Quasigroup

A Groupoid $S$ such that for all $a,b\in S$, there exist unique $x,y\in S$ such that

$$ax = b$$

$$ya = b.$$

No other restrictions are applied; thus a quasigroup need not have an Identity Element, not be associative, etc. Quasigroups are precisely Groupoids whose multiplication tables are Latin Squares. A quasigroup can be empty.

See also Binary Operator, Groupoid, Latin Square, Loop (Algebra), Monoid, Semigroup

References

van Lint, J. H. and Wilson, R. M. A Course in Combinatorics. New York: Cambridge University Press, 1992.