In abstract algebra, a quasigroup is a 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. Abstract algebra is the field of mathematics concerned with the study of algebraic structures such as groups, rings and fields. ... In abstract algebra, an algebraic structure consists of a set together with a collection of operations or relations defined on it which satisfy certain axioms. ... In mathematics, a group is a set, together with a binary operation, such as multiplication or addition, satisfying certain axioms, detailed below. ... In mathematics, especially in elementary arithmetic, division is an arithmetic operation which is the inverse of multiplication, and sometimes it can be interpreted as repeated subtraction. ... In mathematics, associativity is a property that a binary operation can have. ...

Definitions

Formally, a quasigroup (Q, *) is a set Q with a binary operation * : Q × Q → Q (that is, it is a groupoid or magma), such that for all a and b in Q there are unique elements x and y in Q such that In mathematics, a set can be thought of as any collection of distinct things considered as a whole. ... In mathematics, a binary operation, or binary operator, is a calculation involving two input quantities and one kind of a specific operation. ... In mathematics, especially in category theory and homotopy theory, a groupoid is a concept (first developed by Heinrich Brandt in 1926) that simultaneously generalises groups, equivalence relations on sets, and actions of groups on sets. ... In abstract algebra, a magma (also called a groupoid) is a particularly basic kind of algebraic structure. ...

• 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. In this encyclopedia, it will be assumed that a quasigroup is nonempty.

Two quasigroups Q and R are said to have the same order if there is a one-to-one correspondence between their underlying sets. Such quasigroups can be regarded as consisting of the same elements and thus differing only in their multiplications.

Now let Q and R be quasigroups of the same order and denote the set of their elements by M. Q and R are said to be isotopic if there exist permutations A, B, C on M such that // Mathematics In mathematics, especially in abstract algebra and related areas, a permutation is a bijection from a finite set X onto itself. ...

• (x, y) = A[Bx, Cy]

where ( , ) and [ , ] are the products in Q and R respectively.

A loop is a quasigroup with an identity element. It follows that each element of a loop has both a unique left inverse and a unique right inverse. 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 set. ... In mathematics, the inverse of an element x, with respect to an operation *, is an element x such that their compose gives a neutral element. ... In mathematics, the inverse of an element x, with respect to an operation *, is an element x such that their compose gives a neutral element. ...

A Moufang loop (named after Ruth Moufang) is a quasigroup (L, *) satisfying Ruth Moufang (1905-1977) was a German mathematician whose work in projective geometry built upon the work of David Hilbert. ...

• (a*b)*(c*a) = (a*(b*c))*a

for all a, b and c in L. As the name suggests, Moufang loops are actually loops (a proof is given below).

Examples

• Every group is a quasigroup, because a * x = b iff x = a⁻¹ * b, and y * a = b iff y = b * a⁻¹. Since groups are associative, they are also Moufang loops.
• The integers Z with subtraction (−) form a quasigroup.
• The nonzero rationals Q (or the reals R) with division (÷) form a quasigroup.
• 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 16. See hyperbolic quaternions for its application.

For example: i = i j² ≠ (i j) j = k j = − i

• 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 nonzero octonions form a Moufang loop under multiplication.However, unlike the quaternions, they are not associative, and do not form a group. The subset of unit octonions (i.e. those with norm 1) is closed under multiplication and therefore gives the 7-sphere the structure of a Moufang loop.
• More generally, the set of nonzero elements of any finite-dimensional algebra with no zero divisors forms a quasigroup.

