Quasigroup
Encyclopedia
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.
s. 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
), such that for each a and b in Q, there exist unique elements x and y in Q such that:
(In other words: For two elements a and b, b can be found in row a and in column a of the quasigroup's Cayley table
. So the Cayley tables of quasigroups are simply latin square
s.)
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.
, an identity is an equation in which all variables are tacitly universally quantified, and in which all operation
s 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:
Hence if (Q, *) is a quasigroup according to the first definition, then (Q, *, \, /) is the same quasigroup in the sense of universal algebra.
e such that:
It follows that the identity element e is unique, and that every element of Q has a unique left
and right inverse
.
A Moufang loop
is a loop that satisfies the Moufang identity:
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.
The definition says that both mappings are bijection
s 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.
: 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
.
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 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.
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).
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.
, (Q, f) with f: Qn → Q, such that the equation f(x1,...,xn) = 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 = x1 · x2 · ··· · xn; 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 an n-ary group
.
Similarly, a left-quasigroup (Q, *, \) is a type (2,2) algebra satisfying the identities:
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, especially in abstract algebra
Abstract algebra
Abstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras...
, a quasigroup is an algebraic structure
Algebraic structure
In abstract algebra, an algebraic structure consists of one or more sets, called underlying sets or carriers or sorts, closed under one or more operations, satisfying some axioms. Abstract algebra is primarily the study of algebraic structures and their properties...
resembling a group
Group (mathematics)
In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...
in the sense that "division
Division (mathematics)
right|thumb|200px|20 \div 4=5In mathematics, especially in elementary arithmetic, division is an arithmetic operation.Specifically, if c times b equals a, written:c \times b = a\,...
" 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 operationBinary operation
In mathematics, a binary operation is a calculation involving two operands, in other words, an operation whose arity is two. Examples include the familiar arithmetic operations of addition, subtraction, multiplication and division....
s. We begin with the first definition, which is easier to follow.
A quasigroup (Q, *) is a set Q with a binary operation
Binary operation
In mathematics, a binary operation is a calculation involving two operands, in other words, an operation whose arity is two. Examples include the familiar arithmetic operations of addition, subtraction, multiplication and division....
* (that is, a magma
Magma (algebra)
In abstract algebra, a magma is a basic kind of algebraic structure. Specifically, a magma consists of a set M equipped with a single binary operation M \times M \rightarrow M....
), 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 other words: For two elements a and b, b can be found in row a and in column a of the quasigroup's Cayley table
Cayley table
A Cayley table, after the 19th century British mathematician Arthur Cayley, describes the structure of a finite group by arranging all the possible products of all the group's elements in a square table reminiscent of an addition or multiplication table...
. So the Cayley tables of quasigroups are simply latin square
Latin square
In combinatorics and in experimental design, a Latin square is an n × n array filled with n different symbols, each occurring exactly once in each row and exactly once in each column...
s.)
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.
Universal algebra
Given some algebraic structureAlgebraic structure
In abstract algebra, an algebraic structure consists of one or more sets, called underlying sets or carriers or sorts, closed under one or more operations, satisfying some axioms. Abstract algebra is primarily the study of algebraic structures and their properties...
, an identity is an equation in which all variables are tacitly universally quantified, and in which all operation
Operation (mathematics)
The general operation as explained on this page should not be confused with the more specific operators on vector spaces. For a notion in elementary mathematics, see arithmetic operation....
s are among the primitive operations proper to the structure. Algebraic structures axiomatized solely by identities are called varieties
Variety (universal algebra)
In mathematics, specifically universal algebra, a variety of algebras is the class of all algebraic structures of a given signature satisfying a given set of identities. Equivalently, a variety is a class of algebraic structures of the same signature which is closed under the taking of homomorphic...
. Many standard results in universal algebra
Universal algebra
Universal algebra is the field of mathematics that studies algebraic structures themselves, not examples of algebraic structures....
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 elementIdentity element
In mathematics, an identity element is a special type of element of a set with respect to a binary operation on that set. It leaves other elements unchanged when combined with them...
e such that:
- x*e = x = e*x .
