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Involutory decomposition of groups into twisted subgroups and subgroups
 J. Group Theory
, 2000
"... Gyrogroups are generalized groups modelled on the Einstein groupoid of all relativistically admissible velocities with their Einstein’s velocity addition as a binary operation. Einstein’s gyrogroup fails to form a group since it is nonassociative. The breakdown of associativity in the Einstein addit ..."
Abstract

Cited by 7 (2 self)
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Gyrogroups are generalized groups modelled on the Einstein groupoid of all relativistically admissible velocities with their Einstein’s velocity addition as a binary operation. Einstein’s gyrogroup fails to form a group since it is nonassociative. The breakdown of associativity in the Einstein addition does not result in loss of mathematical regularity owing to the presence of the relativistic effect known as the Thomas precession which, by abstraction, becomes an automorphism called the Thomas gyration. The Thomas gyration turns out to be the missing link that gives rise to analogies shared by gyrogroups and groups. In particular, it gives rise to the gyroassociative and the gyrocommuttive laws that Einstein’s addition possesses, in full analogy with the associative and the commutative laws that vector addition possesses in a vector space. The existence of striking analogies shared by gyrogroups
Department of Mathematics,
"... This study digs out some new algebraic properties of an Osborn loop that will help in the future to unveil the mystery behind the middle inner mappings T (x) of an Osborn loop. These new algebraic properties, will open our eyes more to the study of Osborn loops like CCloops which has received a tre ..."
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This study digs out some new algebraic properties of an Osborn loop that will help in the future to unveil the mystery behind the middle inner mappings T (x) of an Osborn loop. These new algebraic properties, will open our eyes more to the study of Osborn loops like CCloops which has received a tremendious attention in this 21 st and VDloops whose study is yet to be explored. In this study, some algebraic properties of nonWIP Osborn loops have been investigated in a broad manner. Huthnance was able to deduce some algebraic properties of Osborn loops with the WIP i.e universal weak WIPLs. So this work exempts the WIP. Two new loop identities, namely left self inverse property loop(LSIPL) identity and right self inverse property loop(RSLPL) are introduced for the first time and it is shown that in an Osborn loop, they are equivalent. A CCloop is shown to be power associative if and only if it is a RSLPL or LSIPL. Among the few identities that have been established for Osborn loops, one of them is recognized and recommended for cryptography in a similar spirit in which the cross inverse property has been used by Keedwell following the fact that it was observed that Osborn loops that do not have the LSIP or RSIP or 3PAPL or weaker forms of inverse property, power associativity and diassociativity to mention a few, will have cycles(even long ones). These identity is called an Osborn cryptographic identity(or just a cryptographic identity). 1
Comment.Math.Univ.Carolinae 41,2 (2000)415{427 415 Loops and quasigroups: Aspects of current work and
"... prospects for the future ..."