# 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 at least two equivalent formal definitions of quasigroup. One definition casts quasigroups as a set with one binary operation, and the other is a version from universal algebra which describes a quasigroup by using three primitive 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), obeying the Latin square property. This states 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: Each element of the set occurs exactly once in each row and exactly once in each column of the quasigroup's multiplication table, or Cayley table. This property ensures that the Cayley table of a finite quasigroup is a Latin square.)

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**.

The empty set equipped with the empty binary operation satisfies this definition of a quasigroup. Some authors accept the empty quasigroup but others explicitly exclude it.^{[1]}^{[2]}

### 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, that is, an element *e* such that:

*x***e*=*x*and*e***x*=*x*for all*x*in*Q*.

It follows that the identity element *e* is unique, and that every element of *Q* has a unique left and right inverse. Since the presence of an identity element is essential, a loop cannot be empty.

A loop which is associative is a group. There are some weaker associativity-like properties which have been given special names.

A **Bol loop** is a loop that satisfies, for each *x*, *y* and *z* in *Q*, one of the two identifies:

*x** (*y** (*x***z*)) = (*x** (*y***x*)) **z*Template:Quad(a*left Bol loop*), or- ((
*z***x*) **y*) **x*=*z** ((*x***y*) **z*)Template:Quad(a*right Bol loop*).

A loop that is both a left and right Bol loop is a **Moufang loop**. This is equivalent to any one of the following single Moufang identities:

*x** (*y** (*x***z*)) = ((*x***y*) **x*) **z*,*z** (*x** (*y***x*)) = ((*z***x*) **y*) **x*,- (
*x***y*) * (*z***x*) =*x** ((*y***z*) **x*), or - (
*x***y*) * (*z***x*) = (*x** (*y***z*)) **x*.

### Total antisymmetry

A quasigroup (*Q*, ∗) is called **totally anti-symmetric** if for all *c*, *x*, *y* ∈ *Q*, the following implications hold:^{[3]}

- (
*c*∗*x*) ∗*y*= (*c*∗*y*) ∗*x*⇒*x*=*y* *x*∗*y*=*y*∗*x*⇒*x*=*y*,

and it is called **weakly totally anti-symmetric** if only the first implication holds.^{[3]}

This property is required, for example, in the Damm algorithm.

## 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*. These quasigroups also satisfy (*x***y*) **y*=*x*for all*x*and*y*in the quasigroup. These quasigroups are known as*Steiner quasigroups*.^{[4]} - 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.
- An associative quasigroup is either empty or is a group, since if there is at least one element, the existence of inverses and associativity imply the existence of an identity.
- The following construction is due to Hans Zassenhaus. On the underlying set of the four-dimensional vector space
**F**^{4}over the 3-element Galois field**F**=**Z**/3**Z**define

- (
*x*_{1},*x*_{2},*x*_{3},*x*_{4}) * (*y*_{1},*y*_{2},*y*_{3},*y*_{4}) = (*x*_{1},*x*_{2},*x*_{3},*x*_{4}) + (*y*_{1},*y*_{2},*y*_{3},*y*_{4}) + (0, 0, 0, (*x*_{3}−*y*_{3})(*x*_{1}*y*_{2}−*x*_{2}*y*_{1})). - Then, (
**F**^{4}, *) is a commutative Moufang loop that is not a group.^{[5]}

- 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*): *Q* → *Q*, defined by

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,

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

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

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

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

- A loop has the
*left inverse property*if for all and . Equivalently, or . - A loop has the
*right inverse property*if for all and . Equivalently, or . - A loop has the
*antiautomorphic inverse property*if or, equivalently, if . - A loop has the
*weak inverse property*when if and only if . This may be stated in terms of inverses via or equivalently .

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

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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

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. Every medial quasigroup is isotopic to an abelian group by the Bruck–Toyoda theorem.

### 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: *x* o *y* := *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*: *Q*^{n} → *Q*, such that the equation *f*(*x*_{1},...,*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, 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* = *x*_{1} · *x*_{2} · ··· · *x*_{n}; 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:

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 called an n-ary group.

### Right- and left-quasigroups

Template:Expand section
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*).

## Number of small quasigroups and loops

The number of isomorphism classes of small quasigroups (sequence A057991 in OEIS) and loops (sequence A057771 in OEIS) is given here:^{[6]}

Order | Number of quasigroups | Number of loops |
---|---|---|

0 | 1 | 0 |

1 | 1 | 1 |

2 | 1 | 1 |

3 | 5 | 1 |

4 | 35 | 2 |

5 | 1,411 | 6 |

6 | 1,130,531 | 109 |

7 | 12,198,455,835 | 23,746 |

8 | 2,697,818,331,680,661 | 106,228,849 |

9 | 15,224,734,061,438,247,321,497 | 9,365,022,303,540 |

10 | 2,750,892,211,809,150,446,995,735,533,513 | 20,890,436,195,945,769,617 |

11 | 19,464,657,391,668,924,966,791,023,043,937,578,299,025 | 1,478,157,455,158,044,452,849,321,016 |

## See also

- Bol loop
- Division ring – a ring in which every non-zero element has a multiplicative inverse
- Semigroup – an algebraic structure consisting of a set together with an associative binary operation
- Monoid – a semigroup with an identity element
- Planar ternary ring – has an additive and multiplicative loop structure
- Small Latin squares and quasigroups
- Problems in loop theory and quasigroup theory
- Mathematics of Sudoku

## Notes

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## 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. - {{#invoke:citation/CS1|citation

|CitationClass=citation }}

- 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.