Quasigroup Totally Explained

Everything about Quasigroup totally explained

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.

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 or groupoid), 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 . The unique solutions to these equations are written x = a b and y = b / a. '' and '/' denote, respectively, the defined binary operations of left and right division. This axiomatization of quasigroups requires existential quantification and hence first-order logic.

The second definition of a quasigroup is grounded in universal algebra, which prefers that algebraic structures be varieties, for example, that structures be axiomatized solely by identities. An identity is an equation in which all variables are tacitly universally quantified, and the only operations are the primitive operations proper to the structure. Quasigroups can be axiomatized in this manner 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 an equivalent quasigroup in the universal algebra sense.
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 all elements of Q have 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,...,x_n),

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.

Further Information

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