the only justification for the einstein velocity addition law appeared to be its empirical adequacy, so that the intrinsic beauty and harmony in einstein addition remained for a long timea mystery to be conquered. accordingly, the aim of this expository article is to present(i) the einstein relativistic vector addition,(ii) the resulting einstein scalar multiplication,(iii) the einstein relativ...

The concept of gyrogroups is a generalization groups which do not explicitly have associativity. Recently, Atiponrat extended the idea topological (paratopological) to gyrogroups. In this paper, we prove that every regular (Hausdorff) locally gyroscopic invariant paratopological gyrogroup G completely (function Hausdorff). These results improve theorems Banakh and Ravsky for groups. Also, exten...

in this article we review an algebraic definition of the gyrogroup and a simplified version of the gyrovector space with two fundamental examples on the open ball of finite-dimensional euclidean space, which are the einstein and möbius gyrovector spaces. we introduce the structure of gyrovector space and the gyroline on the open convex cone of positive definite matrices and see its interest...

the aim of this paper is to show the importance of analytic hyperbolic geometry introduced in [9]. in [1], ungar and chen showed that the algebra of the group sl(2,c) naturally leads to the notion of gyrogroups and gyrovector spaces for dealing with the lorentz group and its underlying hyperbolic geometry. they defined the chen addition and then chen model of hyperbolic geometry. in this paper,...

Suppose that (T;*) is a groupoid with left identity such each element 2 T has inverse. Then called gyrogroup if and only (i) there exists function gyr : x -Aut(T) for all a; b; c T, * (b c) = (a b) ? gyr[a; b]c, where b]c gyr(a; b)(c); (ii) b b] gyr[a b]. In this paper, the structure of normal subgyrogroups certain gyrogroups are investigated.

Using Klein's approach, geometry can be studied in terms of a space points and group transformations that space. This allows us to apply algebraic tools studying mathematical structures. In this article, we follow approach study the $(G, \mathcal{T})$, where $G$ is an abstract gyrogroup $\mathcal{T}$ appropriate containing all gyroautomorphisms $G$. We focus on $n$-transitivity gyrogroups also ...

The concept of gyrogroups, with a weaker algebraic structure without associative law, was introduced under the background c -ball relativistically admissible velocities Einstein velocity addition. A topological gyrogroup is just endowed compatible topology such that multiplication jointly continuous and inverse continuous. This generalizes groups. In this paper, we are going to establish for lo...

‎In this article we review an algebraic definition of the gyrogroup and a simplified version of the gyrovector space with two fundamental examples on the open ball of finite-dimensional Euclidean spaces‎, ‎which are the Einstein and M"{o}bius gyrovector spaces‎. ‎We introduce the structure of gyrovector space and the gyroline on the open convex cone of positive definite matrices and explore its...

This autobiography presents the scientific living of Abraham Ungar and his role in Gyrogroups and Gyrovector spaces.‎