined. When working with Moufang loops, it is common to drop the parenthesis in expressions with only two distinct elements. For example, the Moufang identities may be written unambigously as

  1. z(x(zy)) = (zxz)y
  2. ((xz)y)z = x(zyz)
  3. (zx)(yz) = z(xy)z
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Left and right multiplication

The Moufang identites can be written in terms of the left and right multiplication operaters on Q. The first two identities state that

  • <math>L_xL_yL_x = L_{xyx}</math>
  • <math>R_xR_yR_x = R_{xyx}</math>

while the third identity says

  • <math>L_z(x)R_z(y) = B_z(xy)</math>

for all <math>x,y,z</math> in <math>Q</math>. Here <math>B_z = L_zR_z = R_zL_z</math> is bimultiplication by <math>z</math>. The third Moufang identity is therefore equivalent to the statement that the triple <math>(L_z, R_z, B_z)</math> is an autotopy of <math>Q</math> for all <math>z</math> in <math>Q</math>.

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Inverse properties

All Moufang loops have the inverse property, which means that each element x has a two-sided inverse x−1 which satisfies the identities:

<math>x^{-1}(xy) = y = (yx)x^{-1}</math>

for all x and y. It follows that <math>(xy)^{-1} = y^{-1}x^{-1}</math> and <math>x(yz) = e</math> if and only if <math>(xy)z = e</math>.

Moufang loops are universal among inverse property loops; that is, a loop Q is a Moufang loop iff every loop isotope of Q has the inverse property. If follows that every loop isotope of a Moufang loop is a Moufang loop.

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Moufang quasigroups

Any quasigroup which satisfies one of the Moufang identities must, in fact, have an identity element and therefore be a Moufang loop. We give a proof here for the third identity:

Let a be any element of Q, and let e be the unique element such that ae = a. Then for any x in Q, (xa)x = (x(ae))x = (xa)(ex). Cancelling gives x = ex so that e is a left identity element. Now let f be the element such that fe = e. Then (yf)e = (e(yf))e = (ey)(fe) = (ey)e = ye. Cancelling gives yf = y, so f is a right identity element. Lastly, e = ef = f, so e is a two-sided identity element.

The proofs for first two identities are somewhat more difficult.

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See also

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References

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