from the lorentz transformation group in pseudo-euclidean spaces to bi-gyrogroups

‎the lorentz transformation of order (m=1,n) ‎, ‎ ninnb ‎, ‎is the well-known ‎lorentz transformation of special relativity theory‎. ‎it is a transformation of time-space coordinates of the ‎pseudo-euclidean space rb^{m=1,n} of one time dimension and ‎ n space dimensions ( n=3 in physical applications)‎. ‎a lorentz transformation without rotations is called a {it boost}‎. ‎commonly‎, ‎the special relativistic boost is ‎parametrized by a relativistically admissible velocity parameter vb ‎, ‎ vbinrcn ‎, ‎whose domain is the c -ball rcn of all ‎relativistically admissible velocities‎, ‎ rcn={vbinrn:|vb|0 is an arbitrarily fixed ‎positive constant that represents the vacuum speed of light‎. ‎the study of the lorentz transformation composition law in terms of ‎parameter composition reveals that the group structure of the ‎lorentz transformation of order (m=1,n) induces a gyrogroup and ‎a gyrovector space structure that regulate ‎the parameter space rcn ‎. ‎the gyrogroup and gyrovector space structure ‎of the ball rcn ‎, ‎in turn‎, ‎form the algebraic setting for the beltrami-klein ball model ‎of hyperbolic geometry‎, ‎which underlies the ball rcn ‎. ‎the aim of this article is to extend the study of the ‎lorentz transformation of order (m,n) from m=1 and nge1 to ‎all m,ninnb ‎, ‎obtaining algebraic structures called ‎a {it bi-gyrogroup} and a {it bi-gyrovector space}‎. ‎a bi-gyrogroup is ‎a gyrogroup each gyration of which is a pair of ‎a left gyration and a right gyration‎. ‎a bi-gyrovector space is constructed from a bi-gyrocommutative bi-gyrogroup ‎that admits a scalar multiplication‎.