C-Loops Obtained From Hypercomplex Numbers
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Date
2020
Authors
Kioko, Eugene Moses
Njuguna, Lydia
Journal Title
Journal ISSN
Volume Title
Publisher
IJRIAS
Abstract
Hypercomplex numbers have played a notable and
critical role in the study and exploration of Loop Theory.
Researchers have made numerous studies in this area especially
in the investigation and construction of different loops. This
paper has extended the research to C-loops where we are
investigating the formation of C-loops obtained from
hypercomplex numbers of dimension 2n ; 1 ≥ n ≤ 4. We are
specifically working with the 24- dimensional algebra, called the
sedenions. In constructing the C-loops, we have used the frame
multiplication of hypercomplex numbers using the Cayley-
Dickson construction. We have tested the satisfaction of the left,
(x x) (y z) = (x (x y)) z and right, x ((y z) z) = (x y) (z z) C-loop
identities by the sedenions. We have also formed split extension
of sedenions and equally tested the satisfaction of the C-loop
identities on them. We have found that the sedenions satisfy the
C-loop identities hence forming C-loops. However, the split
extension of sedenions satisfies the right C-loop identity only.
AMS Subject Classification: 20B05
Description
article
Keywords
Loops, C-loops, Hypercomplex numbers, Cayley- Dickson, Split extension