QUASIGROUP Meaning and
Definition
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A quasigroup is a mathematical structure that falls under the category of algebraic systems. It is a set equipped with a binary operation that satisfies specific properties. More formally, a quasigroup is defined as an algebraic structure (Q, •) where Q is a non-empty set and • is a binary operation on Q such that for any two elements a and b in Q, there exist unique elements x and y in Q which satisfy the equations a • x = b and y • a = b.
The defining characteristic of a quasigroup is that it guarantees the existence and uniqueness of solutions to the above equations for any pair of elements in the set. This property is known as the Latin square property. In a quasigroup, for every element a, there exists a unique element x such that a • x is equal to any other element b in the set, and similarly, there exists a unique element y such that y • a is equal to b.
Unlike other algebraic structures such as groups or rings, quasigroups do not require the operation to be associative. Thus, the quasigroup is a more general and flexible structure that encompasses a wider range of mathematical systems. The study of quasigroups has applications in various fields such as coding theory, cryptography, and combinatorics.
Etymology of QUASIGROUP
The word "quasigroup" was coined by mathematician Garrett Birkhoff in 1935. The term combines the Latin word "quasi" (meaning "resembling" or "almost") with the word "group", which is a mathematical structure defined by a set and an associative binary operation.
Birkhoff introduced the concept of quasigroups as a generalization of groups that relaxes the requirement of having an identity element. The name "quasigroup" reflects the fact that these structures resemble groups in many ways but lack certain properties that groups possess.
