Quasigroup - Citizendia

In mathematics, especially in abstract algebra, a quasigroup is an algebraic structure resembling a group in the sense that "division" is always possible. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and Abstract algebra is the subject area of Mathematics that studies Algebraic structures such as groups, rings, fields, modules In Algebra, a branch of Pure mathematics, an algebraic structure consists of one or more sets closed under one or more operations, In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element In Mathematics, especially in elementary Arithmetic, division is an arithmetic operation which is the inverse of Multiplication. Quasigroups differ from groups mainly in that they need not be associative. In Mathematics, associativity is a property that a Binary operation can have

1 Definitions
2 Examples
3 Properties
4 Morphisms
- 4.1 Homotopy and isotopy
5 Generalizations
- 5.1 Polyadic or multary quasigroups
6 See also
7 References
8 External links

Definitions

There are two equivalent formal definitions of quasigroup with, respectively, one and three primitive binary operations. In Mathematics, a binary operation is a calculation involving two Operands, in other words an operation whose Arity is two 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 . In Mathematics, a binary operation is a calculation involving two Operands, in other words an operation whose Arity is two In Abstract algebra, a magma (or groupoid) is a basic kind of Algebraic structure.

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. This axiomatization of quasigroups requires existential quantification and hence first-order logic. In Predicate logic, an existential quantification is the predication of a property or relation to at least one member of the domain First-order logic (FOL is a formal Deductive system used in mathematics philosophy linguistics and computer science

The second definition of a quasigroup is grounded in universal algebra, which prefers that algebraic structures be varieties, i. Universal algebra (sometimes called general algebra) is the field of Mathematics that studies Algebraic structures themselves not examples ("models" In Algebra, a branch of Pure mathematics, an algebraic structure consists of one or more sets closed under one or more operations, In Universal algebra, a variety of algebras is the class of all Algebraic structures of a given signature satisfying a given set of identities 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 . In Mathematics, an identity element (or neutral element) is a special type of element of a set with respect to a Binary operation on that

It follows that the identity element e is unique, and that all elements of Q have a unique left and right inverse. In Mathematics, the idea of inverse element generalises the concepts of negation, in relation to Addition, and reciprocal, in relation to In Mathematics, the idea of inverse element generalises the concepts of negation, in relation to Addition, and reciprocal, in relation to

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. In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element ↔
The integers Z with subtraction (−) form a quasigroup. The integers (from the Latin integer, literally "untouched" hence "whole" the word entire comes from the same origin but via French Subtraction is one of the four basic Arithmetic operations it is the inverse of Addition, meaning that if we start with any number and add any number and then subtract
The nonzero rationals Q* (or the nonzero reals R*) with division (÷) form a quasigroup. In Mathematics, a rational number is a number which can be expressed as a Ratio of two Integers Non-integer rational numbers (commonly called fractions In Mathematics, the real numbers may be described informally in several different ways In Mathematics, especially in elementary Arithmetic, division is an arithmetic operation which is the inverse of Multiplication.
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. In Mathematics, a vector space (or linear space) is a collection of objects (called vectors) that informally speaking may be scaled and added In Abstract algebra, a field is an Algebraic structure in which the operations of Addition, Subtraction, Multiplication and division In Mathematics, the characteristic of a ring R, often denoted char( R) is defined to be the smallest number of times one must add the ring's Idempotence ˌaɪdɨmˈpoʊtəns describes the property of operations in Mathematics and Computer science which means that multiple applications of the operation In Mathematics, commutativity is the ability to change the order of something without changing the end result
Every Steiner triple system defines an idempotent, commutative quasigroup: a * b is the third element of the triple containing a and b. In combinatorial Mathematics, a Steiner system (named after Jakob Steiner) is a type of Block design. Idempotence ˌaɪdɨmˈpoʊtəns describes the property of operations in Mathematics and Computer science which means that multiple applications of the operation In Mathematics, commutativity is the ability to change the order of something without changing the end result
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. In Group theory, the quaternion group is a non-abelian group of order 8 See hyperbolic quaternions for its application. In Mathematics, a hyperbolic quaternion is a mathematical concept first suggested by Alexander MacFarlane in 1891 in a speech to the American Association (The hyperbolic quaternions themselves do not form a loop or quasigroup).
The nonzero octonions form a nonassociative loop under multiplication. In Mathematics, the octonions are a nonassociative extension of the Quaternions Their 8-dimensional Normed division algebra over the Real The octonions are a special type of loop known as a Moufang loop. In Mathematics, a Moufang loop is a special kind of Algebraic structure.
More generally, the set of nonzero elements of any division algebra form a quasigroup. In the field of Mathematics called Abstract algebra, a division algebra is roughly speaking an Algebra over a field in which division is possible

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. In Mathematics, the notion of cancellative is a generalization of the notion of Invertible. 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. In Mathematics, a bijection, or a bijective function is a function f from a set X to a set Y with the property 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

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

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. A Latin square is 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

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. Below the Latin squares and Quasigroups of some small orders are considered

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

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). In Abstract algebra, a homomorphism is a structure-preserving map between two Algebraic structures (such as groups rings or Vector 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. In Mathematics, a bijection, or a bijective function is a function f from a set X to a set Y with the property 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. In Mathematics, an automorphism is an Isomorphism from a Mathematical object to itself

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 multary 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 multary 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 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 the 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. 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.

References

Akivis, M. In Mathematics, a Moufang loop is a special kind of Algebraic structure. In Mathematics, a Bol loop is an algebraic structure generalizing the notion of group. In Mathematics, a semigroup is an Algebraic structure consisting of a nonempty set S together with an Associative Binary operation In Abstract algebra, a branch of Mathematics, a monoid is an Algebraic structure with a single Associative Binary operation Below the Latin squares and Quasigroups of some small orders are considered In Mathematics, especially Abstract algebra, loop theory and Quasigroup theory are active research areas with many Open problems. 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.

External links

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