Quasigroup - Knowlpedia

Identity: There exists e ∈ L e \in L e ∈ L with e a = a e = a ea = ae = a e a = a e = a for all a ∈ L a \in L a ∈ L
Divisibility: For all a , b ∈ L a, b \in L a , b ∈ L , the equations a x = b ax = b a x = b and y a = b ya = b y a = b have unique solutions

A quasigroup is a set $Q$ with a binary operation $\cdot$ such that for all $a,b\in Q$ , the equations

ax=b\quad\text{and}\quad ya=b

each have unique solutions $x,y\in Q$ .

A quasigroup with an identity element is called a loop.

Equivalent characterizations

Equivalently, a quasigroup is a magma whose Cayley table forms a Latin square—each element appears exactly once in each row and column.

Examples