Quasigroup is a binary algebraic structure in which every equation of the form or of the form has a unique solution for , resp. . The notion of quasigroup is a generalization of the notion of group without the associativity law or identity element. A quasigroup with identity is called a loop (French la boucle, Russian лупа).
Note that, in the absence of associativity, it is not enough (even for a loop) to say that every element has an inverse element (on either side); instead, you must say that division is always possible. This is because the definition won't work right without associativity.
The usual definition is this:
A quasigroup is a set equipped with a binary operation (which we will write with concatenation) such that:
Then is called the left quotient ( divided by , over ) and is called the right quotient ( dividing , under ).
Note that we must specify, in the definition, that and are unique; without associativity, we cannot prove this.
As with the inverse elements of a group, we can make the quotients into operations so that all axioms are equations:
A quasigroup is a set equipped with three binary operations (product, left quotient, and right quotient) such that these equations always hold:
Also, without the right quotient we have left quasigroups, and without the left quotient the right quasigroups. Thus quasigroups are described by a Lawvere theory and can therefore be internalized into any cartesian monoidal category. There are weaker structures, say left and right quasigroups in which either or is well defined.
In any case:
A loop is a quasigroup with an identity element.
Loops are also described by a Lawvere theory.
Note that, even in a loop, left and right inverses need not agree. See the discussion on the English Wikipedia for convenient inverse properties.
For good measure, here is another special kind of quasigroup:
A group is an associative loop.
See also Moufang loop.
Local analytic loops have interesting induced structure on the tangent space at the identity, generalizing the Lie algebra of a group, see Sabinin algebra. Sabinin algebras are closely related to the local study of affine connections on manifolds. They include some known important classes of nonassociative algebras, namely Lie algebras, Mal’cev algebras, Lie triple systems (related to the study of symmetric spaces), Bol algebras as simplest cases.
There are interesting subvarieties of quasigroups and loops (which are still not associative). Also, left racks (and quandles in particular) are precisely left distributive left quasigroups, with abundance of recent applications in the study of knots and links. Finite racks have been studied in the connection to classification of finite dimensional pointed Hopf algebras. Local augmented Lie racks appeared as integration objects in the local integration theory of Leibniz algebras.
TS-quasigroups are related to Steiner triple systems.
As a sample of centipede mathematics, we have the following result on smooth quasigroups, i.e., quasigroups internal to the category of smooth manifolds:
The tangent bundle of a smooth quasigroup is trivial.
Suppose WLOG that is inhabited by an element , and let be the tangent space at . Define a map
where is the smooth map and is the map . The map commutes with the bundle projections , . The map has an inverse and each map has an inverse . We may therefore write down an inverse to :
This shows is isomorphic to the product bundle .