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What Are Gyrocommutative Gyrogroups?
Table of Links
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Proposed Approach
C. Formulation of MLR from the Perspective of Distances to Hyperplanes
H. Computation of Canonical Representation
G SOME RELATED DEFINITIONS
G.1 GYROGROUPS AND GYROVECTOR SPACES
Gyrovector spaces form the setting for hyperbolic geometry in the same way that vector spaces form the setting for Euclidean geometry (Ungar, 2002; 2005; 2014). We recap the definitions of gyrogroups and gyrocommutative gyrogroups proposed in Ungar (2002; 2005; 2014). For greater mathematical detail and in-depth discussion, we refer the interested reader to these papers.
Definition G.1 (Gyrogroups (Ungar, 2014)). A pair (G, ⊕*) is a groupoid in the sense that it is a nonempty set, G, with a binary operation, ⊕. A groupoid (G,* ⊕*) is a gyrogroup if its binary operation satisfies the following axioms for a, b, c ∈ G:*
(G1) There is at least one element e ∈ G called a left identity such that e ⊕ a = a.
(G2) There is an element ⊖a ∈ G called a left inverse of a such that ⊖a ⊕ a = e.
(G3) There is an automorphism gyr[a, b] : G → G for each a, b ∈ G such that
a ⊕ (b ⊕ c) = (a ⊕ b) ⊕ gyr[a, b]c (Left Gyroassociative Law).
The automorphism gyr[a, b] is called the gyroautomorphism, or the gyration of G generated by a, b. (G4) gyr[a, b] = gyr[a ⊕ b, b] (Left Reduction Property).
Definition G.2 (Gyrocommutative Gyrogroups (Ungar, 2014)). A gyrogroup (G, ⊕*) is gyrocommutative if it satisfies*
a ⊕ b = gyr[a, b](b ⊕ a) (Gyrocommutative Law).
The following definition of gyrovector spaces is slightly different from Definition 3.2 in Ungar (2014).
Definition G.3 (Gyrovector Spaces). A gyrocommutative gyrogroup (G, ⊕) equipped with a scalar multiplication
Definition G.3 (Gyrovector Spaces). A gyrocommutative gyrogroup (G, ⊕*) equipped with a scalar multiplication*
(t, x) → t ⊙ x : R × G → G
is called a gyrovector space if it satisfies the following axioms for s, t ∈ R and a, b, c ∈ G*:*
(V1) 1 ⊙ a = a, 0 ⊙ a = t ⊙ e = e, and (−1) ⊙ a = ⊖a.
(V2) (s + t) ⊙ a = s ⊙ a ⊕ t ⊙ a.
(V3) (st) ⊙ a = s ⊙ (t ⊙ a).
(V4) gyr[a, b](t ⊙ c) = t ⊙ gyr[a, b]c.
(V5) gyr[s ⊙ a, t ⊙ a] = Id, where Id is the identity map.
G.2 AI GYROVECTOR SPACES
G.3 LE GYROVECTOR SPACES
G.4 LC GYROVECTOR SPACES
G.5 GRASSMANN MANIFOLDS IN THE PROJECTOR PERSPECTIVE
G.6 GRASSMANN MANIFOLDS IN THE ONB PERSPECTIVE
G.7 THE SPD AND GRASSMANN INNER PRODUCTS
G.9 THE GYRODISTANCE FUNCTION IN STRUCTURE SPACES
G.10 THE PSEUDO-GYRODISTANCE FUNCTION IN STRUCTURE SPACES
Authors:
(1) Xuan Son Nguyen, ETIS, UMR 8051, CY Cergy Paris University, ENSEA, CNRS, France ([email protected]);
(2) Shuo Yang, ETIS, UMR 8051, CY Cergy Paris University, ENSEA, CNRS, France ([email protected]);
(3) Aymeric Histace, ETIS, UMR 8051, CY Cergy Paris University, ENSEA, CNRS, France ([email protected]).
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