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

$\displaystyle ax$ $\textstyle =$ $\displaystyle b$  
$\displaystyle ya$ $\textstyle =$ $\displaystyle b.$  

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


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

© 1996-9 Eric W. Weisstein