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.

Definitions

Formally, a quasigroup (Q, *) is a set Q with a binary operation * : Q × Q → Q (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. We shall always assume that a quasigroup is nonempty.

In universal algebra, a quasigroup (Q, *, \, /) can defined as a set Q with three binary operations (*, \, /) satisfying the following identities:

• [y = x*(x\backslash y)\,]
• [y = x\backslash(x*y)\,]
• [y = (y/x)*x\,]
• [y = (y*x)/x\,]

A loop is a quasigroup with an identity element e:

• [x*e=x=e*x.\,]

It follows that there is exactly one identity, and that each element of a loop has both a unique left inverse and a unique 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 reals R) with division (÷) form a quasigroup.
• 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 set 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. Actually, the octonions are a special type of loop known as a Moufang loop.
• More generally, the set of nonzero elements of any finite-dimensional algebra with no zero divisors forms a quasigroup.

Properties

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

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)^y = x\backslash y\,]

[R(x)^y = y/x\,]

In this notation the quasigroup identities are

[L(x)L(x)^ = 1\qquad x(x\backslash y) = y\,]

[L(x)^L(x) = 1\qquad x\backslash(xy) = y\,]

[R(x)R(x)^ = 1\qquad (y/x)x = y\,]

[R(x)^R(x) = 1\qquad (yx)/x = y\,]

Quasigroups have the cancellation property: if ab = ac, then b = c. This is because x = b is certainly a solution of the equation ab = ax, and the solution is required to be unique. Similarly, if ba = ca, then b = c.

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 every permutation of the elements, see small Latin squares and quasigroups.

Inverse properties

Every loop has a unique left and right inverse given by

[x^ = e/x \qquad x^x = e]

[x^ = x\backslash e \qquad xx^ = e]

A loop is said to have (two-sided) inverses if [x^ = x^] for all x. In this case the inverse element is usually denoted by [x^]. There are some stronger notions of inverses in loops which are often useful:

• A loop has the left inverse property if [x^(xy) = y] for all [x] and [y]. This is equivalent to saying [L(x)^ = L(x^)] or [x\backslash y = x^y].
• A loop has the right inverse property if [(yx)x^ = y] for all [x] and [y]. This is equivalent to saying [R(x)^ = R(x^)] or [y/x = yx^].

A loop has the inverse property if it has both the left and right inverse properties. Any loop which satisfies the left or right inverse properties automatically has two-sided inverses.

Two other inverse properties are:

• A loop has the antiautomorphic inverse property if [(xy)^ = y^x^] or, equivalently, if [(xy)^ = y^x^]. Every loop with this property has two-sided inverses.
• 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)^x = y^] or equivalently [x(yx)^ = y^].

Every inverse property loop has both of these properties. Moreover, any loop which satisfies any two of the left, right, antiautomorphic, or weak inverse properties satisfies the inverse property.

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

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. Multary means n-ary for some nonnegative n.

An example of a multary 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 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

• Moufang loop
• magma
• semigroup
• monoid
• small Latin squares and quasigroups

References

• R.H. Bruck (1958), A Survey of Binary Systems, Springer.
• O. Chein, H. O. Pflugfelder and J. D. H. Smith (eds.) (1990), Quasigroups and Loops: Theory and Applications, Heldermann. ISBN 3885380080.
• H.O. Pflugfelder (1990), Quasigroups and Loops: Introduction, Heldermann. ISBN 3885380072.
• J.D.H. Smith and Anna B. Romanowska (1999) Post-Modern Algebra, Wiley-Interscience. ISBN 0471127388.

External links

• [quasigroups]
• [Problems in Loop Theory and Quasigroup Theory]

 

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