Authors: T.G. Jaiyéolá and J.O. Adéníran
Advances in Theoretical and Applied Mathematics ISSN 0973-4554 Vol.1 No.3 (2006), pp. 233-244 © Research India Publications
Abstract
The right (left) derivative, a−1, e−and e, a−1−isotopes of a C-loop are shown to be C-loops.Furthermore, for a central loop (L,F),it is shown that {F,Fa−1,Fa−1,e} and {F,Fa−1 ,Fe,a−1 } are systems of isotopic C-loops that obey a form of generalised distributive law.
It is proved that for a loop (L,θ) to be an LC(RC,C)-loop,it is necessary and sufficient for the parastrophe (L,θ∗) to be a RC(LC,C)-loop.Hence, isotopes (L,⊗) and (L,) of (L,θ) and (L,θ∗) respectively are proved to be isotopic if either (L,⊗) or (L,) is commutative.It is shown that C-loops are isotopic to some finite in decomposable groups of the classes Di ,i = 1, 2, 3, 4, 5 and that the center of such C-loops have a rank of 1,2 or 3.