Quasigroup

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

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

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

A quasigroup with an identity element is called a loop .

Examples:

• $(\mathbb{Z}, -)$ — integers under subtraction
• $(\mathbb{R}^+, \div)$ — positive reals under division
• Any Latin square defines a quasigroup