quasigroup

A quasigroup is an algebraic structure consisting of a nonempty set Q with a binary operation, usually denoted by •, that is closed on Q. The defining property is that for all a and b in Q there exist unique x and y in Q such that a • x = b and y • a = b. Equivalently, the systems of equations a • x = b and y • a = b have unique solutions for every a, b in Q. This ensures that the multiplication table of a finite quasigroup is a Latin square, meaning each element appears exactly once in every row and every column.

Quasigroups generalize groups. Every group is a quasigroup with an identity element and associativity. A quasigroup that has an identity element is called a loop; if such a loop is also associative, it is a group. Quasigroups may lack both associativity and identity, which distinguishes them from groups.

Examples and connections: any group yields a quasigroup, but many quasigroups are nonassociative. Finite quasigroups correspond to Latin squares, and conversely, any Latin square defines a finite quasigroup up to isomorphism. This connection makes quasigroups central in combinatorics and design theory.

Variants and properties: there are many distinguished subclasses of quasigroups that satisfy extra identities, such as Moufang quasigroups and medial quasigroups. Steiner quasigroups form another notable class with special symmetry properties and links to combinatorial designs.

Applications and context: quasigroups appear in cryptography, coding theory, and the study of Latin squares used in experimental design. They also serve as nonassociative models in algebra, with topics including isotopy, autotopisms, and various algebraic identities studied in abstract algebra.