Quasigroups

A quasigroup is a set Q with a closed binary operation ◦ such that for every a,b in Q there exist unique x,y in Q satisfying a ◦ x = b and y ◦ a = b. Equivalently, the equations a ◦ x = b and y ◦ a = b have unique solutions for all a,b in Q. The multiplication table of a quasigroup is a Latin square, meaning that each element of Q occurs exactly once in every row and in every column.

Quasigroups generalize groups; every group is a quasigroup because division is always possible and associativity holds. If a quasigroup is associative, it is a group. A quasigroup that possesses an identity element is called a loop; if it is also associative, it is a group in the usual sense.

Examples and variations: any group is a quasigroup. There exist non-associative quasigroups; a standard finite example is a Steiner quasigroup, defined on a set with a∘a = a and a∘b = c where c is the third element of the unique Steiner triple containing a and b. Steiner quasigroups are commutative, idempotent, and not generally associative, hence not groups.

Related concepts and applications: researchers study isotopes (structures obtained by renaming elements in the operation), homomorphisms between quasigroups, and their connections to Latin squares and combinatorial designs. Quasigroups have applications in cryptography and in constructions that rely on Latin-square structures, as well as in certain areas of error-correcting codes.