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Magma obeying the Latin square property

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 the associative and identity element properties are optional. A quasigroup with an identity element is called a loop.

There are at least two structurally equivalent formal definitions of quasigroup. One defines a quasigroup as a set with one binary operation, and the other, from universal algebra, defines a quasigroup as having three primitive operations. The homomorphic image of a quasigroup defined with a single binary operation, however, need not be a quasigroup. We begin with the first definition.

A quasigroup may exhibit semisymmetric triality.

A narrower class is a totally symmetric quasigroup (sometimes abbreviated TS quasigroup) in which all conjugates coincide as one operation: 1=x ∗ y = x / y = x \ y. Another way to define (the same notion of) totally symmetric quasigroup is as a semisymmetric quasigroup that is commutative, i. e. 1=x ∗ y = y ∗ x.

Idempotent total symmetric quasigroups are precisely (i. e. in a bijection with) Steiner triples, so such a quasigroup is also called a Steiner quasigroup, and sometimes the latter is even abbreviated as squag. The term sloop refers to an analogue for loops, namely, totally symmetric loops that satisfy 1=x ∗ x = 1 instead of 1=x ∗ x = x. Without idempotency, total symmetric quasigroups correspond to the geometric notion of extended Steiner triple, also called Generalized Elliptic Cubic Curve (GECC).

Every group is a loop, because 1=a ∗ x = b if and only if 1=x = a−1 ∗ b, and 1=y ∗ a = b if and only if 1=y = b ∗ a−1. The integers Z (or the rationals Q or the reals R) with subtraction (−) form a quasigroup. These quasigroups are not loops because there is no identity element (0 is a right identity because 1=a − 0 = a, but not a left identity because, in general, 1=0 − a ≠ a). 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 1=x ∗ y = (x + y) / 2. Every Steiner triple system defines an idempotent, commutative quasigroup: 1=a ∗ b is the third element of the triple containing a and b. These quasigroups also satisfy 1=(x ∗ y) ∗ y = x for all x and y in the quasigroup. These quasigroups are known as Steiner quasigroups. The set where 1=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 invertibility of the quasigroup binary operation combined with associativity implies the existence of an identity element, which then implies the existence of inverse elements, thus satisfying all three requirements of a group. The following construction is due to Hans Zassenhaus. On the underlying set of the four dimensional vector space F4 over the 3 element Galois field 1=F = Z/3Z define
(x1, x2, x3, x4) ∗ (y1, y2, y3, y4) = (x1, x2, x3, x4) + (y1, y2, y3, y4) + (0, 0, 0, (x3 − y3)(x1y2 − x2y1)). Then, (F4, ∗) is a commutative Moufang loop that is not a group. More generally, the nonzero elements of any division algebra form a quasigroup with the operation of multiplication in the algebra.

In the remainder of the article we shall denote quasigroup multiplication simply by juxtaposition. Quasigroups have the cancellation property: if 1=ab = ac, then 1=b = c. This follows from the uniqueness of left division of ab or ac by a. Similarly, if 1=ba = ca, then 1=b = c.

The Latin square property of quasigroups implies that, given any two of the three variables in 1=xy = z, the third variable is uniquely determined.

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.

For a countably infinite quasigroup Q, it is possible to imagine an infinite array in which every row and every column corresponds to some element q of Q, and where the element a ∗ b is in the row corresponding to a and the column responding to b. In this situation too, the Latin square property says that each row and each column of the infinite array will contain every possible value precisely once. For an uncountably infinite quasigroup, such as the group of non zero real numbers under multiplication, the Latin square property still holds, although the name is somewhat unsatisfactory, as it is not possible to produce the array of combinations to which the above idea of an infinite array extends since the real numbers cannot all be written in a sequence. (This is somewhat misleading however, as the reals can be written in a sequence of length , assuming the well ordering theorem.)

A quasigroup or loop homomorphism is a map f : Q → P between two quasigroups such that 1=f(xy) = f(x)f(y). Quasigroup homomorphisms necessarily preserve left and right division, as well as identity elements (if they exist).

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 forms a group with the automorphism group as a subgroup. Every 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 that is isotopic to a group need not be a group. For example, the quasigroup on R with multiplication given by (x, y) ↦ (x + y)/2 is isotopic to the additive group (R, +), but is not itself a group as it has no identity element. Every medial quasigroup is isotopic to an abelian group by the Bruck–Toyoda theorem.

Left and right division are examples of forming a quasigroup by permuting the variables in the defining equation. From the original operation ∗ (i. e., 1=x ∗ y = z) we can form five new operations: 1=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).

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 isostrophic to each other. There are also many other names for this relation of "isostrophe", e. g., paratopy.

An n ary quasigroup is a set with an n ary operation, (Q, f) with f : Qn → Q, such that the equation 1=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, 1=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 called an n ary group.