Quasigroup


Quasigroup

In mathematics, especially in abstract algebra, a quasigroup is an 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. A quasigroup with an identity element is called a loop.

Contents

Definitions

There are two equivalent formal definitions of quasigroup with, respectively, one and three primitive binary operations. We begin with the first definition, which is easier to follow.

A quasigroup (Q, *) is a set Q with a binary operation * (that is, a magma), such that for each a and b in Q, there exist unique elements x and y in Q such that:

  • a*x = b ;
  • y*a = b .

(In other words: For two elements a and b, b can be found in row a and in column a of the quasigroup's Cayley table. So the Cayley tables of quasigroups are simply latin squares.)
The unique solutions to these equations are written x = a \ b and y = b / a. The operations '\' and '/' are called, respectively, left and right division.

Universal algebra

Given some algebraic structure, an identity is an equation in which all variables are tacitly universally quantified, and in which all operations are among the primitive operations proper to the structure. Algebraic structures axiomatized solely by identities are called varieties. Many standard results in universal algebra hold only for varieties. Quasigroups are varieties if left and right division are taken as primitive.

A quasigroup (Q, *, \, /) is a type (2,2,2) algebra satisfying the identities:

  • y = x * (x \ y) ;
  • y = x \ (x * y) ;
  • y = (y / x) * x ;
  • y = (y * x) / x .

Hence if (Q, *) is a quasigroup according to the first definition, then (Q, *, \, /) is the same quasigroup in the sense of universal algebra.

Loop

A loop is a quasigroup with an identity element e such that:

  • x*e = x = e*x .

It follows that the identity element e is unique, and that every element of Q has a unique left and right inverse.

Examples

  • Every group is a loop, because a * x = b if and only if x = a−1 * b, and y * a = b if and only if y = b * a−1.
  • The integers Z with subtraction (−) form a quasigroup.
  • The nonzero rationals Q* (or the nonzero reals R*) with division (÷) form a quasigroup.
  • Any vector space over a field of characteristic not equal to 2 forms an idempotent, commutative quasigroup under the operation x * y = (x + y) / 2.
  • Every Steiner triple system defines an idempotent, commutative quasigroup: a * b is the third element of the triple containing a and b.
  • The set {±1, ±i, ±j, ±k} where ii = jj = kk = +1 and with all other products as in the quaternion group forms a nonassociative loop of order 8. See hyperbolic quaternions for its application. (The hyperbolic quaternions themselves do not form a loop or quasigroup).
  • The nonzero octonions form a nonassociative loop under multiplication. The octonions are a special type of loop known as a Moufang loop.
  • More generally, the set of nonzero elements of any division algebra form a quasigroup.

Properties

In the remainder of the article we shall denote quasigroup multiplication simply by juxtaposition.

Quasigroups have the cancellation property: if ab = ac, then b = c. This follows from the uniqueness of left division of ab or ac by a. Similarly, if ba = ca, then b = c.

Multiplication operators

The definition of a quasigroup can be treated as conditions on the left and right multiplication operators L(x), R(y): QQ, defined by

L(x)y = xy\, ,
R(x)y = yx\, .

The definition says that both mappings are bijections from Q to itself. A magma Q is a quasigroup precisely when all these operators, for every x in Q, are bijective. The inverse mappings are left and right division, that is,

L(x)^{-1}y = x\backslash y\, ,
R(x)^{-1}y = y/x\, .

In this notation the identities among the quasigroup's multiplication and division operations (stated in the section on universal algebra) are

\begin{align}
L(x)L(x)^{-1} &= 1\qquad&\text{corresponding to}\qquad x(x\backslash y) &= y,\\
L(x)^{-1}L(x) &= 1\qquad&\text{corresponding to}\qquad x\backslash(xy) &= y,\\
R(x)R(x)^{-1} &= 1\qquad&\text{corresponding to}\qquad (y/x)x &= y,\\
R(x)^{-1}R(x) &= 1\qquad&\text{corresponding to}\qquad (yx)/x &= y,
\end{align}

where 1 denotes the identity mapping on Q.

Latin squares

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 in many ways: the border row (containing the column headers) and the border column (containing the row headers) can each be any permutation of the elements. See small Latin squares and quasigroups.

Inverse properties

Every loop element has a unique left and right inverse given by

x^{\lambda} = e/x \qquad x^{\lambda}x = e
x^{\rho} = x\backslash e \qquad xx^{\rho} = e

A loop is said to have (two-sided) inverses if xλ = xρ for all x. In this case the inverse element is usually denoted by x − 1.

There are some stronger notions of inverses in loops which are often useful:

  • A loop has the left inverse property if xλ(xy) = y for all x and y. Equivalently, L(x) − 1 = L(xλ) or x\backslash y = x^{\lambda}y.
  • A loop has the right inverse property if (yx)xρ = y for all x and y. Equivalently, R(x) − 1 = R(xρ) or y / x = yxρ.
  • A loop has the antiautomorphic inverse property if (xy)λ = yλxλ or, equivalently, if (xy)ρ = yρxρ.
  • A loop has the weak inverse property when (xy)z = e if and only if x(yz) = e. This may be stated in terms of inverses via (xy)λx = yλ or equivalently x(yx)ρ = yρ.

