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.

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 or groupoid), 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 .

The unique solutions to these equations are written x = a \ b and y = b / a. '\' and '/' denote, respectively, the defined binary operations of left and right division (sometimes called parastrophe). This axiomatization of quasigroups requires existential quantification and hence first-order logic.

The second definition of a quasigroup is grounded in universal algebra, which prefers that algebraic structures be varieties, i.e., that structures be axiomatized solely by identities. An identity is an equation in which all variables are tacitly universally quantified, and the only operations are the primitive operations proper to the structure. Quasigroups can be axiomatized in this manner 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 an equivalent quasigroup in the universal algebra sense.

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 all elements of Q have 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.

Left and right multiplication

The definition of a quasigroup Q says that the left and right multiplication operators defined by

L(x)y=xy\,

R(x)y=yx\,

are bijections from Q to itself. A magma Q is a quasigroup precisely when these operators are bijective. The inverse maps are given in terms of left and right division by

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

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

In this notation the quasigroup identities are

{\begin{aligned}L(x)L(x)^{-1}&=1\qquad &x(x\backslash y)&=y\\L(x)^{-1}L(x)&=1\qquad &x\backslash (xy)&=y\\R(x)R(x)^{-1}&=1\qquad &(y/x)x&=y\\R(x)^{-1}R(x)&=1\qquad &(yx)/x&=y\end{aligned}}

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 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 $x^{\lambda }=x^{\rho }$ 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 $x^{\lambda }(xy)=y$ for all $x$ and $y$ . Equivalently, $L(x)^{-1}=L(x^{\lambda })$ or $x\backslash y=x^{\lambda }y$ .
A loop has the right inverse property if $(yx)x^{\rho }=y$ for all $x$ and $y$ . Equivalently, $R(x)^{-1}=R(x^{\rho })$ or $y/x=yx^{\rho }$ .
A loop has the antiautomorphic inverse property if $(xy)^{\lambda }=y^{\lambda }x^{\lambda }$ or, equivalently, if $(xy)^{\rho }=y^{\rho }x^{\rho }$ .
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 $(xy)^{\lambda }x=y^{\lambda }$ or equivalently $x(yx)^{\rho }=y^{\rho }$ .

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 : Q → P 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.

Generalizations

Polyadic or multiary quasigroups

An n-ary quasigroup is a set with an n-ary operation, (Q, f) with f: Qⁿ → Q, such that the equation f(x₁,...,x_n) = 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 quasigroup is just a constant element of Q. A 1-ary quasigroup is a bijection of Q to itself.

An example of a multiary quasigroup is an iterated group operation, y = x₁ · x₂ · ··· · x_n; then it is not necessary to use parentheses because the group is associative. One can also carry out a 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},...,x_{n})=g(x_{1},...,x_{i-1},\,h(x_{i},...,x_{j}),\,x_{j+1},...,x_{n}),

where 1 ≤ i < j ≤ n 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.

References

Akivis, M. A., and Goldberg, Vladislav V. (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.
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. ISBN 1-58488-537-8.
-------- and Anna B. Romanowska (1999) Post-Modern Algebra. Wiley-Interscience. ISBN 0-471-12738-8.

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