quasigroup

As such, it can be given the structure of a quasigroup.

A quasigroup with an identity element is called a loop.

There are at least two equivalent formal definitions of quasigroup.

An autotopy is an isotopy from a quasigroup to itself.

For example, every quasigroup, and thus every group, is cancellative.

However, a quasigroup which is isotopic to a group need not be a group.

In particular, every medial quasigroup is isotopic to an abelian group.

(The hyperbolic quaternions themselves do not form a loop or quasigroup).

The integers Z with subtraction ( ) form a quasigroup.

Cracovians are an example of a quasigroup.

The totally anti-symmetric quasigroup is taken from Damm's doctoral dissertation page 111.

More generally, the set of nonzero elements of any division algebra form a quasigroup.

Left and right division are examples of forming a quasigroup by permuting the variables in the defining equation.

Each quasigroup is isotopic to a loop.

This makes S an idempotent, commutative quasigroup.

Since a gyrogroup has inverses and an identity it qualifies as a quasigroup and a loop.

See Quasigroup.

The validity of a digit sequence containing a check digit is defined over a quasigroup.

This approach gives rise to notions of left and right division in a partially ordered magma, additionally endowing it with a quasigroup structure.

An n-ary quasigroup is irreducible if its operation cannot be factored into the composition of two operations in the following way:

The latter carries the structure of a Lie quasigroup (a nonassociative group), which can be identified with the set of unit octonions.

The concepts of quasigroup and algebra over a field are examples of mathematical structures describing hyperbolic quaternions.

A binary, or 2-ary, quasigroup is an ordinary quasigroup.

A magma Q is a quasigroup precisely when all these operators, for every x in Q, are bijective.

An n-ary quasigroup with an n-ary version of associativity is called an n-ary group.