Quasigroup

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.