In mathematics, especially in abstract algebra, a quasigroup is an algebraic structure that resembles 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. In fact, a nonempty associative quasigroup is a group. A quasigroup that has an identity element is called a loop.
Definitions There are at least two structurally equivalent formal definitions of a quasigroup:
One defines a quasigroup as a set with one binary operation. The other, from universal algebra, defines a quasigroup as having three primitive operations. The homomorphic image of a quasigroup that is defined with a single binary operation, however, need not be a quasigroup, in contrast to a quasigroup as having three primitive operations. We begin with the first definition.
Algebra A quasigroup (Q, ∗) is a set Q with a binary operation ∗ (that is, a magma, indicating that a quasigroup has to satisfy the closure property), 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 both
a ∗ x = b {\displaystyle a\ast x=b}
y ∗ a = b {\displaystyle y\ast a=b} hold. (In other words: Each element of the set occurs exactly once in each row and exactly once in each column of the quasigroup's multiplication table, or Cayley table. This property ensures that the Cayley table of a finite quasigroup, and, in particular, a finite group, is a Latin square.) The requirement that x and y be unique can be replaced by the requirement that the magma be cancellative. The unique solutions to these equations are written x = a \ b and y = b / a. The operations '\' and '/' are called, respectively, left division and right division. With regard to the Cayley table, the first equation (left division) means that the b entry in the a row is in the x column while the second equation (right division) means that the b entry in the a column is in the y row. The empty set equipped with the empty binary operation satisfies this definition of a quasigroup. Some authors accept the empty quasigroup, but others explicitly exclude it.
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 that satisfy axioms that are given solely by identities are called varieties. Many standard results in universal algebra hold only for varieties. Quasigroups form a variety if left and right division are taken as primitive. A right-quasigroup (Q, ∗, /) is a type (2, 2) algebra that satisfies the identities:
y = ( y / x ) ∗ x {\displaystyle y=(y/x)\ast x}
y = ( y ∗ x ) / x {\displaystyle y=(y\ast x)/x}
A left-quasigroup (Q, ∗, \) is a type (2, 2) algebra that satisfies the identities:
y = x ∗ ( x ∖ y ) {\displaystyle y=x\ast (x\backslash y)}
y = x ∖ ( x ∗ y ) {\displaystyle y=x\backslash (x\ast y)}
A quasigroup (Q, ∗, \, /) is a type (2, 2, 2) algebra (i.e., equipped with three binary operations) that satisfies the identities:
y = ( y / x ) ∗ x {\displaystyle y=(y/x)\ast x}
y = ( y ∗ x ) / x {\displaystyle y=(y\ast x)/x}
y = x ∗ ( x ∖ y ) {\displaystyle y=x\ast (x\backslash y)}
y = x ∖ ( x ∗ y ) {\displaystyle y=x\backslash (x\ast y)}
In other words: Multiplication and division in either order, one after the other, on the same side by the same element, have no net effect. Hence if (Q, ∗) is a quasigroup according to the definition of the previous section, then (Q, ∗, \, /) is the same quasigroup in the sense of universal algebra. And vice versa: if (Q, ∗, \, /) is a quasigroup according to the sense of universal algebra, then (Q, ∗) is a quasigroup according to the first definition.
Loops A loop is a quasigroup with an identity element; that is, an element, e, such that
x ∗ e = x and e ∗ x = x for all x in Q. It follows that the identity element, e, is unique, and that every element of Q has unique left and right inverses (which need not be the same). Since the presence of an identity element is essential, a loop cannot be empty. A quasigroup with an idempotent element is called a pique ("pointed idempotent quasigroup"); this is a weaker notion than a loop but common nonetheless because, for example, given an abelian group, (A, +), taking its subtraction operation as quasigroup multiplication yields a pique (A, −) with the group identity (zero) turned into a "pointed idempotent". (That is, there is a principal isotopy (x, y, z) ↦ (x, −y, z).) A loop that is associative is a group. A group can have a strictly nonassociative pique isotope, but it cannot have a strictly nonassociative loop isotope. There are weaker associativity properties that have been given special names. For instance, a Bol loop is a loop that satisfies either:
x ∗ (y ∗ (x ∗ z)) = (x ∗ (y ∗ x)) ∗ z for each x, y and z in Q (a left Bol loop), or else
((z ∗ x) ∗ y) ∗ x = z ∗ ((x ∗ y) ∗ x) for each x, y and z in Q (a right Bol loop). A loop that is both a left and right Bol loop is a Moufang loop. This is equivalent to any one of the following single Moufang identities holding for all x, y, z:
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