AUTHOR(S): Cebrai̇l Ozan Oktar

TITLE 
ABSTRACT Gyrogroups are generalized groups, which are best motivated by the algebra of Möbius transformations of the complex open unit disc. Groups are classified into commutative and noncommutative groups and, in full analogy, gyrogroups are classified into gyrocommutative and nongyrocommutative gyrogroups. Some commutative groups admit scalar multiplication, giving rise to vector spaces. In full analogy, some gyrocommutative gyrogroups admit scalar multiplication, giving rise to gyrovector spaces. Furthermore, vector spaces form the algebraic setting for the standard model of Euclidean geometry and, in full analogy, gyrovector spaces form the algebraic setting for various models of the hyperbolic geometry of Bolyai and Lobachevsky.The special grouplike loops, known as gyrocommutative gyrogroups, have thrust the Einstein velocity addition law, which previously has operated mostly in the shadows, into the spotlight. We will find that Einstein (Möbius) addition is a gyrocommutative gyrogroup operation that forms the setting for the BeltramiKlein (Poincaré) ball model of hyperbolic geometry just as the common vector addition is a commutative group operation that forms the setting for the standard model of Euclidean geometry. The resulting analogies to which the grouplike loops give rise lead us to new results in (i) hyperbolic geometry; (ii) relativistic physics; and (iii) quantum information and computation.Time Tensors functions have been used to describe the flows of time. The magnitude of the value of time tensor function means the temporal coordinates in a flow of time. We also use a function to describe the motion of particles in quantum mechanics but it has different meanings. The function is called time tensor function. Time tensor imposes space and time measurements and space and time probing. Although using optimised space and time probe fields will allow to deep probing in a position and time measurement beyond the space and time measurements of the probe field stil result in a time tensor. 
KEYWORDS Grouplike Loops, Gyrogroups, Gyrovector Spaces, Hyperbolic Geometry, Special Relativity,Time Tensors 

Cite this paper Cebrai̇l Ozan Oktar. (2024) Gyrogroup. International Journal of Instrumentation and Measurement, 9, 1119 
