Gyrogroup
A gyrovector space is a mathematical concept proposed by Abraham A. Ungar for studying hyperbolic geometry in analogy to the way vector spaces are used in Euclidean geometry. Ungar introduced the concept of gyrovectors that have addition based on gyrogroups instead of vectors which have addition based on groups. Ungar developed his concept as a tool for the formulation of special relativity as an alternative to the use of Lorentz transformations to represent compositions of velocities (also called boosts - "boosts" are aspects of relative velocities, and should not be conflated with "translations"). This is achieved by introducing "gyro operators"; two 3d velocity vectors are used to construct an operator, which acts on another 3d velocity.
Gyrogroups are weakly-associative-grouplike-structure. Ungar proposed the term gyrogroup was for what he called a gyrocommutative-gyrogroup with the term gyrogroup being reserved for the non-gyrocommutative case in analogy with groups vs commutative-groups. Gyrogroups are a type of Bol loop. Gyrocommutative gyrogroups are equivalent to K-loops although defined differently. The terms Bruck loop and dyadic symset are also in use.
A groupoid (G, ) is a gyrogroup if its binary operation satisfies the following axioms:
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