Quasigroup

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QUICK PHRASES: group without identity and associativity, magma with unique left and right quotients

Definition

Definition with symbols

A quasigroup is a magma (set with binary operation) (S,*) with the following properties:

  • For every a,b \in S, there is a unique x \in S such that a * x = b
  • For every a,b \in S, there is a unique y \in S such that y * a = b

Note that x,y may be equal or different for a particular choice of a and b.

Equivalently, a quasigroup is a magma where every element is cancellative and where every element is left-accessible and right-accessible from every other element.

Relation with other notions

Stronger notions

Any nonempty quasigroup that is also a semigroup is a group. For full proof, refer: Associative quasigroup implies group

Weaker notions

A finite magma is a quasigroup if and only if its multiplication table is a Latin square.