JAIYEOLA Temitope Gbolahan and ADENIRAN John Olushola
Abstract
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 of an Osborn loop. These new algebraic
properties, will open our eyes more to the study of Osborn loops
like CC-loops which has received a tremendious attention in this
and VD-loops whose study is yet to be explored. In
this study, some algebraic properties of non-WIP 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
CC-loop 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 3-PAPL 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).
Note: The most popular identity that is cryptographic in nature is the cross inverse property identity. The identity in this paper is new.