Quasigroup

Summary

A quasigroup is a set with a binary operation in which one can always divide on both sides. For all $a, b$ the equations $a x = b$ and $y a = b$ have exactly one solution each. The operation need not be associative and need not have an identity element. A quasigroup with an identity element is a loop, and an associative quasigroup that is nonempty is a group. The multiplication tables of finite quasigroups are exactly the Latin squares.