A loop has the inverse property if it has both the left and right inverse properties. Inverse property loops also have the antiautomorphic and weak inverse properties. In fact, any loop which satisfies any two of the above four identities has the inverse property and therefore satisfies all four.

Any loop which satisfies the left, right, or antiautomorphic inverse properties automatically has two-sided inverses.

Morphisms

A quasigroup or loop homomorphism is a map f : QP between two quasigroups such that f(xy) = f(x)f(y). Quasigroup homomorphisms necessarily preserve left and right division, as well as identity elements (if they exist).

Homotopy and isotopy

Let Q and P be quasigroups. A quasigroup homotopy from Q to P is a triple (α, β, γ) of maps from Q to P such that

\alpha(x)\beta(y) = \gamma(xy)\,

for all x, y in Q. A quasigroup homomorphism is just a homotopy for which the three maps are equal.

An isotopy is a homotopy for which each of the three maps (α, β, γ) is a bijection. Two quasigroups are isotopic if there is an isotopy between them. In terms of Latin squares, an isotopy (α, β, γ) is given by a permutation of rows α, a permutation of columns β, and a permutation on the underlying element set γ.

An autotopy is an isotopy from a quasigroup to itself. The set of all autotopies of a quasigroup form a group with the automorphism group as a subgroup.

Each quasigroup is isotopic to a loop. If a loop is isotopic to a group, then it is isomorphic to that group and thus is itself a group. However, a quasigroup which is isotopic to a group need not be a group. For example, the quasigroup on R with multiplication given by (x+y)/2 is isotopic to the additive group (R,+), but is not itself a group.

Conjugation (parastrophe)

Left and right division are examples of forming a quasigroup by permuting the variables in the defining equation. From the original operation * (i.e., x*y = z) we can form five new operations: xoy := y*x (the opposite operation), / and \, and their opposites. That makes a total of six quasigroup operations, which are called the conjugates or parastrophes of *. Any two of these operations are said to be "conjugate" or "parastrophic" to each other (and to themselves).

Paratopy

If the set Q has two quasigroup operations, * and ·, and one of them is isotopic to a conjugate of the other, the operations are said to be paratopic to each other. There are also many other names for this relation of "paratopy", e.g., isostrophe".

Generalizations

Polyadic or multiary 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. Polyadic or multiary means n-ary for some nonnegative integer n.

A 0-ary, or nullary, quasigroup is just a constant element of Q. A 1-ary, or unary, quasigroup is a bijection of Q to itself. A binary, or 2-ary, quasigroup is an ordinary quasigroup.

An example of a multiary quasigroup is an iterated group operation, y = x1 · x2 · ··· · xn; it is not necessary to use parentheses to specify the order of operations because the group is associative. One can also form a multiary quasigroup by carrying out any sequence of the same or different group or quasigroup operations, if the order of operations is specified.

There exist multiary quasigroups that cannot be represented in any of these ways. An n-ary quasigroup is irreducible if its operation cannot be factored into the composition of two operations in the following way:

 f(x_1,\dots,x_n) = g(x_1,\dots,x_{i-1},\,h(x_i,\dots,x_j),\,x_{j+1},\dots,x_n),

where 1 ≤ i < jn and (i, j) ≠ (1, n). Finite irreducible n-ary quasigroups exist for all n > 2; see Akivis and Goldberg (2001) for details.

An n-ary quasigroup with an n-ary version of associativity is an n-ary group.

Right- and left-quasigroups

A right-quasigroup (Q, *, /) is a type (2,2) algebra satisfying the identities:

  • y = (y / x) * x;
  • y = (y * x) / x.

Similarly, a left-quasigroup (Q, *, \) is a type (2,2) algebra satisfying the identities:

  • y = x * (x \ y);
  • y = x \ (x * y).

See also

References

  • Akivis, M. A., and Vladislav V. Goldberg (2001), "Solution of Belousov's problem," Discussiones Mathematicae. General Algebra and Applications 21: 93–103.
  • Bruck, R.H. (1958), A Survey of Binary Systems. Springer-Verlag.
  • Chein, O., H. O. Pflugfelder, and J.D.H. Smith, eds. (1990), Quasigroups and Loops: Theory and Applications. Berlin: Heldermann. ISBN 3-88538-008-0.
  • Dudek, W.A., and Glazek, K. (2008), "Around the Hosszu-Gluskin Theorem for n-ary groups," Discrete Math. 308: 4861-4876.
  • Pflugfelder, H.O. (1990), Quasigroups and Loops: Introduction. Berlin: Heldermann. ISBN 3-88538-007-2.
  • Smith, J.D.H. (2007), An Introduction to Quasigroups and their Representations. Chapman & Hall/CRC Press. ISBN 1-58488-537-8.
  • Smith, J.D.H. and Anna B. Romanowska (1999), Post-Modern Algebra. Wiley-Interscience. ISBN 0-471-12738-8.

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