Quasigroup

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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.

1 Definitions
2 Examples
3 Properties
- 3.1 Latin squares
- 3.2 Inverse properties
4 Morphisms
- 4.1 Homotopy and isotopy
5 Generalizations
6 See also
7 References
8 External links

[edit] Definitions

There are two formal definitions of quasigroup in common use, one of which defines a quasigroup to be a set with one binary operation, and the other which defines a quasigroup to be a set with three binary operations. We begin with the former definition, which is easier to follow.

A quasigroup (Q, *) is a set Q with a binary operation * : Q × Q → Q (that is, it 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 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 the universal algebra approach, a quasigroup (Q, *, \, /) is defined to be a set Q with three binary operations (*, \, /) satisfying the following identities:

y = x * (x \ y) ,
y = x \ (x * y) ,
y = (y / x) * x ,
y = (y * x) / x .

If (Q, *) is a quasigroup according to the first definition, then with its induced left and right division operations, (Q, *, \, /) is a quasigroup in the universal algebra sense. Conversely, if (Q, *, \, /) is a quasigroup in the second sense, then by simply "forgetting" the left and right division operations, we see that (Q, *) is a quasigroup in the first sense.

A loop is a quasigroup with an identity element e:

x*e = x = e*x .

It follows that there is exactly one identity element, and that each element of a loop has both a unique left inverse and a unique right inverse.

[edit] 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 {±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. 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.

[edit] 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

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

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

In this notation the quasigroup identities are

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

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

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

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

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.

[edit] 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.

[edit] 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 $x λ = x ρ$ 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 $x λ (x y) = y$ for all $x$ and $y$ . This is equivalent to saying $L (x) - 1 = L (x λ)$ or $x\backslash y = x^{\lambda}y$ .
A loop has the right inverse property if $(y x) x ρ = y$ for all $x$ and $y$ . This is equivalent to saying $R (x) - 1 = R (x ρ)$ or $y / x = y x ρ$ .

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 $(x y) λ = y λ x λ$ or, equivalently, if $(x y) ρ = y ρ x ρ$ . Every loop with this property has two-sided inverses.
A loop has the weak inverse property when $(x y) z = e$ if and only if $x (y z) = e$ . This may be stated in terms of inverses via $(x y) λ x = y λ$ or equivalently $x (y x) ρ = y ρ$ .

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.

[edit] 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).

[edit] 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.

[edit] 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.

[edit] See also

[edit] 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 3-88538-008-0 .
H.O. Pflugfelder (1990), Quasigroups and Loops: Introduction, Heldermann. ISBN 3-88538-007-2 .
J.D.H. Smith and Anna B. Romanowska (1999) Post-Modern Algebra, Wiley-Interscience. ISBN 0-471-12738-8 .

[edit] External links

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Categories: Nonassociative algebra | Group theory | Latin squares