 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 two equivalent formal definitions of quasigroup with, respectively, one and three primitive binary 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), 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. So the Cayley tables of quasigroups are simply latin squares.)
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 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 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 and 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 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.
 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.
 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 (twosided) 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}(xy) = y for all x and y. Equivalently, L(x) ^{− 1} = L(x^{λ}) or .
 A loop has the right inverse property if (yx)x^{ρ} = y for all x and y. Equivalently, R(x) ^{− 1} = R(x^{ρ}) or y / x = yx^{ρ}.
 A loop has the antiautomorphic inverse property if (xy)^{λ} = y^{λ}x^{λ} or, equivalently, if (xy)^{ρ} = y^{ρ}x^{ρ}.
 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)^{λ}x = y^{λ} or equivalently x(yx)^{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 twosided 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
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.
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".
Generalizations
Polyadic or multiary quasigroups
An nary quasigroup is a set with an nary 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 nary for some nonnegative integer n.
A 0ary, or nullary, quasigroup is just a constant element of Q. A 1ary, or unary, quasigroup is a bijection of Q to itself. A binary, or 2ary, 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 nary 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 nary quasigroups exist for all n > 2; see Akivis and Goldberg (2001) for details.
An nary quasigroup with an nary version of associativity is an nary group.
Right and leftquasigroups
A rightquasigroup (Q, *, /) is a type (2,2) algebra satisfying the identities:
 y = (y / x) * x;
 y = (y * x) / x.
Similarly, a leftquasigroup (Q, *, \) is a type (2,2) algebra satisfying the identities:
 y = x * (x \ y);
 y = x \ (x * y).
See also
 Moufang loop
 Bol loop
 Semigroup
 Monoid
 Small Latin squares and quasigroups
 Problems in loop theory and quasigroup theory
 Mathematics of Sudoku
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. SpringerVerlag.
 Chein, O., H. O. Pflugfelder, and J.D.H. Smith, eds. (1990), Quasigroups and Loops: Theory and Applications. Berlin: Heldermann. ISBN 3885380080.
 Dudek, W.A., and Glazek, K. (2008), "Around the HosszuGluskin Theorem for nary groups," Discrete Math. 308: 48614876.
 Pflugfelder, H.O. (1990), Quasigroups and Loops: Introduction. Berlin: Heldermann. ISBN 3885380072.
 Smith, J.D.H. (2007), An Introduction to Quasigroups and their Representations. Chapman & Hall/CRC Press. ISBN 1584885378.
 Smith, J.D.H. and Anna B. Romanowska (1999), PostModern Algebra. WileyInterscience. ISBN 0471127388.
External links
Categories: Nonassociative algebra
 Group theory
 Latin squares
Wikimedia Foundation. 2010.
Look at other dictionaries:
quasigroup — noun An algebraic structure, resembling a group, whose arithmetic may not be associative … Wiktionary
Problems in loop theory and quasigroup theory — In 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. Many … Wikipedia
CHquasigroup — In mathematics, a CH quasigroup, introduced by Manin (1986, definition 1.3), is a symmetric quasigroup in which any three elements generate an abelian quasigroup. CH stands for cubic hypersurface. References Manin, Yuri Ivanovich (1986) [1972],… … Wikipedia
Outline of algebraic structures — In universal algebra, a branch of pure mathematics, an algebraic structure is a variety or quasivariety. Abstract algebra is primarily the study of algebraic structures and their properties. Some axiomatic formal systems that are neither… … Wikipedia
Algebraic structure — In algebra, a branch of pure mathematics, an algebraic structure consists of one or more sets closed under one or more operations, satisfying some axioms. Abstract algebra is primarily the study of algebraic structures and their properties. The… … Wikipedia
Hyperbolic quaternion — In mathematics, a hyperbolic quaternion is a mathematical concept first suggested by Alexander MacFarlane in 1891 in a speech to the American Association for the Advancement of Science. The idea was criticized for its failure to conform to… … Wikipedia
List of algebraic structures — In universal algebra, a branch of pure mathematics, an algebraic structure is a variety or quasivariety. Abstract algebra is primarily the study of algebraic structures and their properties. Some axiomatic formal systems that are neither… … Wikipedia
Isotopy of loops — Isotopy of quasigroups = Let (Q,cdot) and (P,circ) 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)circeta(y) = gamma(xcdot y),for all x , y in Q . A quasigroup homomorphism is just… … Wikipedia
Medial — This article is about medial in mathematics. For other uses, see medial (disambiguation). Contents 1 Medial magmas 2 Bruck Toyoda theorem 3 Generalizations 4 See also … Wikipedia
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. Contents 1 Definition 2 Examples 3 Properties … Wikipedia