TABLE OF CONTENTS Vectors in a vector space form equivalence classes, where two vectors are equivalent if they are parallel and possess equal lengths. Gyrovectors, in contrast, do not admit parallelism. Yet, they do form analogous equivalence classes even at the primitive level of gyrocommutative gyrogroups where, in general, the concepts of length and parallelism do not exist. In the more advanced level of gyrovector spaces, gyrovectors will be found fully analogous to vectors, where they regulate algebraically the hyperbolic geometry of Bolyai and Lobachevsky just as vectors regulate algebraically Euclidean geometry.
The definition of gyrovectors and cogyrovectors in gyrocommutative gyrogroups will be presented in Secs. 5.2 and 5.6 in terms of equivalence classes of pairs of points.
A (binary) relation on a nonempty set S is a subset R of S S, written as a ~ b if (a, b) ? R. A relation ~ on a set S is
Reflexive if a ~ a for all a ? S.
Symmetric if a ~ b implies b ~ a for all a, b ? S.
Transitive if a ~ b and b ~ c implies a ~ c for all a, b, c ? S.
A relation is an equivalence relation if it is reflexive, symmetric and transitive.
An equivalence relation ~ on a set S gives rise to equivalence classes. The equivalence class of a ? S is the subset { x
