Quasigroup

From Wikipedia, the free encyclopedia - View original article

 
Jump to: navigation, search

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.

Definitions[edit]

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:

(In other words: For two elements a and b, b can be found in row a and in column a of the quasigroup's multiplication table, or Cayley table. So the Cayley tables of finite 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.

The empty set equipped with the empty binary operation satisfies the definition of a quasigroup, but some authors explicitly exclude this case.[1][2]

Universal algebra[edit]

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:

Hence if (Q, *) is a quasigroup according to the first definition, then (Q, *, \, /) is the same quasigroup in the sense of universal algebra.

Loop[edit]

A loop is a quasigroup with an identity element, that is, an element 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. Since the presence of an identity element is essential, a loop cannot be empty.

A Moufang loop is a loop that satisfies the Moufang identity:

Examples[edit]

(x1, x2, x3, x4) * (y1, y2, y3, y4) = (x1, x2, x3, x4) + (y1, y2, y3, y4) + (0, 0, 0, (x3y3)(x1y2x2y1)).
Then, (F4, *) is a commutative Moufang loop that is not a group.[citation needed]

Properties[edit]

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[edit]

The definition of a quasigroup can be treated as conditions on the left and right multiplication operators L(x), R(y): QQ, defined by

\begin{align}   L(x)y &= xy \\   R(x)y &= yx \end{align}

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,

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

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

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

where 1 denotes the identity mapping on Q.

Latin squares[edit]

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[edit]

Every loop element 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^{\lambda} = x^{\rho} 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 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[edit]

A quasigroup or loop homomorphism is a map f : QP 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[edit]

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. 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)[edit]

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[edit]

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[edit]

Polyadic or multiary quasigroups[edit]

An n-ary quasigroup is a set with an n-ary operation, (Q, f) with f: QnQ, 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:

 f(x_1,\dots,x_n) = g(x_1,\dots,x_{i-1},\,h(x_i,\dots,x_j),\,x_{j+1},\dots,x_n),

where 1 ≤ i < jn 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[edit]

A right-quasigroup (Q, *, /) is a type (2,2) algebra satisfying the identities:

Similarly, a left-quasigroup (Q, *, \) is a type (2,2) algebra satisfying the identities:

Number of small quasigroups and loops[edit]

The number of isomorphism classes of small quasigroups (sequence A057991 in OEIS) and loops (sequence A057771 in OEIS) is given here:[4]

OrderNumber of quasigroupsNumber of loops
010
111
211
351
4352
51,4116
61,130,531109
712,198,455,83523,746
82,697,818,331,680,661106,228,849
915,224,734,061,438,247,321,4979,365,022,303,540
102,750,892,211,809,150,446,995,735,533,51320,890,436,195,945,769,617
1119,464,657,391,668,924,966,791,023,043,937,578,299,0251,478,157,455,158,044,452,849,321,016

See also[edit]

Notes[edit]

  1. ^ Hala O. Pflugfelder (1990). Quasigroups and loops: introduction. Heldermann Verlag. p. 2. 
  2. ^ Richard Hubert Bruck (1971). A survey of binary systems. Springer. p. 1. 
  3. ^ Colbourn & Dinitz 2007, pg. 497, definition 28.12
  4. ^ McKay, Meynert, Myrvold, Small Latin Squares, Quasigroups and Loops

References[edit]

External links[edit]