In mathematics, a group is a set, together with a binary operation, such as multiplication or addition, satisfying certain axioms, detailed below. ... ↔ ⇔ ≡ For other possible meanings of iff, see IFF. In mathematics, philosophy, logic and technical fields that depend on them, iff is used as an abbreviation for if and only if. Common alternative phrases to iff or if and only if include Q is necessary and sufficient for P and P... In mathematics, associativity is a property that a binary operation can have. ... The integers consist of the positive natural numbers (1, 2, 3, …), their negatives (−1, −2, −3, ...) and the number zero. ... 5 - 2 = 3 Subtraction is one of the four basic arithmetic operations; it is essentially the opposite of addition. ... In mathematics, a rational number (or informally fraction) is a ratio of two integers, usually written as the vulgar fraction a/b, where b is not zero. ... In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite line—the number line. ... In mathematics, especially in elementary arithmetic, division is an arithmetic operation which is the inverse of multiplication, and sometimes it can be interpreted as repeated subtraction. ... Cycle diagram of Q. Each color specifies a series of powers of any element connected to the identity element (1). ... 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. ... A vector space (or linear space) is the basic object of study in the branch of mathematics called linear algebra. ... In mathematics, an idempotent element is an element which, intuitively, leaves something unchanged. ... In mathematics, especially abstract algebra, a binary operation * on a set S is commutative if x * y = y * x for all x and y in S. Otherwise * is noncommutative. ... In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ... In mathematics, the characteristic of a ring R with identity element 1R is defined to be the smallest positive integer n such that n1R = 0, where n1R is defined as 1R + ... + 1R with n summands. ... In mathematics, a Steiner system is a type of block design. ... In mathematics, an idempotent element is an element which, intuitively, leaves something unchanged. ... In mathematics, especially abstract algebra, a binary operation * on a set S is commutative if x * y = y * x for all x and y in S. Otherwise * is noncommutative. ... In mathematics, the octonions are a nonassociative extension of the quaternions. ... In mathematics, the quaternions are a non-commutative extension of the complex numbers. ... In mathematics, associativity is a property that a binary operation can have. ... A sphere is a perfectly symmetrical geometrical object. ... In mathematics, the dimension of a vector space V is the cardinality (i. ... In mathematics, an algebra over a field K, or a K-algebra, is a vector space A over K equipped with a compatible notion of multiplication of elements of A. A straightforward generalisation allows K to be any commutative ring. ... In abstract algebra, a non-zero element a of a ring R is a left zero divisor if there exists a non-zero b such that ab = 0. ...

Properties

Quasigroups have the cancellation property: if a * b = a * c, then b = c. This is because x = b is certainly a solution of the equation a * b = a * x, and the solution is required to be unique. Similarly, if a * b = c * b, then a = c. In mathematics, an element a in a magma (M,*) has the left cancellation property (or is left-cancellative) if for all b and c in M, a * b = a * c always implies b = c. ...

Each quasigroup is isotopic to a loop, and if a loop is isotopic to a group, then it is isomorphic to that group and thus is itself a group.

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

Moufang loops

We stated earlier that Moufang loops are loops, which is to say that they have a unique identity element.

Proof. Let a be any element of M, and let e be the element such that a * e = a. Then for any x in Q, (x * a) * x = (x * (a * e)) * x = (x * a) * (e * x), and cancelling gives x = e * x. So e is a left identity element. Now let b be the element such that b * e = e. Then y * b = e * (y * b), as e is a left identity, so (y * b) * e = (e * (y * b)) * e = (e * y) * (b * e) = (e * y) * e = y * e. Cancelling gives y * b = y, so b is a right identity element. Lastly, e = e * b = b, so e is a two-sided identity element. □

Any associative quasigroup must be a Moufang loop, and an associative loop must clearly be a group. This shows that groups are precisely the associative quasigroups. The structure theory of loops is quite analogous to that of groups. In mathematics, associativity is a property that a binary operation can have. ...

Although Moufang loops are not generally associative, they do satisfy weaker forms of associativity. One can show that the defining Moufang identity (multiplication denoted by juxtaposition)

• (ab)(ca) = (a(bc))a

is equivalent to each of:

• a(b(ac)) = ((ab)a)c
• a(b(cb)) = ((ab)c)b

All three of these are called Moufang identities. Any one of them can serve to define a Moufang loop. By setting various elements to the identity one can show that these laws imply

• a(ab) = (aa)b
• (ab)b = a(bb)
• a(ba) = (ab)a

Thus all Moufang loops are alternative. Moufang showed moreover that the subloop generated by any two elements of a Moufang loop is associative (and therefore a group). In particular, Moufang loops are power associative. When working with Moufang loops, it is common to drop the parenthesis in expressions with only two distinct elements. In abstract algebra, a magma G is said to be left alternative if (xx)y=x(xy) for all x and y in G and right alternative if y(xx)=(yx)x for all x and y in G. A magma that is both left and right alternative is said... In abstract algebra, power associativity is a weak form of associativity. ...

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. 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.

See also

• magma
• semigroup
• monoid
• small Latin squares and quasigroups

In abstract algebra, a magma (also called a groupoid) is a particularly basic kind of algebraic structure. ... In mathematics, a semigroup is a set with an associative binary operation on it. ... In abstract algebra, a branch of mathematics, a monoid is an algebraic structure with a single, associative binary operation and an identity element. ... Below the Latin squares and quasigroups of some small orders are considered. ...

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 3885380080.

• H.O. Pflugfelder (1990), Quasigroups and Loops: Introduction, Heldermann. ISBN 3885380072.

• J.D.H. Smith and Anna B. Romanowska (1999) Post-Modern Algebra, Wiley-Interscience. ISBN 0471127388.

External link

• quasigroups