It follows that the identity element e is unique, and that every element of Q has a unique left
Inverse element
In abstract algebra, the idea of an inverse element generalises the concept of a negation, in relation to addition, and a reciprocal, in relation to multiplication. The intuition is of an element that can 'undo' the effect of combination with another given element...
and right inverse
Inverse element
In abstract algebra, the idea of an inverse element generalises the concept of a negation, in relation to addition, and a reciprocal, in relation to multiplication. The intuition is of an element that can 'undo' the effect of combination with another given element...
.
A Moufang loop
Moufang loop
In mathematics, a Moufang loop is a special kind of algebraic structure. It is similar to a group in many ways but need not be associative. Moufang loops were introduced by Ruth Moufang.-Definition:...
is a loop that satisfies the Moufang identity:
- (x*y)*(z*x) = x*((y*z)*x) .
Examples
- Every groupGroup (mathematics)In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...
is a loop, because a * x = b if and only ifIf and only ifIn logic and related fields such as mathematics and philosophy, if and only if is a biconditional logical connective between statements....
x = a−1 * b, and y * a = b if and only if y = b * a−1. - The integerIntegerThe integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...
s Z with subtractionSubtractionIn arithmetic, subtraction is one of the four basic binary operations; it is the inverse of addition, meaning that if we start with any number and add any number and then subtract the same number we added, we return to the number we started with...
(−) form a quasigroup. - The nonzero rationals Q* (or the nonzero realsReal numberIn mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
R*) with divisionDivision (mathematics)right|thumb|200px|20 \div 4=5In mathematics, especially in elementary arithmetic, division is an arithmetic operation.Specifically, if c times b equals a, written:c \times b = a\,...
(÷) form a quasigroup. - Any vector spaceVector spaceA vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...
over a fieldField (mathematics)In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...
of characteristicCharacteristic (algebra)In mathematics, the characteristic of a ring R, often denoted char, is defined to be the smallest number of times one must use the ring's multiplicative identity element in a sum to get the additive identity element ; the ring is said to have characteristic zero if this repeated sum never reaches...
not equal to 2 forms an idempotent, commutative quasigroup under the operation x * y = (x + y) / 2. - Every Steiner triple systemSteiner system250px|right|thumbnail|The [[Fano plane]] is an S Steiner triple system. The blocks are the 7 lines, each containing 3 points. Every pair of points belongs to a unique line....
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 groupQuaternion groupIn group theory, the quaternion group is a non-abelian group of order eight, isomorphic to a certain eight-element subset of the quaternions under multiplication...
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 loopMoufang loopIn mathematics, a Moufang loop is a special kind of algebraic structure. It is similar to a group in many ways but need not be associative. Moufang loops were introduced by Ruth Moufang.-Definition:...
. - The following construction is due to Hans Julius ZassenhausHans Julius ZassenhausHans Julius Zassenhaus was a German mathematician, known for work in many parts of abstract algebra, and as a pioneer of computer algebra....
. On the underlying set of the four dimensional vector spaceVector spaceA vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...
F4 over the 3-element Galois field F = Z/3Z define * (y1, y2, y3, y4) = (0, 0, 0, (x3 - y3)(x1y2 - x2y1)) .
- Then, (F4, *) is a commutative Moufang loopMoufang loopIn mathematics, a Moufang loop is a special kind of algebraic structure. It is similar to a group in many ways but need not be associative. Moufang loops were introduced by Ruth Moufang.-Definition:...
that is not a groupGroupA group is a number of things or persons being in some relation to one another.- Mathematics :* Group , a set together with a binary operation satisfying certain algebraic conditions** Group theory, the study of groups...
.- More generally, the set of nonzero elements of any division algebraDivision algebraIn the field of mathematics called abstract algebra, a division algebra is, roughly speaking, an algebra over a field, in which division is possible.- Definitions :...
form a quasigroup.
- More generally, the set of nonzero elements of any division algebra
Properties
- In the remainder of the article we shall denote quasigroup multiplication simply by juxtaposition.
Quasigroups have the cancellation property
Cancellation property
In mathematics, the notion of cancellative is a generalization of the notion of invertible.An element a in a magma has the left cancellation property if for all b and c in M, a * b = a * c always implies b = c.An element a in a magma has the right cancellation...
: 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 byThe definition says that both mappings are bijection
Bijection
A bijection is a function giving an exact pairing of the elements of two sets. A bijection from the set X to the set Y has an inverse function from Y to X. If X and Y are finite sets, then the existence of a bijection means they have the same number of elements...
s 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 squareLatin square
In combinatorics and in experimental design, a Latin square is an n × n array filled with n different symbols, each occurring exactly once in each row and exactly once in each column...
: 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
Small Latin squares and quasigroups
Below the Latin squares and quasigroups of some small orders are considered.-Order 1:...
.
Inverse properties
Every loop element has a unique left and right inverse given byA 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 homomorphismHomomorphism
In abstract algebra, a homomorphism is a structure-preserving map between two algebraic structures . The word homomorphism comes from the Greek language: ὁμός meaning "same" and μορφή meaning "shape".- Definition :The definition of homomorphism depends on the type of algebraic structure under...
is a map
Map (mathematics)
In most of mathematics and in some related technical fields, the term mapping, usually shortened to map, is either a synonym for function, or denotes a particular kind of function which is important in that branch, or denotes something conceptually similar to a function.In graph theory, a map is a...
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 thatfor 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
Bijection
A bijection is a function giving an exact pairing of the elements of two sets. A bijection from the set X to the set Y has an inverse function from Y to X. If X and Y are finite sets, then the existence of a bijection means they have the same number of elements...
. 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.
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: xoy := 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".Polyadic or multiary quasigroups
An n-ary quasigroup is a set with an n-ary operationArity
In logic, mathematics, and computer science, the arity of a function or operation is the number of arguments or operands that the function takes. The arity of a relation is the dimension of the domain in the corresponding Cartesian product...
, (Q, f) with f: Qn → Q, such that the equation f(x1,...,xn) = 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 = x1 · x2 · ··· · xn; 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
Associativity
In mathematics, associativity is a property of some binary operations. It means that, within an expression containing two or more occurrences in a row of the same associative operator, the order in which the operations are performed does not matter as long as the sequence of the operands is not...
is an n-ary group
N-ary group
In mathematics, an n-ary group is a generalization of a group to a set G with a n-ary operation instead of a binary operation. The axioms for an n-ary group are defined in such a way as to reduce to those of a group in the case .-Associativity:The easiest axiom to generalize is the associative law...
.
Right- and left-quasigroups
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).
See also
- Moufang loopMoufang loopIn mathematics, a Moufang loop is a special kind of algebraic structure. It is similar to a group in many ways but need not be associative. Moufang loops were introduced by Ruth Moufang.-Definition:...
- Bol loopBol loopIn mathematics and abstract algebra, a Bol loop is an algebraic structure generalizing the notion of group. Bol loops are named for the Dutch mathematician Gerrit Bol who introduced them in ....
- SemigroupSemigroupIn mathematics, a semigroup is an algebraic structure consisting of a set together with an associative binary operation. A semigroup generalizes a monoid in that there might not exist an identity element...
- MonoidMonoidIn abstract algebra, a branch of mathematics, a monoid is an algebraic structure with a single associative binary operation and an identity element. Monoids are studied in semigroup theory as they are naturally semigroups with identity. Monoids occur in several branches of mathematics; for...
- Small Latin squares and quasigroupsSmall Latin squares and quasigroupsBelow the Latin squares and quasigroups of some small orders are considered.-Order 1:...
- Problems in loop theory and quasigroup theoryProblems in loop theory and quasigroup theoryIn mathematics, especially abstract algebra, loop theory and quasigroup theory are active research areas with many open problems. As in other areas of mathematics, such problems are often made public at professional conferences and meetings...
- Mathematics of SudokuMathematics of SudokuThe class of Sudoku puzzles consists of a partially completed row-column grid of cells partitioned into N regions each of size N cells, to be filled in using a prescribed set of N distinct symbols , so that each row, column and region contains exactly one of each element of the set...